Project acronym 1stProposal
Project An alternative development of analytic number theory and applications
Researcher (PI) ANDREW Granville
Host Institution (HI) UNIVERSITY COLLEGE LONDON
Call Details Advanced Grant (AdG), PE1, ERC-2014-ADG
Summary The traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods.
Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject.
We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small.
We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration.
Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further.
Much of this is joint work with K Soundararajan of Stanford University.
Summary
The traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods.
Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject.
We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small.
We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration.
Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further.
Much of this is joint work with K Soundararajan of Stanford University.
Max ERC Funding
2 011 742 €
Duration
Start date: 2015-08-01, End date: 2020-07-31
Project acronym 2-3-AUT
Project Surfaces, 3-manifolds and automorphism groups
Researcher (PI) Nathalie Wahl
Host Institution (HI) KOBENHAVNS UNIVERSITET
Call Details Starting Grant (StG), PE1, ERC-2009-StG
Summary The scientific goal of the proposal is to answer central questions related to diffeomorphism groups of manifolds of dimension 2 and 3, and to their deformation invariant analogs, the mapping class groups. While the classification of surfaces has been known for more than a century, their automorphism groups have yet to be fully understood. Even less is known about diffeomorphisms of 3-manifolds despite much interest, and the objects here have only been classified recently, by the breakthrough work of Perelman on the Poincar\'e and geometrization conjectures. In dimension 2, I will focus on the relationship between mapping class groups and topological conformal field theories, with applications to Hochschild homology. In dimension 3, I propose to compute the stable homology of classifying spaces of diffeomorphism groups and mapping class groups, as well as study the homotopy type of the space of diffeomorphisms. I propose moreover to establish homological stability theorems in the wider context of automorphism groups and more general families of groups. The project combines breakthrough methods from homotopy theory with methods from differential and geometric topology. The research team will consist of 3 PhD students, and 4 postdocs, which I will lead.
Summary
The scientific goal of the proposal is to answer central questions related to diffeomorphism groups of manifolds of dimension 2 and 3, and to their deformation invariant analogs, the mapping class groups. While the classification of surfaces has been known for more than a century, their automorphism groups have yet to be fully understood. Even less is known about diffeomorphisms of 3-manifolds despite much interest, and the objects here have only been classified recently, by the breakthrough work of Perelman on the Poincar\'e and geometrization conjectures. In dimension 2, I will focus on the relationship between mapping class groups and topological conformal field theories, with applications to Hochschild homology. In dimension 3, I propose to compute the stable homology of classifying spaces of diffeomorphism groups and mapping class groups, as well as study the homotopy type of the space of diffeomorphisms. I propose moreover to establish homological stability theorems in the wider context of automorphism groups and more general families of groups. The project combines breakthrough methods from homotopy theory with methods from differential and geometric topology. The research team will consist of 3 PhD students, and 4 postdocs, which I will lead.
Max ERC Funding
724 992 €
Duration
Start date: 2009-11-01, End date: 2014-10-31
Project acronym 3DWATERWAVES
Project Mathematical aspects of three-dimensional water waves with vorticity
Researcher (PI) Erik Torsten Wahlén
Host Institution (HI) LUNDS UNIVERSITET
Call Details Starting Grant (StG), PE1, ERC-2015-STG
Summary The goal of this project is to develop a mathematical theory for steady three-dimensional water waves with vorticity. The mathematical model consists of the incompressible Euler equations with a free surface, and vorticity is important for modelling the interaction of surface waves with non-uniform currents. In the two-dimensional case, there has been a lot of progress on water waves with vorticity in the last decade. This progress has mainly been based on the stream function formulation, in which the problem is reformulated as a nonlinear elliptic free boundary problem. An analogue of this formulation is not available in three dimensions, and the theory has therefore so far been restricted to irrotational flow. In this project we seek to go beyond this restriction using two different approaches. In the first approach we will adapt methods which have been used to construct three-dimensional ideal flows with vorticity in domains with a fixed boundary to the free boundary context (for example Beltrami flows). In the second approach we will develop methods which are new even in the case of a fixed boundary, by performing a detailed study of the structure of the equations close to a given shear flow using ideas from infinite-dimensional bifurcation theory. This involves handling infinitely many resonances.
Summary
The goal of this project is to develop a mathematical theory for steady three-dimensional water waves with vorticity. The mathematical model consists of the incompressible Euler equations with a free surface, and vorticity is important for modelling the interaction of surface waves with non-uniform currents. In the two-dimensional case, there has been a lot of progress on water waves with vorticity in the last decade. This progress has mainly been based on the stream function formulation, in which the problem is reformulated as a nonlinear elliptic free boundary problem. An analogue of this formulation is not available in three dimensions, and the theory has therefore so far been restricted to irrotational flow. In this project we seek to go beyond this restriction using two different approaches. In the first approach we will adapt methods which have been used to construct three-dimensional ideal flows with vorticity in domains with a fixed boundary to the free boundary context (for example Beltrami flows). In the second approach we will develop methods which are new even in the case of a fixed boundary, by performing a detailed study of the structure of the equations close to a given shear flow using ideas from infinite-dimensional bifurcation theory. This involves handling infinitely many resonances.
Max ERC Funding
1 203 627 €
Duration
Start date: 2016-03-01, End date: 2021-02-28
Project acronym AAMOT
Project Arithmetic of automorphic motives
Researcher (PI) Michael Harris
Host Institution (HI) INSTITUT DES HAUTES ETUDES SCIENTIFIQUES
Call Details Advanced Grant (AdG), PE1, ERC-2011-ADG_20110209
Summary The primary purpose of this project is to build on recent spectacular progress in the Langlands program to study the arithmetic properties of automorphic motives constructed in the cohomology of Shimura varieties. Because automorphic methods are available to study the L-functions of these motives, which include elliptic curves and certain families of Calabi-Yau varieties over totally real fields (possibly after base change), they represent the most accessible class of varieties for which one can hope to verify fundamental conjectures on special values of L-functions, including Deligne's conjecture and the Main Conjecture of Iwasawa theory. Immediate goals include the proof of irreducibility of automorphic Galois representations; the establishment of period relations for automorphic and potentially automorphic realizations of motives in the cohomology of distinct Shimura varieties; the construction of p-adic L-functions for these and related motives, notably adjoint and tensor product L-functions in p-adic families; and the geometrization of the p-adic and mod p Langlands program. All four goals, as well as the others mentioned in the body of the proposal, are interconnected; the final goal provides a bridge to related work in geometric representation theory, algebraic geometry, and mathematical physics.
Summary
The primary purpose of this project is to build on recent spectacular progress in the Langlands program to study the arithmetic properties of automorphic motives constructed in the cohomology of Shimura varieties. Because automorphic methods are available to study the L-functions of these motives, which include elliptic curves and certain families of Calabi-Yau varieties over totally real fields (possibly after base change), they represent the most accessible class of varieties for which one can hope to verify fundamental conjectures on special values of L-functions, including Deligne's conjecture and the Main Conjecture of Iwasawa theory. Immediate goals include the proof of irreducibility of automorphic Galois representations; the establishment of period relations for automorphic and potentially automorphic realizations of motives in the cohomology of distinct Shimura varieties; the construction of p-adic L-functions for these and related motives, notably adjoint and tensor product L-functions in p-adic families; and the geometrization of the p-adic and mod p Langlands program. All four goals, as well as the others mentioned in the body of the proposal, are interconnected; the final goal provides a bridge to related work in geometric representation theory, algebraic geometry, and mathematical physics.
Max ERC Funding
1 491 348 €
Duration
Start date: 2012-06-01, End date: 2018-05-31
Project acronym AAS
Project Approximate algebraic structure and applications
Researcher (PI) Ben Green
Host Institution (HI) THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary This project studies several mathematical topics with a related theme, all of them part of the relatively new discipline known as additive combinatorics.
We look at approximate, or rough, variants of familiar mathematical notions such as group, polynomial or homomorphism. In each case we seek to describe the structure of these approximate objects, and then to give applications of the resulting theorems. This endeavour has already lead to groundbreaking results in the theory of prime numbers, group theory and combinatorial number theory.
Summary
This project studies several mathematical topics with a related theme, all of them part of the relatively new discipline known as additive combinatorics.
We look at approximate, or rough, variants of familiar mathematical notions such as group, polynomial or homomorphism. In each case we seek to describe the structure of these approximate objects, and then to give applications of the resulting theorems. This endeavour has already lead to groundbreaking results in the theory of prime numbers, group theory and combinatorial number theory.
Max ERC Funding
1 000 000 €
Duration
Start date: 2011-10-01, End date: 2016-09-30
Project acronym ACCOPT
Project ACelerated COnvex OPTimization
Researcher (PI) Yurii NESTEROV
Host Institution (HI) UNIVERSITE CATHOLIQUE DE LOUVAIN
Call Details Advanced Grant (AdG), PE1, ERC-2017-ADG
Summary The amazing rate of progress in the computer technologies and telecommunications presents many new challenges for Optimization Theory. New problems are usually very big in size, very special in structure and possibly have a distributed data support. This makes them unsolvable by the standard optimization methods. In these situations, old theoretical models, based on the hidden Black-Box information, cannot work. New theoretical and algorithmic solutions are urgently needed. In this project we will concentrate on development of fast optimization methods for problems of big and very big size. All the new methods will be endowed with provable efficiency guarantees for large classes of optimization problems, arising in practical applications. Our main tool is the acceleration technique developed for the standard Black-Box methods as applied to smooth convex functions. However, we will have to adapt it to deal with different situations.
The first line of development will be based on the smoothing technique as applied to a non-smooth functions. We propose to substantially extend this approach to generate approximate solutions in relative scale. The second line of research will be related to applying acceleration techniques to the second-order methods minimizing functions with sparse Hessians. Finally, we aim to develop fast gradient methods for huge-scale problems. The size of these problems is so big that even the usual vector operations are extremely expensive. Thus, we propose to develop new methods with sublinear iteration costs. In our approach, the main source for achieving improvements will be the proper use of problem structure.
Our overall aim is to be able to solve in a routine way many important problems, which currently look unsolvable. Moreover, the theoretical development of Convex Optimization will reach the state, when there is no gap between theory and practice: the theoretically most efficient methods will definitely outperform any homebred heuristics.
Summary
The amazing rate of progress in the computer technologies and telecommunications presents many new challenges for Optimization Theory. New problems are usually very big in size, very special in structure and possibly have a distributed data support. This makes them unsolvable by the standard optimization methods. In these situations, old theoretical models, based on the hidden Black-Box information, cannot work. New theoretical and algorithmic solutions are urgently needed. In this project we will concentrate on development of fast optimization methods for problems of big and very big size. All the new methods will be endowed with provable efficiency guarantees for large classes of optimization problems, arising in practical applications. Our main tool is the acceleration technique developed for the standard Black-Box methods as applied to smooth convex functions. However, we will have to adapt it to deal with different situations.
The first line of development will be based on the smoothing technique as applied to a non-smooth functions. We propose to substantially extend this approach to generate approximate solutions in relative scale. The second line of research will be related to applying acceleration techniques to the second-order methods minimizing functions with sparse Hessians. Finally, we aim to develop fast gradient methods for huge-scale problems. The size of these problems is so big that even the usual vector operations are extremely expensive. Thus, we propose to develop new methods with sublinear iteration costs. In our approach, the main source for achieving improvements will be the proper use of problem structure.
Our overall aim is to be able to solve in a routine way many important problems, which currently look unsolvable. Moreover, the theoretical development of Convex Optimization will reach the state, when there is no gap between theory and practice: the theoretically most efficient methods will definitely outperform any homebred heuristics.
Max ERC Funding
2 090 038 €
Duration
Start date: 2018-09-01, End date: 2023-08-31
Project acronym ADAPTIVES
Project Algorithmic Development and Analysis of Pioneer Techniques for Imaging with waVES
Researcher (PI) Chrysoula Tsogka
Host Institution (HI) IDRYMA TECHNOLOGIAS KAI EREVNAS
Call Details Starting Grant (StG), PE1, ERC-2009-StG
Summary The proposed work concerns the theoretical and numerical development of robust and adaptive methodologies for broadband imaging in clutter. The word clutter expresses our uncertainty on the wave speed of the propagation medium. Our results are expected to have a strong impact in a wide range of applications, including underwater acoustics, exploration geophysics and ultrasound non-destructive testing. Our machinery is coherent interferometry (CINT), a state-of-the-art statistically stable imaging methodology, highly suitable for the development of imaging methods in clutter. We aim to extend CINT along two complementary directions: novel types of applications, and further mathematical and numerical development so as to assess and extend its range of applicability. CINT is designed for imaging with partially coherent array data recorded in richly scattering media. It uses statistical smoothing techniques to obtain results that are independent of the clutter realization. Quantifying the amount of smoothing needed is difficult, especially when there is no a priori knowledge about the propagation medium. We intend to address this question by coupling the imaging process with the estimation of the medium's large scale features. Our algorithms rely on the residual coherence in the data. When the coherent signal is too weak, the CINT results are unsatisfactory. We propose two ways for enhancing the resolution of CINT: filter the data prior to imaging (noise reduction) and waveform design (optimize the source distribution). Finally, we propose to extend the applicability of our imaging-in-clutter methodologies by investigating the possibility of utilizing ambient noise sources to perform passive sensor imaging, as well as by studying the imaging problem in random waveguides.
Summary
The proposed work concerns the theoretical and numerical development of robust and adaptive methodologies for broadband imaging in clutter. The word clutter expresses our uncertainty on the wave speed of the propagation medium. Our results are expected to have a strong impact in a wide range of applications, including underwater acoustics, exploration geophysics and ultrasound non-destructive testing. Our machinery is coherent interferometry (CINT), a state-of-the-art statistically stable imaging methodology, highly suitable for the development of imaging methods in clutter. We aim to extend CINT along two complementary directions: novel types of applications, and further mathematical and numerical development so as to assess and extend its range of applicability. CINT is designed for imaging with partially coherent array data recorded in richly scattering media. It uses statistical smoothing techniques to obtain results that are independent of the clutter realization. Quantifying the amount of smoothing needed is difficult, especially when there is no a priori knowledge about the propagation medium. We intend to address this question by coupling the imaging process with the estimation of the medium's large scale features. Our algorithms rely on the residual coherence in the data. When the coherent signal is too weak, the CINT results are unsatisfactory. We propose two ways for enhancing the resolution of CINT: filter the data prior to imaging (noise reduction) and waveform design (optimize the source distribution). Finally, we propose to extend the applicability of our imaging-in-clutter methodologies by investigating the possibility of utilizing ambient noise sources to perform passive sensor imaging, as well as by studying the imaging problem in random waveguides.
Max ERC Funding
690 000 €
Duration
Start date: 2010-06-01, End date: 2015-11-30
Project acronym ADDECCO
Project Adaptive Schemes for Deterministic and Stochastic Flow Problems
Researcher (PI) Remi Abgrall
Host Institution (HI) INSTITUT NATIONAL DE RECHERCHE ENINFORMATIQUE ET AUTOMATIQUE
Call Details Advanced Grant (AdG), PE1, ERC-2008-AdG
Summary The numerical simulation of complex compressible flow problem is still a challenge nowaday even for simple models. In our opinion, the most important hard points that need currently to be tackled and solved is how to obtain stable, scalable, very accurate, easy to code and to maintain schemes on complex geometries. The method should easily handle mesh refinement, even near the boundary where the most interesting engineering quantities have to be evaluated. Unsteady uncertainties in the model, for example in the geometry or the boundary conditions should represented efficiently.This proposal goal is to design, develop and evaluate solutions to each of the above problems. Our work program will lead to significant breakthroughs for flow simulations. More specifically, we propose to work on 3 connected problems: 1-A class of very high order numerical schemes able to easily deal with the geometry of boundaries and still can solve steep problems. The geometry is generally defined by CAD tools. The output is used to generate a mesh which is then used by the scheme. Hence, any mesh refinement process is disconnected from the CAD, a situation that prevents the spread of mesh adaptation techniques in industry! 2-A class of very high order numerical schemes which can utilize possibly solution dependant basis functions in order to lower the number of degrees of freedom, for example to compute accurately boundary layers with low resolutions. 3-A general non intrusive technique for handling uncertainties in order to deal with irregular probability density functions (pdf) and also to handle pdf that may evolve in time, for example thanks to an optimisation loop. The curse of dimensionality will be dealt thanks Harten's multiresolution method combined with sparse grid methods. Currently, and up to our knowledge, no scheme has each of these properties. This research program will have an impact on numerical schemes and industrial applications.
Summary
The numerical simulation of complex compressible flow problem is still a challenge nowaday even for simple models. In our opinion, the most important hard points that need currently to be tackled and solved is how to obtain stable, scalable, very accurate, easy to code and to maintain schemes on complex geometries. The method should easily handle mesh refinement, even near the boundary where the most interesting engineering quantities have to be evaluated. Unsteady uncertainties in the model, for example in the geometry or the boundary conditions should represented efficiently.This proposal goal is to design, develop and evaluate solutions to each of the above problems. Our work program will lead to significant breakthroughs for flow simulations. More specifically, we propose to work on 3 connected problems: 1-A class of very high order numerical schemes able to easily deal with the geometry of boundaries and still can solve steep problems. The geometry is generally defined by CAD tools. The output is used to generate a mesh which is then used by the scheme. Hence, any mesh refinement process is disconnected from the CAD, a situation that prevents the spread of mesh adaptation techniques in industry! 2-A class of very high order numerical schemes which can utilize possibly solution dependant basis functions in order to lower the number of degrees of freedom, for example to compute accurately boundary layers with low resolutions. 3-A general non intrusive technique for handling uncertainties in order to deal with irregular probability density functions (pdf) and also to handle pdf that may evolve in time, for example thanks to an optimisation loop. The curse of dimensionality will be dealt thanks Harten's multiresolution method combined with sparse grid methods. Currently, and up to our knowledge, no scheme has each of these properties. This research program will have an impact on numerical schemes and industrial applications.
Max ERC Funding
1 432 769 €
Duration
Start date: 2008-12-01, End date: 2013-11-30
Project acronym ADORA
Project Asymptotic approach to spatial and dynamical organizations
Researcher (PI) Benoit PERTHAME
Host Institution (HI) SORBONNE UNIVERSITE
Call Details Advanced Grant (AdG), PE1, ERC-2016-ADG
Summary The understanding of spatial, social and dynamical organization of large numbers of agents is presently a fundamental issue in modern science. ADORA focuses on problems motivated by biology because, more than anywhere else, access to precise and many data has opened the route to novel and complex biomathematical models. The problems we address are written in terms of nonlinear partial differential equations. The flux-limited Keller-Segel system, the integrate-and-fire Fokker-Planck equation, kinetic equations with internal state, nonlocal parabolic equations and constrained Hamilton-Jacobi equations are among examples of the equations under investigation.
The role of mathematics is not only to understand the analytical structure of these new problems, but it is also to explain the qualitative behavior of solutions and to quantify their properties. The challenge arises here because these goals should be achieved through a hierarchy of scales. Indeed, the problems under consideration share the common feature that the large scale behavior cannot be understood precisely without access to a hierarchy of finer scales, down to the individual behavior and sometimes its molecular determinants.
Major difficulties arise because the numerous scales present in these equations have to be discovered and singularities appear in the asymptotic process which yields deep compactness obstructions. Our vision is that the complexity inherent to models of biology can be enlightened by mathematical analysis and a classification of the possible asymptotic regimes.
However an enormous effort is needed to uncover the equations intimate mathematical structures, and bring them at the level of conceptual understanding they deserve being given the applications motivating these questions which range from medical science or neuroscience to cell biology.
Summary
The understanding of spatial, social and dynamical organization of large numbers of agents is presently a fundamental issue in modern science. ADORA focuses on problems motivated by biology because, more than anywhere else, access to precise and many data has opened the route to novel and complex biomathematical models. The problems we address are written in terms of nonlinear partial differential equations. The flux-limited Keller-Segel system, the integrate-and-fire Fokker-Planck equation, kinetic equations with internal state, nonlocal parabolic equations and constrained Hamilton-Jacobi equations are among examples of the equations under investigation.
The role of mathematics is not only to understand the analytical structure of these new problems, but it is also to explain the qualitative behavior of solutions and to quantify their properties. The challenge arises here because these goals should be achieved through a hierarchy of scales. Indeed, the problems under consideration share the common feature that the large scale behavior cannot be understood precisely without access to a hierarchy of finer scales, down to the individual behavior and sometimes its molecular determinants.
Major difficulties arise because the numerous scales present in these equations have to be discovered and singularities appear in the asymptotic process which yields deep compactness obstructions. Our vision is that the complexity inherent to models of biology can be enlightened by mathematical analysis and a classification of the possible asymptotic regimes.
However an enormous effort is needed to uncover the equations intimate mathematical structures, and bring them at the level of conceptual understanding they deserve being given the applications motivating these questions which range from medical science or neuroscience to cell biology.
Max ERC Funding
2 192 500 €
Duration
Start date: 2017-09-01, End date: 2022-08-31
Project acronym AF and MSOGR
Project Automorphic Forms and Moduli Spaces of Galois Representations
Researcher (PI) Toby Gee
Host Institution (HI) IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
Call Details Starting Grant (StG), PE1, ERC-2012-StG_20111012
Summary I propose to establish a research group to develop completely new tools in order to solve three important problems on the relationships between automorphic forms and Galois representations, which lie at the heart of the Langlands program. The first is to prove Serre’s conjecture for real quadratic fields. I will use automorphic induction to transfer the problem to U(4) over the rational numbers, where I will use automorphy lifting theorems and results on the weight part of Serre's conjecture that I established in my earlier work to reduce the problem to proving results in small weight and level. I will prove these base cases via integral p-adic Hodge theory and discriminant bounds.
The second is to develop a geometric theory of moduli spaces of mod p and p-adic Galois representations, and to use it to establish the Breuil–Mézard conjecture in arbitrary dimension, by reinterpreting the conjecture in geometric terms. This will transform the subject by building the first connections between the p-adic Langlands program and the geometric Langlands program, providing an entirely new world of techniques for number theorists. As a consequence of the Breuil-Mézard conjecture, I will be able to deduce far stronger automorphy lifting theorems (in arbitrary dimension) than those currently available.
The third is to completely determine the reduction mod p of certain two-dimensional crystalline representations, and as an application prove a strengthened version of the Gouvêa–Mazur conjecture. I will do this by means of explicit computations with the p-adic local Langlands correspondence for GL_2(Q_p), as well as by improving existing arguments which prove multiplicity one theorems via automorphy lifting theorems. This work will show that the existence of counterexamples to the Gouvêa-Mazur conjecture is due to a purely local phenomenon, and that when this local obstruction vanishes, far stronger conjectures of Buzzard on the slopes of the U_p operator hold.
Summary
I propose to establish a research group to develop completely new tools in order to solve three important problems on the relationships between automorphic forms and Galois representations, which lie at the heart of the Langlands program. The first is to prove Serre’s conjecture for real quadratic fields. I will use automorphic induction to transfer the problem to U(4) over the rational numbers, where I will use automorphy lifting theorems and results on the weight part of Serre's conjecture that I established in my earlier work to reduce the problem to proving results in small weight and level. I will prove these base cases via integral p-adic Hodge theory and discriminant bounds.
The second is to develop a geometric theory of moduli spaces of mod p and p-adic Galois representations, and to use it to establish the Breuil–Mézard conjecture in arbitrary dimension, by reinterpreting the conjecture in geometric terms. This will transform the subject by building the first connections between the p-adic Langlands program and the geometric Langlands program, providing an entirely new world of techniques for number theorists. As a consequence of the Breuil-Mézard conjecture, I will be able to deduce far stronger automorphy lifting theorems (in arbitrary dimension) than those currently available.
The third is to completely determine the reduction mod p of certain two-dimensional crystalline representations, and as an application prove a strengthened version of the Gouvêa–Mazur conjecture. I will do this by means of explicit computations with the p-adic local Langlands correspondence for GL_2(Q_p), as well as by improving existing arguments which prove multiplicity one theorems via automorphy lifting theorems. This work will show that the existence of counterexamples to the Gouvêa-Mazur conjecture is due to a purely local phenomenon, and that when this local obstruction vanishes, far stronger conjectures of Buzzard on the slopes of the U_p operator hold.
Max ERC Funding
1 131 339 €
Duration
Start date: 2012-10-01, End date: 2017-09-30
Project acronym AFMIDMOA
Project "Applying Fundamental Mathematics in Discrete Mathematics, Optimization, and Algorithmics"
Researcher (PI) Alexander Schrijver
Host Institution (HI) UNIVERSITEIT VAN AMSTERDAM
Call Details Advanced Grant (AdG), PE1, ERC-2013-ADG
Summary "This proposal aims at strengthening the connections between more fundamentally oriented areas of mathematics like algebra, geometry, analysis, and topology, and the more applied oriented and more recently emerging disciplines of discrete mathematics, optimization, and algorithmics.
The overall goal of the project is to obtain, with methods from fundamental mathematics, new effective tools to unravel the complexity of structures like graphs, networks, codes, knots, polynomials, and tensors, and to get a grip on such complex structures by new efficient characterizations, sharper bounds, and faster algorithms.
In the last few years, there have been several new developments where methods from representation theory, invariant theory, algebraic geometry, measure theory, functional analysis, and topology found new applications in discrete mathematics and optimization, both theoretically and algorithmically. Among the typical application areas are networks, coding, routing, timetabling, statistical and quantum physics, and computer science.
The project focuses in particular on:
A. Understanding partition functions with invariant theory and algebraic geometry
B. Graph limits, regularity, Hilbert spaces, and low rank approximation of polynomials
C. Reducing complexity in optimization by exploiting symmetry with representation theory
D. Reducing complexity in discrete optimization by homotopy and cohomology
These research modules are interconnected by themes like symmetry, regularity, and complexity, and by common methods from algebra, analysis, geometry, and topology."
Summary
"This proposal aims at strengthening the connections between more fundamentally oriented areas of mathematics like algebra, geometry, analysis, and topology, and the more applied oriented and more recently emerging disciplines of discrete mathematics, optimization, and algorithmics.
The overall goal of the project is to obtain, with methods from fundamental mathematics, new effective tools to unravel the complexity of structures like graphs, networks, codes, knots, polynomials, and tensors, and to get a grip on such complex structures by new efficient characterizations, sharper bounds, and faster algorithms.
In the last few years, there have been several new developments where methods from representation theory, invariant theory, algebraic geometry, measure theory, functional analysis, and topology found new applications in discrete mathematics and optimization, both theoretically and algorithmically. Among the typical application areas are networks, coding, routing, timetabling, statistical and quantum physics, and computer science.
The project focuses in particular on:
A. Understanding partition functions with invariant theory and algebraic geometry
B. Graph limits, regularity, Hilbert spaces, and low rank approximation of polynomials
C. Reducing complexity in optimization by exploiting symmetry with representation theory
D. Reducing complexity in discrete optimization by homotopy and cohomology
These research modules are interconnected by themes like symmetry, regularity, and complexity, and by common methods from algebra, analysis, geometry, and topology."
Max ERC Funding
2 001 598 €
Duration
Start date: 2014-01-01, End date: 2018-12-31
Project acronym AGALT
Project Asymptotic Geometric Analysis and Learning Theory
Researcher (PI) Shahar Mendelson
Host Institution (HI) TECHNION - ISRAEL INSTITUTE OF TECHNOLOGY
Call Details Starting Grant (StG), PE1, ERC-2007-StG
Summary In a typical learning problem one tries to approximate an unknown function by a function from a given class using random data, sampled according to an unknown measure. In this project we will be interested in parameters that govern the complexity of a learning problem. It turns out that this complexity is determined by the geometry of certain sets in high dimension that are connected to the given class (random coordinate projections of the class). Thus, one has to understand the structure of these sets as a function of the dimension - which is given by the cardinality of the random sample. The resulting analysis leads to many theoretical questions in Asymptotic Geometric Analysis, Probability (most notably, Empirical Processes Theory) and Combinatorics, which are of independent interest beyond the application to Learning Theory. Our main goal is to describe the role of various complexity parameters involved in a learning problem, to analyze the connections between them and to investigate the way they determine the geometry of the relevant high dimensional sets. Some of the questions we intend to tackle are well known open problems and making progress towards their solution will have a significant theoretical impact. Moreover, this project should lead to a more complete theory of learning and is likely to have some practical impact, for example, in the design of more efficient learning algorithms.
Summary
In a typical learning problem one tries to approximate an unknown function by a function from a given class using random data, sampled according to an unknown measure. In this project we will be interested in parameters that govern the complexity of a learning problem. It turns out that this complexity is determined by the geometry of certain sets in high dimension that are connected to the given class (random coordinate projections of the class). Thus, one has to understand the structure of these sets as a function of the dimension - which is given by the cardinality of the random sample. The resulting analysis leads to many theoretical questions in Asymptotic Geometric Analysis, Probability (most notably, Empirical Processes Theory) and Combinatorics, which are of independent interest beyond the application to Learning Theory. Our main goal is to describe the role of various complexity parameters involved in a learning problem, to analyze the connections between them and to investigate the way they determine the geometry of the relevant high dimensional sets. Some of the questions we intend to tackle are well known open problems and making progress towards their solution will have a significant theoretical impact. Moreover, this project should lead to a more complete theory of learning and is likely to have some practical impact, for example, in the design of more efficient learning algorithms.
Max ERC Funding
750 000 €
Duration
Start date: 2009-03-01, End date: 2014-02-28
Project acronym AlgTateGro
Project Constructing line bundles on algebraic varieties --around conjectures of Tate and Grothendieck
Researcher (PI) François CHARLES
Host Institution (HI) UNIVERSITE PARIS-SUD
Call Details Starting Grant (StG), PE1, ERC-2016-STG
Summary The goal of this project is to investigate two conjectures in arithmetic geometry pertaining to the geometry of projective varieties over finite and number fields. These two conjectures, formulated by Tate and Grothendieck in the 1960s, predict which cohomology classes are chern classes of line bundles. They both form an arithmetic counterpart of a theorem of Lefschetz, proved in the 1940s, which itself is the only known case of the Hodge conjecture. These two long-standing conjectures are one of the aspects of a more general web of questions regarding the topology of algebraic varieties which have been emphasized by Grothendieck and have since had a central role in modern arithmetic geometry. Special cases of these conjectures, appearing for instance in the work of Tate, Deligne, Faltings, Schneider-Lang, Masser-Wüstholz, have all had important consequences.
My goal is to investigate different lines of attack towards these conjectures, building on recent work on myself and Jean-Benoît Bost on related problems. The two main directions of the proposal are as follows. Over finite fields, the Tate conjecture is related to finiteness results for certain cohomological objects. I want to understand how to relate these to hidden boundedness properties of algebraic varieties that have appeared in my recent geometric proof of the Tate conjecture for K3 surfaces. The existence and relevance of a theory of Donaldson invariants for moduli spaces of twisted sheaves over finite fields seems to be a promising and novel direction. Over number fields, I want to combine the geometric insight above with algebraization techniques developed by Bost. In a joint project, we want to investigate how these can be used to first understand geometrically major results in transcendence theory and then attack the Grothendieck period conjecture for divisors via a number-theoretic and complex-analytic understanding of universal vector extensions of abelian schemes over curves.
Summary
The goal of this project is to investigate two conjectures in arithmetic geometry pertaining to the geometry of projective varieties over finite and number fields. These two conjectures, formulated by Tate and Grothendieck in the 1960s, predict which cohomology classes are chern classes of line bundles. They both form an arithmetic counterpart of a theorem of Lefschetz, proved in the 1940s, which itself is the only known case of the Hodge conjecture. These two long-standing conjectures are one of the aspects of a more general web of questions regarding the topology of algebraic varieties which have been emphasized by Grothendieck and have since had a central role in modern arithmetic geometry. Special cases of these conjectures, appearing for instance in the work of Tate, Deligne, Faltings, Schneider-Lang, Masser-Wüstholz, have all had important consequences.
My goal is to investigate different lines of attack towards these conjectures, building on recent work on myself and Jean-Benoît Bost on related problems. The two main directions of the proposal are as follows. Over finite fields, the Tate conjecture is related to finiteness results for certain cohomological objects. I want to understand how to relate these to hidden boundedness properties of algebraic varieties that have appeared in my recent geometric proof of the Tate conjecture for K3 surfaces. The existence and relevance of a theory of Donaldson invariants for moduli spaces of twisted sheaves over finite fields seems to be a promising and novel direction. Over number fields, I want to combine the geometric insight above with algebraization techniques developed by Bost. In a joint project, we want to investigate how these can be used to first understand geometrically major results in transcendence theory and then attack the Grothendieck period conjecture for divisors via a number-theoretic and complex-analytic understanding of universal vector extensions of abelian schemes over curves.
Max ERC Funding
1 222 329 €
Duration
Start date: 2016-12-01, End date: 2021-11-30
Project acronym ALKAGE
Project Algebraic and Kähler geometry
Researcher (PI) Jean-Pierre, Raymond, Philippe Demailly
Host Institution (HI) UNIVERSITE GRENOBLE ALPES
Call Details Advanced Grant (AdG), PE1, ERC-2014-ADG
Summary The purpose of this project is to study basic questions in algebraic and Kähler geometry. It is well known that the structure of projective or Kähler manifolds is governed by positivity or negativity properties of the curvature tensor. However, many fundamental problems are still wide open. Since the mid 1980's, I have developed a large number of key concepts and results that have led to important progress in transcendental algebraic geometry. Let me mention the discovery of holomorphic Morse inequalities, systematic applications of L² estimates with singular hermitian metrics, and a much improved understanding of Monge-Ampère equations and of singularities of plurisuharmonic functions. My first goal will be to investigate the Green-Griffiths-Lang conjecture asserting that an entire curve drawn in a variety of general type is algebraically degenerate. The subject is intimately related to important questions concerning Diophantine equations, especially higher dimensional generalizations of Faltings' theorem - the so-called Vojta program. One can rely here on a breakthrough I made in 2010, showing that all such entire curves must satisfy algebraic differential equations. A second closely related area of research of this project is the analysis of the structure of projective or compact Kähler manifolds. It can be seen as a generalization of the classification theory of surfaces by Kodaira, and of the more recent results for dimension 3 (Kawamata, Kollár, Mori, Shokurov, ...) to other dimensions. My plan is to combine powerful recent results obtained on the duality of positive cohomology cones with an analysis of the instability of the tangent bundle, i.e. of the Harder-Narasimhan filtration. On these ground-breaking questions, I intend to go much further and to enhance my national and international collaborations. These subjects already attract many young researchers and postdocs throughout the world, and the grant could be used to create even stronger interactions.
Summary
The purpose of this project is to study basic questions in algebraic and Kähler geometry. It is well known that the structure of projective or Kähler manifolds is governed by positivity or negativity properties of the curvature tensor. However, many fundamental problems are still wide open. Since the mid 1980's, I have developed a large number of key concepts and results that have led to important progress in transcendental algebraic geometry. Let me mention the discovery of holomorphic Morse inequalities, systematic applications of L² estimates with singular hermitian metrics, and a much improved understanding of Monge-Ampère equations and of singularities of plurisuharmonic functions. My first goal will be to investigate the Green-Griffiths-Lang conjecture asserting that an entire curve drawn in a variety of general type is algebraically degenerate. The subject is intimately related to important questions concerning Diophantine equations, especially higher dimensional generalizations of Faltings' theorem - the so-called Vojta program. One can rely here on a breakthrough I made in 2010, showing that all such entire curves must satisfy algebraic differential equations. A second closely related area of research of this project is the analysis of the structure of projective or compact Kähler manifolds. It can be seen as a generalization of the classification theory of surfaces by Kodaira, and of the more recent results for dimension 3 (Kawamata, Kollár, Mori, Shokurov, ...) to other dimensions. My plan is to combine powerful recent results obtained on the duality of positive cohomology cones with an analysis of the instability of the tangent bundle, i.e. of the Harder-Narasimhan filtration. On these ground-breaking questions, I intend to go much further and to enhance my national and international collaborations. These subjects already attract many young researchers and postdocs throughout the world, and the grant could be used to create even stronger interactions.
Max ERC Funding
1 809 345 €
Duration
Start date: 2015-09-01, End date: 2020-08-31
Project acronym AMAREC
Project Amenability, Approximation and Reconstruction
Researcher (PI) Wilhelm WINTER
Host Institution (HI) WESTFAELISCHE WILHELMS-UNIVERSITAET MUENSTER
Call Details Advanced Grant (AdG), PE1, ERC-2018-ADG
Summary Algebras of operators on Hilbert spaces were originally introduced as the right framework for the mathematical description of quantum mechanics. In modern mathematics the scope has much broadened due to the highly versatile nature of operator algebras. They are particularly useful in the analysis of groups and their actions. Amenability is a finiteness property which occurs in many different contexts and which can be characterised in many different ways. We will analyse amenability in terms of approximation properties, in the frameworks of abstract C*-algebras, of topological dynamical systems, and of discrete groups. Such approximation properties will serve as bridging devices between these setups, and they will be used to systematically recover geometric information about the underlying structures. When passing from groups, and more generally from dynamical systems, to operator algebras, one loses information, but one gains new tools to isolate and analyse pertinent properties of the underlying structure. We will mostly be interested in the topological setting, and in the associated C*-algebras. Amenability of groups or of dynamical systems then translates into the completely positive approximation property. Systems of completely positive approximations store all the essential data about a C*-algebra, and sometimes one can arrange the systems so that one can directly read of such information. For transformation group C*-algebras, one can achieve this by using approximation properties of the underlying dynamics. To some extent one can even go back, and extract dynamical approximation properties from completely positive approximations of the C*-algebra. This interplay between approximation properties in topological dynamics and in noncommutative topology carries a surprisingly rich structure. It connects directly to the heart of the classification problem for nuclear C*-algebras on the one hand, and to central open questions on amenable dynamics on the other.
Summary
Algebras of operators on Hilbert spaces were originally introduced as the right framework for the mathematical description of quantum mechanics. In modern mathematics the scope has much broadened due to the highly versatile nature of operator algebras. They are particularly useful in the analysis of groups and their actions. Amenability is a finiteness property which occurs in many different contexts and which can be characterised in many different ways. We will analyse amenability in terms of approximation properties, in the frameworks of abstract C*-algebras, of topological dynamical systems, and of discrete groups. Such approximation properties will serve as bridging devices between these setups, and they will be used to systematically recover geometric information about the underlying structures. When passing from groups, and more generally from dynamical systems, to operator algebras, one loses information, but one gains new tools to isolate and analyse pertinent properties of the underlying structure. We will mostly be interested in the topological setting, and in the associated C*-algebras. Amenability of groups or of dynamical systems then translates into the completely positive approximation property. Systems of completely positive approximations store all the essential data about a C*-algebra, and sometimes one can arrange the systems so that one can directly read of such information. For transformation group C*-algebras, one can achieve this by using approximation properties of the underlying dynamics. To some extent one can even go back, and extract dynamical approximation properties from completely positive approximations of the C*-algebra. This interplay between approximation properties in topological dynamics and in noncommutative topology carries a surprisingly rich structure. It connects directly to the heart of the classification problem for nuclear C*-algebras on the one hand, and to central open questions on amenable dynamics on the other.
Max ERC Funding
1 596 017 €
Duration
Start date: 2019-10-01, End date: 2024-09-30
Project acronym AMSTAT
Project Problems at the Applied Mathematics-Statistics Interface
Researcher (PI) Andrew Stuart
Host Institution (HI) THE UNIVERSITY OF WARWICK
Call Details Advanced Grant (AdG), PE1, ERC-2008-AdG
Summary Applied mathematics is concerned with developing models with predictive capability, and with probing those models to obtain qualitative and quantitative insight into the phenomena being modelled. Statistics is data-driven and is aimed at the development of methodologies to optimize the information derived from data. The increasing complexity of phenomena that scientists and engineers wish to model, together with our increased ability to gather, store and interrogate data, mean that the subjects of applied mathematics and statistics are increasingly required to work in conjunction. This research proposal is concerned with a research program at the interface between these two disciplines, aimed at problems in differential equations where profusion of data and the sophisticated model combine to produce the mathematical problem of obtaining information from a probability measure on function space. Applications are far-reaching and include the atmospheric sciences, geophysics, chemistry, econometrics and signal processing. The objectives of the research are: (i) to create the systematic foundations for a range of problems at the applied mathematics and statistics interface which share the common mathematical structure underpinning the range of applications described above; (ii) to exploit this common mathematical structure to design effecient algorithms to sample probability measures on function space; (iii) to apply these algorithms to attack a range of significant problems arising in molecular dynamics and in the atmospheric sciences.
Summary
Applied mathematics is concerned with developing models with predictive capability, and with probing those models to obtain qualitative and quantitative insight into the phenomena being modelled. Statistics is data-driven and is aimed at the development of methodologies to optimize the information derived from data. The increasing complexity of phenomena that scientists and engineers wish to model, together with our increased ability to gather, store and interrogate data, mean that the subjects of applied mathematics and statistics are increasingly required to work in conjunction. This research proposal is concerned with a research program at the interface between these two disciplines, aimed at problems in differential equations where profusion of data and the sophisticated model combine to produce the mathematical problem of obtaining information from a probability measure on function space. Applications are far-reaching and include the atmospheric sciences, geophysics, chemistry, econometrics and signal processing. The objectives of the research are: (i) to create the systematic foundations for a range of problems at the applied mathematics and statistics interface which share the common mathematical structure underpinning the range of applications described above; (ii) to exploit this common mathematical structure to design effecient algorithms to sample probability measures on function space; (iii) to apply these algorithms to attack a range of significant problems arising in molecular dynamics and in the atmospheric sciences.
Max ERC Funding
1 693 501 €
Duration
Start date: 2008-12-01, End date: 2014-11-30
Project acronym ANADEL
Project Analysis of Geometrical Effects on Dispersive Equations
Researcher (PI) Danela Oana IVANOVICI
Host Institution (HI) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Call Details Starting Grant (StG), PE1, ERC-2017-STG
Summary We are concerned with localization properties of solutions to hyperbolic PDEs, especially problems with a geometric component: how do boundaries and heterogeneous media influence spreading and concentration of solutions. While our first focus is on wave and Schrödinger equations on manifolds with boundary, strong connections exist with phase space localization for (clusters of) eigenfunctions, which are of independent interest. Motivations come from nonlinear dispersive models (in physically relevant settings), properties of eigenfunctions in quantum chaos (related to both physics of optic fiber design as well as number theoretic questions), or harmonic analysis on manifolds.
Waves propagation in real life physics occur in media which are neither homogeneous or spatially infinity. The birth of radar/sonar technologies (and the raise of computed tomography) greatly motivated numerous developments in microlocal analysis and the linear theory. Only recently toy nonlinear models have been studied on a curved background, sometimes compact or rough. Understanding how to extend such tools, dealing with wave dispersion or focusing, will allow us to significantly progress in our mathematical understanding of physically relevant models. There, boundaries appear naturally and most earlier developments related to propagation of singularities in this context have limited scope with respect to crucial dispersive effects. Despite great progress over the last decade, driven by the study of quasilinear equations, our knowledge is still very limited. Going beyond this recent activity requires new tools whose development is at the heart of this proposal, including good approximate solutions (parametrices) going over arbitrarily large numbers of caustics, sharp pointwise bounds on Green functions, development of efficient wave packets methods, quantitative refinements of propagation of singularities (with direct applications in control theory), only to name a few important ones.
Summary
We are concerned with localization properties of solutions to hyperbolic PDEs, especially problems with a geometric component: how do boundaries and heterogeneous media influence spreading and concentration of solutions. While our first focus is on wave and Schrödinger equations on manifolds with boundary, strong connections exist with phase space localization for (clusters of) eigenfunctions, which are of independent interest. Motivations come from nonlinear dispersive models (in physically relevant settings), properties of eigenfunctions in quantum chaos (related to both physics of optic fiber design as well as number theoretic questions), or harmonic analysis on manifolds.
Waves propagation in real life physics occur in media which are neither homogeneous or spatially infinity. The birth of radar/sonar technologies (and the raise of computed tomography) greatly motivated numerous developments in microlocal analysis and the linear theory. Only recently toy nonlinear models have been studied on a curved background, sometimes compact or rough. Understanding how to extend such tools, dealing with wave dispersion or focusing, will allow us to significantly progress in our mathematical understanding of physically relevant models. There, boundaries appear naturally and most earlier developments related to propagation of singularities in this context have limited scope with respect to crucial dispersive effects. Despite great progress over the last decade, driven by the study of quasilinear equations, our knowledge is still very limited. Going beyond this recent activity requires new tools whose development is at the heart of this proposal, including good approximate solutions (parametrices) going over arbitrarily large numbers of caustics, sharp pointwise bounds on Green functions, development of efficient wave packets methods, quantitative refinements of propagation of singularities (with direct applications in control theory), only to name a few important ones.
Max ERC Funding
1 293 763 €
Duration
Start date: 2018-02-01, End date: 2023-01-31
Project acronym analysisdirac
Project The analysis of the Dirac operator: the hypoelliptic Laplacian and its applications
Researcher (PI) Jean-Michel Philippe Marie-José Bismut
Host Institution (HI) UNIVERSITE PARIS-SUD
Call Details Advanced Grant (AdG), PE1, ERC-2011-ADG_20110209
Summary This proposal is devoted to the applications of a new hypoelliptic Dirac operator,
whose analytic properties have been studied by Lebeau and myself. Its construction connects classical Hodge theory with the geodesic flow, and more generally any geometrically defined Hodge Laplacian with a dynamical system on the cotangent bundle. The proper description of this object can be given in analytic, index theoretic and probabilistic terms, which explains both its potential many applications, and also its complexity.
Summary
This proposal is devoted to the applications of a new hypoelliptic Dirac operator,
whose analytic properties have been studied by Lebeau and myself. Its construction connects classical Hodge theory with the geodesic flow, and more generally any geometrically defined Hodge Laplacian with a dynamical system on the cotangent bundle. The proper description of this object can be given in analytic, index theoretic and probabilistic terms, which explains both its potential many applications, and also its complexity.
Max ERC Funding
1 112 400 €
Duration
Start date: 2012-02-01, End date: 2017-01-31
Project acronym ANALYTIC
Project ANALYTIC PROPERTIES OF INFINITE GROUPS:
limits, curvature, and randomness
Researcher (PI) Gulnara Arzhantseva
Host Institution (HI) UNIVERSITAT WIEN
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The overall goal of this project is to develop new concepts and techniques in geometric and asymptotic group theory for a systematic study of the analytic properties of discrete groups. These are properties depending on the unitary representation theory of the group. The fundamental examples are amenability, discovered by von Neumann in 1929, and property (T), introduced by Kazhdan in 1967.
My main objective is to establish the precise relations between groups recently appeared in K-theory and topology such as C*-exact groups and groups coarsely embeddable into a Hilbert space, versus those discovered in ergodic theory and operator algebra, for example, sofic and hyperlinear groups. This is a first ever attempt to confront the analytic behavior of so different nature. I plan to work on crucial open questions: Is every coarsely embeddable group C*-exact? Is every group sofic? Is every hyperlinear group sofic?
My motivation is two-fold:
- Many outstanding conjectures were recently solved for these groups, e.g. the Novikov conjecture (1965) for coarsely embeddable groups by Yu in 2000 and the Gottschalk surjunctivity conjecture (1973) for sofic groups by Gromov in 1999. However, their group-theoretical structure remains mysterious.
- In recent years, geometric group theory has undergone significant changes, mainly due to the growing impact of this theory on other branches of mathematics. However, the interplay between geometric, asymptotic, and analytic group properties has not yet been fully understood.
The main innovative contribution of this proposal lies in the interaction between 3 axes: (i) limits of groups, in the space of marked groups or metric ultralimits; (ii) analytic properties of groups with curvature, of lacunary or relatively hyperbolic groups; (iii) random groups, in a topological or statistical meaning. As a result, I will describe the above apparently unrelated classes of groups in a unified way and will detail their algebraic behavior.
Summary
The overall goal of this project is to develop new concepts and techniques in geometric and asymptotic group theory for a systematic study of the analytic properties of discrete groups. These are properties depending on the unitary representation theory of the group. The fundamental examples are amenability, discovered by von Neumann in 1929, and property (T), introduced by Kazhdan in 1967.
My main objective is to establish the precise relations between groups recently appeared in K-theory and topology such as C*-exact groups and groups coarsely embeddable into a Hilbert space, versus those discovered in ergodic theory and operator algebra, for example, sofic and hyperlinear groups. This is a first ever attempt to confront the analytic behavior of so different nature. I plan to work on crucial open questions: Is every coarsely embeddable group C*-exact? Is every group sofic? Is every hyperlinear group sofic?
My motivation is two-fold:
- Many outstanding conjectures were recently solved for these groups, e.g. the Novikov conjecture (1965) for coarsely embeddable groups by Yu in 2000 and the Gottschalk surjunctivity conjecture (1973) for sofic groups by Gromov in 1999. However, their group-theoretical structure remains mysterious.
- In recent years, geometric group theory has undergone significant changes, mainly due to the growing impact of this theory on other branches of mathematics. However, the interplay between geometric, asymptotic, and analytic group properties has not yet been fully understood.
The main innovative contribution of this proposal lies in the interaction between 3 axes: (i) limits of groups, in the space of marked groups or metric ultralimits; (ii) analytic properties of groups with curvature, of lacunary or relatively hyperbolic groups; (iii) random groups, in a topological or statistical meaning. As a result, I will describe the above apparently unrelated classes of groups in a unified way and will detail their algebraic behavior.
Max ERC Funding
1 065 500 €
Duration
Start date: 2011-04-01, End date: 2016-03-31
Project acronym ANAMULTISCALE
Project Analysis of Multiscale Systems Driven by Functionals
Researcher (PI) Alexander Mielke
Host Institution (HI) FORSCHUNGSVERBUND BERLIN EV
Call Details Advanced Grant (AdG), PE1, ERC-2010-AdG_20100224
Summary Many complex phenomena in the sciences are described by nonlinear partial differential equations, the solutions of which exhibit oscillations and concentration effects on multiple temporal or spatial scales. Our aim is to use methods from applied analysis to contribute to the understanding of the interplay of effects on different scales. The central question is to determine those quantities on the microscale which are needed to for the correct description of the macroscopic evolution.
We aim to develop a mathematical framework for analyzing and modeling coupled systems with multiple scales. This will include Hamiltonian dynamics as well as different types of dissipation like gradient flows or rate-independent dynamics. The choice of models will be guided by specific applications in material modeling (e.g., thermoplasticity, pattern formation, porous media) and optoelectronics (pulse interaction, Maxwell-Bloch systems, semiconductors, quantum mechanics). The research will address mathematically fundamental issues like existence and stability of solutions but will mainly be devoted to the modeling of multiscale phenomena in evolution systems. We will focus on systems with geometric structures, where the dynamics is driven by functionals. Thus, we can go much beyond the classical theory of homogenization and singular perturbations. The novel features of our approach are
- the combination of different dynamical effects in one framework,
- the use of geometric and metric structures for coupled partial differential equations,
- the exploitation of Gamma-convergence for evolution systems driven by functionals.
Summary
Many complex phenomena in the sciences are described by nonlinear partial differential equations, the solutions of which exhibit oscillations and concentration effects on multiple temporal or spatial scales. Our aim is to use methods from applied analysis to contribute to the understanding of the interplay of effects on different scales. The central question is to determine those quantities on the microscale which are needed to for the correct description of the macroscopic evolution.
We aim to develop a mathematical framework for analyzing and modeling coupled systems with multiple scales. This will include Hamiltonian dynamics as well as different types of dissipation like gradient flows or rate-independent dynamics. The choice of models will be guided by specific applications in material modeling (e.g., thermoplasticity, pattern formation, porous media) and optoelectronics (pulse interaction, Maxwell-Bloch systems, semiconductors, quantum mechanics). The research will address mathematically fundamental issues like existence and stability of solutions but will mainly be devoted to the modeling of multiscale phenomena in evolution systems. We will focus on systems with geometric structures, where the dynamics is driven by functionals. Thus, we can go much beyond the classical theory of homogenization and singular perturbations. The novel features of our approach are
- the combination of different dynamical effects in one framework,
- the use of geometric and metric structures for coupled partial differential equations,
- the exploitation of Gamma-convergence for evolution systems driven by functionals.
Max ERC Funding
1 390 000 €
Duration
Start date: 2011-04-01, End date: 2017-03-31
Project acronym ANGEOM
Project Geometric analysis in the Euclidean space
Researcher (PI) Xavier Tolsa Domenech
Host Institution (HI) UNIVERSITAT AUTONOMA DE BARCELONA
Call Details Advanced Grant (AdG), PE1, ERC-2012-ADG_20120216
Summary "We propose to study different questions in the area of the so called geometric analysis. Most of the topics we are interested in deal with the connection between the behavior of singular integrals and the geometry of sets and measures. The study of this connection has been shown to be extremely helpful in the solution of certain long standing problems in the last years, such as the solution of the Painlev\'e problem or the obtaining of the optimal distortion bounds for quasiconformal mappings by Astala.
More specifically, we would like to study the relationship between the L^2 boundedness of singular integrals associated with Riesz and other related kernels, and rectifiability and other geometric notions. The so called David-Semmes problem is probably the main open problem in this area. Up to now, the techniques used to deal with this problem come from multiscale analysis and involve ideas from Littlewood-Paley theory and quantitative techniques of rectifiability. We propose to apply new ideas that combine variational arguments with other techniques which have connections with mass transportation. Further, we think that it is worth to explore in more detail the connection among mass transportation, singular integrals, and uniform rectifiability.
We are also interested in the field of quasiconformal mappings. We plan to study a problem regarding the quasiconformal distortion of quasicircles. This problem consists in proving that the bounds obtained recently by S. Smirnov on the dimension of K-quasicircles are optimal. We want to apply techniques from quantitative geometric measure theory to deal with this question.
Another question that we intend to explore lies in the interplay of harmonic analysis, geometric measure theory and partial differential equations. This concerns an old problem on the unique continuation of harmonic functions at the boundary open C^1 or Lipschitz domain. All the results known by now deal with smoother Dini domains."
Summary
"We propose to study different questions in the area of the so called geometric analysis. Most of the topics we are interested in deal with the connection between the behavior of singular integrals and the geometry of sets and measures. The study of this connection has been shown to be extremely helpful in the solution of certain long standing problems in the last years, such as the solution of the Painlev\'e problem or the obtaining of the optimal distortion bounds for quasiconformal mappings by Astala.
More specifically, we would like to study the relationship between the L^2 boundedness of singular integrals associated with Riesz and other related kernels, and rectifiability and other geometric notions. The so called David-Semmes problem is probably the main open problem in this area. Up to now, the techniques used to deal with this problem come from multiscale analysis and involve ideas from Littlewood-Paley theory and quantitative techniques of rectifiability. We propose to apply new ideas that combine variational arguments with other techniques which have connections with mass transportation. Further, we think that it is worth to explore in more detail the connection among mass transportation, singular integrals, and uniform rectifiability.
We are also interested in the field of quasiconformal mappings. We plan to study a problem regarding the quasiconformal distortion of quasicircles. This problem consists in proving that the bounds obtained recently by S. Smirnov on the dimension of K-quasicircles are optimal. We want to apply techniques from quantitative geometric measure theory to deal with this question.
Another question that we intend to explore lies in the interplay of harmonic analysis, geometric measure theory and partial differential equations. This concerns an old problem on the unique continuation of harmonic functions at the boundary open C^1 or Lipschitz domain. All the results known by now deal with smoother Dini domains."
Max ERC Funding
1 105 930 €
Duration
Start date: 2013-05-01, End date: 2018-04-30
Project acronym ANOPTSETCON
Project Analysis of optimal sets and optimal constants: old questions and new results
Researcher (PI) Aldo Pratelli
Host Institution (HI) FRIEDRICH-ALEXANDER-UNIVERSITAET ERLANGEN NUERNBERG
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The analysis of geometric and functional inequalities naturally leads to consider the extremal cases, thus
looking for optimal sets, or optimal functions, or optimal constants. The most classical examples are the (different versions of the) isoperimetric inequality and the Sobolev-like inequalities. Much is known about equality cases and best constants, but there are still many questions which seem quite natural but yet have no answer. For instance, it is not known, even in the 2-dimensional space, the answer of a question by Brezis: which set,
among those with a given volume, has the biggest Sobolev-Poincaré constant for p=1? This is a very natural problem, and it appears reasonable that the optimal set should be the ball, but this has never been proved. The interest in problems like this relies not only in the extreme simplicity of the questions and in their classical flavour, but also in the new ideas and techniques which are needed to provide the answers.
The main techniques that we aim to use are fine arguments of symmetrization, geometric constructions and tools from mass transportation (which is well known to be deeply connected with functional inequalities). These are the basic tools that we already used to reach, in last years, many results in a specific direction, namely the search of sharp quantitative inequalities. Our first result, together with Fusco and Maggi, showed what follows. Everybody knows that the set which minimizes the perimeter with given volume is the ball.
But is it true that a set which almost minimizes the perimeter must be close to a ball? The question had been posed in the 1920's and many partial result appeared in the years. In our paper (Ann. of Math., 2007) we proved the sharp result. Many other results of this kind were obtained in last two years.
Summary
The analysis of geometric and functional inequalities naturally leads to consider the extremal cases, thus
looking for optimal sets, or optimal functions, or optimal constants. The most classical examples are the (different versions of the) isoperimetric inequality and the Sobolev-like inequalities. Much is known about equality cases and best constants, but there are still many questions which seem quite natural but yet have no answer. For instance, it is not known, even in the 2-dimensional space, the answer of a question by Brezis: which set,
among those with a given volume, has the biggest Sobolev-Poincaré constant for p=1? This is a very natural problem, and it appears reasonable that the optimal set should be the ball, but this has never been proved. The interest in problems like this relies not only in the extreme simplicity of the questions and in their classical flavour, but also in the new ideas and techniques which are needed to provide the answers.
The main techniques that we aim to use are fine arguments of symmetrization, geometric constructions and tools from mass transportation (which is well known to be deeply connected with functional inequalities). These are the basic tools that we already used to reach, in last years, many results in a specific direction, namely the search of sharp quantitative inequalities. Our first result, together with Fusco and Maggi, showed what follows. Everybody knows that the set which minimizes the perimeter with given volume is the ball.
But is it true that a set which almost minimizes the perimeter must be close to a ball? The question had been posed in the 1920's and many partial result appeared in the years. In our paper (Ann. of Math., 2007) we proved the sharp result. Many other results of this kind were obtained in last two years.
Max ERC Funding
540 000 €
Duration
Start date: 2010-08-01, End date: 2015-07-31
Project acronym ANPROB
Project Analytic-probabilistic methods for borderline singular integrals
Researcher (PI) Tuomas Pentinpoika Hytönen
Host Institution (HI) HELSINGIN YLIOPISTO
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary The proposal consists of an extensive research program to advance the understanding of singular integral operators of Harmonic Analysis in various situations on the borderline of the existing theory. This is to be achieved by a creative combination of techniques from Analysis and Probability. On top of the standard arsenal of modern Harmonic Analysis, the main probabilistic tools are the martingale transform inequalities of Burkholder, and random geometric constructions in the spirit of the random dyadic cubes introduced to Nonhomogeneous Analysis by Nazarov, Treil and Volberg.
The problems to be addressed fall under the following subtitles, with many interconnections and overlap: (i) sharp weighted inequalities; (ii) nonhomogeneous singular integrals on metric spaces; (iii) local Tb theorems with borderline assumptions; (iv) functional calculus of rough differential operators; and (v) vector-valued singular integrals.
Topic (i) is a part of Classical Analysis, where new methods have led to substantial recent progress, culminating in my solution in July 2010 of a celebrated problem on the linear dependence of the weighted operator norm on the Muckenhoupt norm of the weight. The proof should be extendible to several related questions, and the aim is to also address some outstanding open problems in the area.
Topics (ii) and (v) deal with extensions of the theory of singular integrals to functions with more general domain and range spaces, allowing them to be abstract metric and Banach spaces, respectively. In case (ii), I have recently been able to relax the requirements on the space compared to the established theories, opening a new research direction here. Topics (iii) and (iv) are concerned with weakening the assumptions on singular integrals in the usual Euclidean space, to allow certain applications in the theory of Partial Differential Equations. The goal is to maintain a close contact and exchange of ideas between such abstract and concrete questions.
Summary
The proposal consists of an extensive research program to advance the understanding of singular integral operators of Harmonic Analysis in various situations on the borderline of the existing theory. This is to be achieved by a creative combination of techniques from Analysis and Probability. On top of the standard arsenal of modern Harmonic Analysis, the main probabilistic tools are the martingale transform inequalities of Burkholder, and random geometric constructions in the spirit of the random dyadic cubes introduced to Nonhomogeneous Analysis by Nazarov, Treil and Volberg.
The problems to be addressed fall under the following subtitles, with many interconnections and overlap: (i) sharp weighted inequalities; (ii) nonhomogeneous singular integrals on metric spaces; (iii) local Tb theorems with borderline assumptions; (iv) functional calculus of rough differential operators; and (v) vector-valued singular integrals.
Topic (i) is a part of Classical Analysis, where new methods have led to substantial recent progress, culminating in my solution in July 2010 of a celebrated problem on the linear dependence of the weighted operator norm on the Muckenhoupt norm of the weight. The proof should be extendible to several related questions, and the aim is to also address some outstanding open problems in the area.
Topics (ii) and (v) deal with extensions of the theory of singular integrals to functions with more general domain and range spaces, allowing them to be abstract metric and Banach spaces, respectively. In case (ii), I have recently been able to relax the requirements on the space compared to the established theories, opening a new research direction here. Topics (iii) and (iv) are concerned with weakening the assumptions on singular integrals in the usual Euclidean space, to allow certain applications in the theory of Partial Differential Equations. The goal is to maintain a close contact and exchange of ideas between such abstract and concrete questions.
Max ERC Funding
1 100 000 €
Duration
Start date: 2011-11-01, End date: 2016-10-31
Project acronym ANT
Project Automata in Number Theory
Researcher (PI) Boris Adamczewski
Host Institution (HI) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Call Details Consolidator Grant (CoG), PE1, ERC-2014-CoG
Summary Finite automata are fundamental objects in Computer Science, of great importance on one hand for theoretical aspects (formal language theory, decidability, complexity) and on the other for practical applications (parsing). In number theory, finite automata are mainly used as simple devices for generating sequences of symbols over a finite set (e.g., digital representations of real numbers), and for recognizing some sets of integers or more generally of finitely generated abelian groups or monoids. One of the main features of these automatic structures comes from the fact that they are highly ordered without necessarily being trivial (i.e., periodic). With their rich fractal nature, they lie somewhere between order and chaos, even if, in most respects, their rigidity prevails. Over the last few years, several ground-breaking results have lead to a great renewed interest in the study of automatic structures in arithmetics.
A primary objective of the ANT project is to exploit this opportunity by developing new directions and interactions between automata and number theory. In this proposal, we outline three lines of research concerning fundamental number theoretical problems that have baffled mathematicians for decades. They include the study of integer base expansions of classical constants, of arithmetical linear differential equations and their link with enumerative combinatorics, and of arithmetics in positive characteristic. At first glance, these topics may seem unrelated, but, surprisingly enough, the theory of finite automata will serve as a natural guideline. We stress that this new point of view on classical questions is a key part of our methodology: we aim at creating a powerful synergy between the different approaches we propose to develop, placing automata theory and related methods at the heart of the subject. This project provides a unique opportunity to create the first international team focusing on these different problems as a whole.
Summary
Finite automata are fundamental objects in Computer Science, of great importance on one hand for theoretical aspects (formal language theory, decidability, complexity) and on the other for practical applications (parsing). In number theory, finite automata are mainly used as simple devices for generating sequences of symbols over a finite set (e.g., digital representations of real numbers), and for recognizing some sets of integers or more generally of finitely generated abelian groups or monoids. One of the main features of these automatic structures comes from the fact that they are highly ordered without necessarily being trivial (i.e., periodic). With their rich fractal nature, they lie somewhere between order and chaos, even if, in most respects, their rigidity prevails. Over the last few years, several ground-breaking results have lead to a great renewed interest in the study of automatic structures in arithmetics.
A primary objective of the ANT project is to exploit this opportunity by developing new directions and interactions between automata and number theory. In this proposal, we outline three lines of research concerning fundamental number theoretical problems that have baffled mathematicians for decades. They include the study of integer base expansions of classical constants, of arithmetical linear differential equations and their link with enumerative combinatorics, and of arithmetics in positive characteristic. At first glance, these topics may seem unrelated, but, surprisingly enough, the theory of finite automata will serve as a natural guideline. We stress that this new point of view on classical questions is a key part of our methodology: we aim at creating a powerful synergy between the different approaches we propose to develop, placing automata theory and related methods at the heart of the subject. This project provides a unique opportunity to create the first international team focusing on these different problems as a whole.
Max ERC Funding
1 438 745 €
Duration
Start date: 2015-10-01, End date: 2020-09-30
Project acronym ANTEGEFI
Project Analytic Techniques for Geometric and Functional Inequalities
Researcher (PI) Nicola Fusco
Host Institution (HI) UNIVERSITA DEGLI STUDI DI NAPOLI FEDERICO II
Call Details Advanced Grant (AdG), PE1, ERC-2008-AdG
Summary Isoperimetric and Sobolev inequalities are the best known examples of geometric-functional inequalities. In recent years the PI and collaborators have obtained new and sharp quantitative versions of these and other important related inequalities. These results have been obtained by the combined use of classical symmetrization methods, new tools coming from mass transportation theory, deep geometric measure tools and ad hoc symmetrizations. The objective of this project is to further develop thes techniques in order to get: sharp quantitative versions of Faber-Krahn inequality, Gaussian isoperimetric inequality, Brunn-Minkowski inequality, Poincaré and Sobolev logarithm inequalities; sharp decay rates for the quantitative Sobolev inequalities and Polya-Szegö inequality.
Summary
Isoperimetric and Sobolev inequalities are the best known examples of geometric-functional inequalities. In recent years the PI and collaborators have obtained new and sharp quantitative versions of these and other important related inequalities. These results have been obtained by the combined use of classical symmetrization methods, new tools coming from mass transportation theory, deep geometric measure tools and ad hoc symmetrizations. The objective of this project is to further develop thes techniques in order to get: sharp quantitative versions of Faber-Krahn inequality, Gaussian isoperimetric inequality, Brunn-Minkowski inequality, Poincaré and Sobolev logarithm inequalities; sharp decay rates for the quantitative Sobolev inequalities and Polya-Szegö inequality.
Max ERC Funding
600 000 €
Duration
Start date: 2009-01-01, End date: 2013-12-31
Project acronym ANTHOS
Project Analytic Number Theory: Higher Order Structures
Researcher (PI) Valentin Blomer
Host Institution (HI) GEORG-AUGUST-UNIVERSITAT GOTTINGENSTIFTUNG OFFENTLICHEN RECHTS
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary This is a proposal for research at the interface of analytic number theory, automorphic forms and algebraic geometry. Motivated by fundamental conjectures in number theory, classical problems will be investigated in higher order situations: general number fields, automorphic forms on higher rank groups, the arithmetic of algebraic varieties of higher degree. In particular, I want to focus on
- computation of moments of L-function of degree 3 and higher with applications to subconvexity and/or non-vanishing, as well as subconvexity for multiple L-functions;
- bounds for sup-norms of cusp forms on various spaces and equidistribution of Hecke correspondences;
- automorphic forms on higher rank groups and general number fields, in particular new bounds towards the Ramanujan conjecture;
- a proof of Manin's conjecture for a certain class of singular algebraic varieties.
The underlying methods are closely related; for example, rational points on algebraic varieties
will be counted by a multiple L-series technique.
Summary
This is a proposal for research at the interface of analytic number theory, automorphic forms and algebraic geometry. Motivated by fundamental conjectures in number theory, classical problems will be investigated in higher order situations: general number fields, automorphic forms on higher rank groups, the arithmetic of algebraic varieties of higher degree. In particular, I want to focus on
- computation of moments of L-function of degree 3 and higher with applications to subconvexity and/or non-vanishing, as well as subconvexity for multiple L-functions;
- bounds for sup-norms of cusp forms on various spaces and equidistribution of Hecke correspondences;
- automorphic forms on higher rank groups and general number fields, in particular new bounds towards the Ramanujan conjecture;
- a proof of Manin's conjecture for a certain class of singular algebraic varieties.
The underlying methods are closely related; for example, rational points on algebraic varieties
will be counted by a multiple L-series technique.
Max ERC Funding
1 004 000 €
Duration
Start date: 2010-10-01, End date: 2015-09-30
Project acronym APGRAPH
Project Asymptotic Graph Properties
Researcher (PI) Deryk Osthus
Host Institution (HI) THE UNIVERSITY OF BIRMINGHAM
Call Details Starting Grant (StG), PE1, ERC-2012-StG_20111012
Summary Many parts of Graph Theory have witnessed a huge growth over the last years, partly because of their relation to Theoretical Computer Science and Statistical Physics. These connections arise because graphs can be used to model many diverse structures.
The focus of this proposal is on asymptotic results, i.e. the graphs under consideration are large. This often unveils patterns and connections which remain obscure when considering only small graphs.
It also allows for the use of powerful techniques such as probabilistic arguments, which have led to spectacular new developments. In particular, my aim is to make decisive progress on central problems in the following 4 areas:
(1) Factorizations: Factorizations of graphs can be viewed as partitions of the edges of a graph into simple regular structures. They have a rich history and arise in many different settings, such as edge-colouring problems, decomposition problems and in information theory. They also have applications to finding good tours for the famous Travelling salesman problem.
(2) Hamilton cycles: A Hamilton cycle is a cycle which contains all the vertices of the graph. One of the most fundamental problems in Graph Theory/Theoretical Computer Science is to find conditions which guarantee the existence of a Hamilton cycle in a graph.
(3) Embeddings of graphs: This is a natural (but difficult) continuation of the previous question where the aim is to embed more general structures than Hamilton cycles - there has been exciting progress here in recent years which has opened up new avenues.
(4) Resilience of graphs: In many cases, it is important to know whether a graph `strongly’ possesses some property, i.e. one cannot destroy the property by changing a few edges. The systematic study of this notion is a new and rapidly growing area.
I have developed new methods for deep and long-standing problems in these areas which will certainly lead to further applications elsewhere.
Summary
Many parts of Graph Theory have witnessed a huge growth over the last years, partly because of their relation to Theoretical Computer Science and Statistical Physics. These connections arise because graphs can be used to model many diverse structures.
The focus of this proposal is on asymptotic results, i.e. the graphs under consideration are large. This often unveils patterns and connections which remain obscure when considering only small graphs.
It also allows for the use of powerful techniques such as probabilistic arguments, which have led to spectacular new developments. In particular, my aim is to make decisive progress on central problems in the following 4 areas:
(1) Factorizations: Factorizations of graphs can be viewed as partitions of the edges of a graph into simple regular structures. They have a rich history and arise in many different settings, such as edge-colouring problems, decomposition problems and in information theory. They also have applications to finding good tours for the famous Travelling salesman problem.
(2) Hamilton cycles: A Hamilton cycle is a cycle which contains all the vertices of the graph. One of the most fundamental problems in Graph Theory/Theoretical Computer Science is to find conditions which guarantee the existence of a Hamilton cycle in a graph.
(3) Embeddings of graphs: This is a natural (but difficult) continuation of the previous question where the aim is to embed more general structures than Hamilton cycles - there has been exciting progress here in recent years which has opened up new avenues.
(4) Resilience of graphs: In many cases, it is important to know whether a graph `strongly’ possesses some property, i.e. one cannot destroy the property by changing a few edges. The systematic study of this notion is a new and rapidly growing area.
I have developed new methods for deep and long-standing problems in these areas which will certainly lead to further applications elsewhere.
Max ERC Funding
818 414 €
Duration
Start date: 2012-12-01, End date: 2018-11-30
Project acronym AQSER
Project Automorphic q-series and their application
Researcher (PI) Kathrin Bringmann
Host Institution (HI) UNIVERSITAET ZU KOELN
Call Details Starting Grant (StG), PE1, ERC-2013-StG
Summary This proposal aims to unravel mysteries at the frontier of number theory and other areas of mathematics and physics. The main focus will be to understand and exploit “modularity” of q-hypergeometric series. “Modular forms are functions on the complex plane that are inordinately symmetric.” (Mazur) The motivation comes from the wide-reaching applications of modularity in combinatorics, percolation, Lie theory, and physics (black holes).
The interplay between automorphic forms, q-series, and other areas of mathematics and physics is often two-sided. On the one hand, the other areas provide interesting examples of automorphic objects and predict their behavior. Sometimes these even motivate new classes of automorphic objects which have not been previously studied. On the other hand, knowing that certain generating functions are modular gives one access to deep theoretical tools to prove results in other areas. “Mathematics is a language, and we need that language to understand the physics of our universe.”(Ooguri) Understanding this interplay has attracted attention of researchers from a variety of areas. However, proofs of modularity of q-hypergeometric series currently fall far short of a comprehensive theory to describe the interplay between them and automorphic forms. A recent conjecture of W. Nahm relates the modularity of such series to K-theory. In this proposal I aim to fill this gap and provide a better understanding of this interplay by building a general structural framework enveloping these q-series. For this I will employ new kinds of automorphic objects and embed the functions of interest into bigger families
A successful outcome of the proposed research will open further horizons and also answer open questions, even those in other areas which were not addressed in this proposal; for example the new theory could be applied to better understand Donaldson invariants.
Summary
This proposal aims to unravel mysteries at the frontier of number theory and other areas of mathematics and physics. The main focus will be to understand and exploit “modularity” of q-hypergeometric series. “Modular forms are functions on the complex plane that are inordinately symmetric.” (Mazur) The motivation comes from the wide-reaching applications of modularity in combinatorics, percolation, Lie theory, and physics (black holes).
The interplay between automorphic forms, q-series, and other areas of mathematics and physics is often two-sided. On the one hand, the other areas provide interesting examples of automorphic objects and predict their behavior. Sometimes these even motivate new classes of automorphic objects which have not been previously studied. On the other hand, knowing that certain generating functions are modular gives one access to deep theoretical tools to prove results in other areas. “Mathematics is a language, and we need that language to understand the physics of our universe.”(Ooguri) Understanding this interplay has attracted attention of researchers from a variety of areas. However, proofs of modularity of q-hypergeometric series currently fall far short of a comprehensive theory to describe the interplay between them and automorphic forms. A recent conjecture of W. Nahm relates the modularity of such series to K-theory. In this proposal I aim to fill this gap and provide a better understanding of this interplay by building a general structural framework enveloping these q-series. For this I will employ new kinds of automorphic objects and embed the functions of interest into bigger families
A successful outcome of the proposed research will open further horizons and also answer open questions, even those in other areas which were not addressed in this proposal; for example the new theory could be applied to better understand Donaldson invariants.
Max ERC Funding
1 240 500 €
Duration
Start date: 2014-01-01, End date: 2019-04-30
Project acronym AQUAMS
Project Analysis of quantum many-body systems
Researcher (PI) Robert Seiringer
Host Institution (HI) INSTITUTE OF SCIENCE AND TECHNOLOGYAUSTRIA
Call Details Advanced Grant (AdG), PE1, ERC-2015-AdG
Summary The main focus of this project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g., Bose–Einstein condensation and superfluidity, which makes them interesting both from a physical and a mathematical point of view.
The goal of this project is the development of new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and thus increase our understanding of physical systems. From the point of view of mathematical physics, there has been significant progress in the last few years in understanding the interesting phenomena occurring in quantum gases, and the goal of this project is to investigate some of the key issues that remain unsolved. Due to the complex nature of the problems, new mathematical ideas
and methods will have to be developed for this purpose. One of the main question addressed in this proposal is the validity of the Bogoliubov approximation for the excitation spectrum of many-body quantum systems. While its accuracy has been
successfully shown for the ground state energy of various models, its predictions concerning the excitation spectrum have so far only been verified in the Hartree limit, an extreme form of a mean-field limit where the interaction among the particles is very weak and ranges over the whole system. The central part of this project is concerned with the extension of these results to the case of short-range interactions. Apart from being mathematically much more challenging, the short-range case is the
one most relevant for the description of actual physical systems. Hence progress along these lines can be expected to yield valuable insight into the complex behavior of these many-body quantum systems.
Summary
The main focus of this project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g., Bose–Einstein condensation and superfluidity, which makes them interesting both from a physical and a mathematical point of view.
The goal of this project is the development of new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and thus increase our understanding of physical systems. From the point of view of mathematical physics, there has been significant progress in the last few years in understanding the interesting phenomena occurring in quantum gases, and the goal of this project is to investigate some of the key issues that remain unsolved. Due to the complex nature of the problems, new mathematical ideas
and methods will have to be developed for this purpose. One of the main question addressed in this proposal is the validity of the Bogoliubov approximation for the excitation spectrum of many-body quantum systems. While its accuracy has been
successfully shown for the ground state energy of various models, its predictions concerning the excitation spectrum have so far only been verified in the Hartree limit, an extreme form of a mean-field limit where the interaction among the particles is very weak and ranges over the whole system. The central part of this project is concerned with the extension of these results to the case of short-range interactions. Apart from being mathematically much more challenging, the short-range case is the
one most relevant for the description of actual physical systems. Hence progress along these lines can be expected to yield valuable insight into the complex behavior of these many-body quantum systems.
Max ERC Funding
1 497 755 €
Duration
Start date: 2016-10-01, End date: 2021-09-30
Project acronym ARIPHYHIMO
Project Arithmetic and physics of Higgs moduli spaces
Researcher (PI) Tamas Hausel
Host Institution (HI) INSTITUTE OF SCIENCE AND TECHNOLOGYAUSTRIA
Call Details Advanced Grant (AdG), PE1, ERC-2012-ADG_20120216
Summary The proposal studies problems concerning the geometry and topology of moduli spaces of Higgs bundles on a Riemann surface motivated by parallel considerations in number theory and mathematical physics. In this way the proposal bridges various duality theories in string theory with the Langlands program in number theory.
The heart of the proposal is a circle of precise conjectures relating to the topology of the moduli space of Higgs bundles. The formulation and motivations of the conjectures make direct contact with the Langlands program in number theory, various duality conjectures in string theory, algebraic combinatorics, knot theory and low dimensional topology and representation theory of quivers, finite groups and algebras of Lie type and Cherednik algebras.
Summary
The proposal studies problems concerning the geometry and topology of moduli spaces of Higgs bundles on a Riemann surface motivated by parallel considerations in number theory and mathematical physics. In this way the proposal bridges various duality theories in string theory with the Langlands program in number theory.
The heart of the proposal is a circle of precise conjectures relating to the topology of the moduli space of Higgs bundles. The formulation and motivations of the conjectures make direct contact with the Langlands program in number theory, various duality conjectures in string theory, algebraic combinatorics, knot theory and low dimensional topology and representation theory of quivers, finite groups and algebras of Lie type and Cherednik algebras.
Max ERC Funding
1 304 945 €
Duration
Start date: 2013-04-01, End date: 2018-08-31
Project acronym ARITHQUANTUMCHAOS
Project Arithmetic and Quantum Chaos
Researcher (PI) Zeev Rudnick
Host Institution (HI) TEL AVIV UNIVERSITY
Call Details Advanced Grant (AdG), PE1, ERC-2012-ADG_20120216
Summary Quantum Chaos is an emerging discipline which is crossing over from Physics into Pure Mathematics. The recent crossover is driven in part by a connection with Number Theory. This project explores several aspects of this interrelationship and is composed of a number of sub-projects. The sub-projects deal with: statistics of energy levels and wave functions of pseudo-integrable systems, a hitherto unexplored subject in the mathematical community which is not well understood in the physics community; with statistics of zeros of zeta functions over function fields, a purely number theoretic topic which is linked to the subproject on Quantum Chaos through the mysterious connections to Random Matrix Theory and an analogy between energy levels and zeta zeros; and with spatial statistics in arithmetic.
Summary
Quantum Chaos is an emerging discipline which is crossing over from Physics into Pure Mathematics. The recent crossover is driven in part by a connection with Number Theory. This project explores several aspects of this interrelationship and is composed of a number of sub-projects. The sub-projects deal with: statistics of energy levels and wave functions of pseudo-integrable systems, a hitherto unexplored subject in the mathematical community which is not well understood in the physics community; with statistics of zeros of zeta functions over function fields, a purely number theoretic topic which is linked to the subproject on Quantum Chaos through the mysterious connections to Random Matrix Theory and an analogy between energy levels and zeta zeros; and with spatial statistics in arithmetic.
Max ERC Funding
1 714 000 €
Duration
Start date: 2013-02-01, End date: 2019-01-31
Project acronym AROMA-CFD
Project Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics
Researcher (PI) Gianluigi Rozza
Host Institution (HI) SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE
Call Details Consolidator Grant (CoG), PE1, ERC-2015-CoG
Summary The aim of AROMA-CFD is to create a team of scientists at SISSA for the development of Advanced Reduced Order Modelling techniques with a focus in Computational Fluid Dynamics (CFD), in order to face and overcome many current limitations of the state of the art and improve the capabilities of reduced order methodologies for more demanding applications in industrial, medical and applied sciences contexts. AROMA-CFD deals with strong methodological developments in numerical analysis, with a special emphasis on mathematical modelling and extensive exploitation of computational science and engineering. Several tasks have been identified to tackle important problems and open questions in reduced order modelling: study of bifurcations and instabilities in flows, increasing Reynolds number and guaranteeing stability, moving towards turbulent flows, considering complex geometrical parametrizations of shapes as computational domains into extended networks. A reduced computational and geometrical framework will be developed for nonlinear inverse problems, focusing on optimal flow control, shape optimization and uncertainty quantification. Further, all the advanced developments in reduced order modelling for CFD will be delivered for applications in multiphysics, such as fluid-structure interaction problems and general coupled phenomena involving inviscid, viscous and thermal flows, solids and porous media. The advanced developed framework within AROMA-CFD will provide attractive capabilities for several industrial and medical applications (e.g. aeronautical, mechanical, naval, off-shore, wind, sport, biomedical engineering, and cardiovascular surgery as well), combining high performance computing (in dedicated supercomputing centers) and advanced reduced order modelling (in common devices) to guarantee real time computing and visualization. A new open source software library for AROMA-CFD will be created: ITHACA, In real Time Highly Advanced Computational Applications.
Summary
The aim of AROMA-CFD is to create a team of scientists at SISSA for the development of Advanced Reduced Order Modelling techniques with a focus in Computational Fluid Dynamics (CFD), in order to face and overcome many current limitations of the state of the art and improve the capabilities of reduced order methodologies for more demanding applications in industrial, medical and applied sciences contexts. AROMA-CFD deals with strong methodological developments in numerical analysis, with a special emphasis on mathematical modelling and extensive exploitation of computational science and engineering. Several tasks have been identified to tackle important problems and open questions in reduced order modelling: study of bifurcations and instabilities in flows, increasing Reynolds number and guaranteeing stability, moving towards turbulent flows, considering complex geometrical parametrizations of shapes as computational domains into extended networks. A reduced computational and geometrical framework will be developed for nonlinear inverse problems, focusing on optimal flow control, shape optimization and uncertainty quantification. Further, all the advanced developments in reduced order modelling for CFD will be delivered for applications in multiphysics, such as fluid-structure interaction problems and general coupled phenomena involving inviscid, viscous and thermal flows, solids and porous media. The advanced developed framework within AROMA-CFD will provide attractive capabilities for several industrial and medical applications (e.g. aeronautical, mechanical, naval, off-shore, wind, sport, biomedical engineering, and cardiovascular surgery as well), combining high performance computing (in dedicated supercomputing centers) and advanced reduced order modelling (in common devices) to guarantee real time computing and visualization. A new open source software library for AROMA-CFD will be created: ITHACA, In real Time Highly Advanced Computational Applications.
Max ERC Funding
1 656 579 €
Duration
Start date: 2016-05-01, End date: 2021-04-30
Project acronym BeyondA1
Project Set theory beyond the first uncountable cardinal
Researcher (PI) Assaf Shmuel Rinot
Host Institution (HI) BAR ILAN UNIVERSITY
Call Details Starting Grant (StG), PE1, ERC-2018-STG
Summary We propose to establish a research group that will unveil the combinatorial nature of the second uncountable cardinal. This includes its Ramsey-theoretic, order-theoretic, graph-theoretic and topological features. Among others, we will be directly addressing fundamental problems due to Erdos, Rado, Galvin, and Shelah.
While some of these problems are old and well-known, an unexpected series of breakthroughs from the last three years suggest that now is a promising point in time to carry out such a project. Indeed, through a short period, four previously unattainable problems concerning the second uncountable cardinal were successfully tackled: Aspero on a club-guessing problem of Shelah, Krueger on the club-isomorphism problem for Aronszajn trees, Neeman on the isomorphism problem for dense sets of reals, and the PI on the Souslin problem. Each of these results was obtained through the development of a completely new technical framework, and these frameworks could now pave the way for the solution of some major open questions.
A goal of the highest risk in this project is the discovery of a consistent (possibly, parameterized) forcing axiom that will (preferably, simultaneously) provide structure theorems for stationary sets, linearly ordered sets, trees, graphs, and partition relations, as well as the refutation of various forms of club-guessing principles, all at the level of the second uncountable cardinal. In comparison, at the level of the first uncountable cardinal, a forcing axiom due to Foreman, Magidor and Shelah achieves exactly that.
To approach our goals, the proposed project is divided into four core areas: Uncountable trees, Ramsey theory on ordinals, Club-guessing principles, and Forcing Axioms. There is a rich bilateral interaction between any pair of the four different cores, but the proposed division will allow an efficient allocation of manpower, and will increase the chances of parallel success.
Summary
We propose to establish a research group that will unveil the combinatorial nature of the second uncountable cardinal. This includes its Ramsey-theoretic, order-theoretic, graph-theoretic and topological features. Among others, we will be directly addressing fundamental problems due to Erdos, Rado, Galvin, and Shelah.
While some of these problems are old and well-known, an unexpected series of breakthroughs from the last three years suggest that now is a promising point in time to carry out such a project. Indeed, through a short period, four previously unattainable problems concerning the second uncountable cardinal were successfully tackled: Aspero on a club-guessing problem of Shelah, Krueger on the club-isomorphism problem for Aronszajn trees, Neeman on the isomorphism problem for dense sets of reals, and the PI on the Souslin problem. Each of these results was obtained through the development of a completely new technical framework, and these frameworks could now pave the way for the solution of some major open questions.
A goal of the highest risk in this project is the discovery of a consistent (possibly, parameterized) forcing axiom that will (preferably, simultaneously) provide structure theorems for stationary sets, linearly ordered sets, trees, graphs, and partition relations, as well as the refutation of various forms of club-guessing principles, all at the level of the second uncountable cardinal. In comparison, at the level of the first uncountable cardinal, a forcing axiom due to Foreman, Magidor and Shelah achieves exactly that.
To approach our goals, the proposed project is divided into four core areas: Uncountable trees, Ramsey theory on ordinals, Club-guessing principles, and Forcing Axioms. There is a rich bilateral interaction between any pair of the four different cores, but the proposed division will allow an efficient allocation of manpower, and will increase the chances of parallel success.
Max ERC Funding
1 362 500 €
Duration
Start date: 2018-10-01, End date: 2023-09-30
Project acronym BG-BB-AS
Project Birational Geometry, B-branes and Artin Stacks
Researcher (PI) Edward Paul Segal
Host Institution (HI) UNIVERSITY COLLEGE LONDON
Call Details Consolidator Grant (CoG), PE1, ERC-2016-COG
Summary Derived categories of coherent sheaves on a variety are a fundamental tool in algebraic geometry. They also arise in String Theory, as the category of B-branes in a quantum field theory whose target space is the variety. This connection to physics has been extraordinarily fruitful, providing deep insights and conjectures.
An Artin stack is a sophisticated generalization of a variety, they encode the idea of equivariant geometry. A simple example is a vector space carrying a linear action of a Lie group. In String Theory this data defines a Gauged Linear Sigma Model, which is a basic tool in the subject. A GLSM should also give rise to a category of B-branes, but surprisingly it is not yet understood what this should be. An overarching goal of this project is to develop an understanding of this category (more accurately, system of categories), and to extend this understanding to more general Artin stacks.
The basic importance of this question is that in certain limits a GLSM reduces to a sigma model, whose target is a quotient of the vector space by the group. This quotient must be taken using Geometric Invariant Theory. Thus this project is intimately connected with the question of how derived categories change under variation-of-GIT, and birational maps in general.
For GLSMs with abelian groups this approach has already produced spectacular results, in the non-abelian case we understand only a few remarkable examples. We will develop these examples into a wide-ranging general theory.
Our key objectives are to:
- Provide powerful new tools for controlling the behaviour of derived categories under birational maps.
- Understand the category of B-branes on a large class of Artin stacks.
- Prove and apply a striking new duality between GLSMs.
- Construct completely new symmetries of derived categories.
Summary
Derived categories of coherent sheaves on a variety are a fundamental tool in algebraic geometry. They also arise in String Theory, as the category of B-branes in a quantum field theory whose target space is the variety. This connection to physics has been extraordinarily fruitful, providing deep insights and conjectures.
An Artin stack is a sophisticated generalization of a variety, they encode the idea of equivariant geometry. A simple example is a vector space carrying a linear action of a Lie group. In String Theory this data defines a Gauged Linear Sigma Model, which is a basic tool in the subject. A GLSM should also give rise to a category of B-branes, but surprisingly it is not yet understood what this should be. An overarching goal of this project is to develop an understanding of this category (more accurately, system of categories), and to extend this understanding to more general Artin stacks.
The basic importance of this question is that in certain limits a GLSM reduces to a sigma model, whose target is a quotient of the vector space by the group. This quotient must be taken using Geometric Invariant Theory. Thus this project is intimately connected with the question of how derived categories change under variation-of-GIT, and birational maps in general.
For GLSMs with abelian groups this approach has already produced spectacular results, in the non-abelian case we understand only a few remarkable examples. We will develop these examples into a wide-ranging general theory.
Our key objectives are to:
- Provide powerful new tools for controlling the behaviour of derived categories under birational maps.
- Understand the category of B-branes on a large class of Artin stacks.
- Prove and apply a striking new duality between GLSMs.
- Construct completely new symmetries of derived categories.
Max ERC Funding
1 358 925 €
Duration
Start date: 2017-09-01, End date: 2022-08-31
Project acronym BHSandAADS
Project The Black Hole Stability Problem and the Analysis of asymptotically anti-de Sitter spacetimes
Researcher (PI) Gustav HOLZEGEL
Host Institution (HI) IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
Call Details Consolidator Grant (CoG), PE1, ERC-2017-COG
Summary The present proposal is concerned with the analysis of the Einstein equations of general relativity, a non-linear system of geometric partial differential equations describing phenomena from the bending of light to the dynamics of black holes. The theory has recently been confirmed in a spectacular fashion with the detection of gravitational waves.
The main objective of the proposal is to consolidate my research group based at Imperial College by developing novel mathematical techniques that will fundamentally advance our understanding of the Einstein equations. Here the proposal builds on mathematical progress in the last decade resulting from achievements in the fields of partial differential equations, differential geometry, microlocal analysis and theoretical physics.
The Black Hole Stability Problem
A major open problem in general relativity is to prove the non-linear stability of the Kerr family of black hole solutions. Recent advances in the problem of linear stability made by the PI and collaborators open the door to finally address a complete resolution of the stability problem. In this proposal we will describe what non-linear techniques will need to be developed in addition to achieve this goal. A successful resolution of this program would conclude an almost 50-year-old problem.
The Analysis of asymptotically anti-de Sitter (aAdS) spacetimes
We propose to prove the stability of pure AdS if so-called dissipative boundary conditions are imposed at the boundary. This result would align with the well-known stability results for the other maximally-symmetric solutions of the Einstein equations, Minkowski space and de Sitter space.
As a second -- related -- theme we propose to formulate and prove a unique continuation principle for the full non-linear Einstein equations on aAdS spacetimes. This goal will be achieved by advancing techniques that have recently been developed by the PI and collaborators for non-linear wave equations on aAdS spacetimes.
Summary
The present proposal is concerned with the analysis of the Einstein equations of general relativity, a non-linear system of geometric partial differential equations describing phenomena from the bending of light to the dynamics of black holes. The theory has recently been confirmed in a spectacular fashion with the detection of gravitational waves.
The main objective of the proposal is to consolidate my research group based at Imperial College by developing novel mathematical techniques that will fundamentally advance our understanding of the Einstein equations. Here the proposal builds on mathematical progress in the last decade resulting from achievements in the fields of partial differential equations, differential geometry, microlocal analysis and theoretical physics.
The Black Hole Stability Problem
A major open problem in general relativity is to prove the non-linear stability of the Kerr family of black hole solutions. Recent advances in the problem of linear stability made by the PI and collaborators open the door to finally address a complete resolution of the stability problem. In this proposal we will describe what non-linear techniques will need to be developed in addition to achieve this goal. A successful resolution of this program would conclude an almost 50-year-old problem.
The Analysis of asymptotically anti-de Sitter (aAdS) spacetimes
We propose to prove the stability of pure AdS if so-called dissipative boundary conditions are imposed at the boundary. This result would align with the well-known stability results for the other maximally-symmetric solutions of the Einstein equations, Minkowski space and de Sitter space.
As a second -- related -- theme we propose to formulate and prove a unique continuation principle for the full non-linear Einstein equations on aAdS spacetimes. This goal will be achieved by advancing techniques that have recently been developed by the PI and collaborators for non-linear wave equations on aAdS spacetimes.
Max ERC Funding
1 999 755 €
Duration
Start date: 2018-11-01, End date: 2023-10-31
Project acronym BIOSMA
Project Mathematics for Shape Memory Technologies in Biomechanics
Researcher (PI) Ulisse Stefanelli
Host Institution (HI) CONSIGLIO NAZIONALE DELLE RICERCHE
Call Details Starting Grant (StG), PE1, ERC-2007-StG
Summary Shape Memory Alloys (SMAs) are nowadays widely exploited for the realization of innovative devices and have a great impact on the development of a variety of biomedical applications ranging from orthodontic archwires to vascular stents. The design, realization, and optimization of such devices are quite demanding tasks. Mathematics is involved in this process as a major tool in order to let the modeling more accurate, the numerical simulations more reliable, and the design more effective. Many material properties of SMAs such as martensitic reorientation, training, and ferromagnetic behavior, are still to be properly and efficiently addressed. Therefore, new modeling ideas, along with original analytical and numerical techniques, are required. This project is aimed at addressing novel mathematical issues in order to move from experimental materials results toward the solution of real-scale biomechanical Engineering problems. The research focus will be multidisciplinary and include modeling, analytic, numerical, and computational issues. A progress in the macroscopic description of SMAs, the computational simulation of real-scale SMA devices, and the optimization of the production processes will contribute to advance in the direction of innovative applications.
Summary
Shape Memory Alloys (SMAs) are nowadays widely exploited for the realization of innovative devices and have a great impact on the development of a variety of biomedical applications ranging from orthodontic archwires to vascular stents. The design, realization, and optimization of such devices are quite demanding tasks. Mathematics is involved in this process as a major tool in order to let the modeling more accurate, the numerical simulations more reliable, and the design more effective. Many material properties of SMAs such as martensitic reorientation, training, and ferromagnetic behavior, are still to be properly and efficiently addressed. Therefore, new modeling ideas, along with original analytical and numerical techniques, are required. This project is aimed at addressing novel mathematical issues in order to move from experimental materials results toward the solution of real-scale biomechanical Engineering problems. The research focus will be multidisciplinary and include modeling, analytic, numerical, and computational issues. A progress in the macroscopic description of SMAs, the computational simulation of real-scale SMA devices, and the optimization of the production processes will contribute to advance in the direction of innovative applications.
Max ERC Funding
700 000 €
Duration
Start date: 2008-09-01, End date: 2013-08-31
Project acronym BIOSTRUCT
Project Multiscale mathematical modelling of dynamics of structure formation in cell systems
Researcher (PI) Anna Marciniak-Czochra
Host Institution (HI) RUPRECHT-KARLS-UNIVERSITAET HEIDELBERG
Call Details Starting Grant (StG), PE1, ERC-2007-StG
Summary The aim of this transdisciplinary project is to develop and analyse multiscale mathematical models of pattern formation in multicellular systems controlled by the dynamics of intracellular signalling pathways and cell-to-cell communication and to develop new mathematical methods for the modelling of such complex processes. This aim will be achieved through a close collaboration with experimental groups and comprehensive analytical investigations of the mathematical problems arising in the modelling of these biological processes. The mathematical methods and techniques to be employed will be the analysis of systems of partial differential equations, asymptotic analysis, as well as methods of dynamical systems. These techniques will be used to formulate the models and to study the spatio-temporal behaviour of solutions, especially stability and dependence on characteristic scales, geometry, initial data and key parameters. Advanced numerical methods will be applied to simulate the models. This comprehensive methodology goes beyond the state-of-the-art, since usually the analyses are limited to a single aspect of model behaviour. Groundbreaking impacts envisioned are threefold: (i) The project will contribute to the understanding of mechanisms of structure formation in the developmental process, in the context of recently discovered signalling pathways. In addition, some of the factors and mechanisms playing a role in developmental processes, such as Wnt signalling, are implicated in carcinogenesis, for instance colon and lung cancer. (ii) Accurate quantitative and predictive mathematical models of cell proliferation and differentiation are important for the control of tumour growth and tissue egeneration; (iii) Qualitative analysis of multiscale mathematical models of biological phenomena generates challenging mathematical problems and, therefore, the project will lead to the development of new mathematical theories and tools.
Summary
The aim of this transdisciplinary project is to develop and analyse multiscale mathematical models of pattern formation in multicellular systems controlled by the dynamics of intracellular signalling pathways and cell-to-cell communication and to develop new mathematical methods for the modelling of such complex processes. This aim will be achieved through a close collaboration with experimental groups and comprehensive analytical investigations of the mathematical problems arising in the modelling of these biological processes. The mathematical methods and techniques to be employed will be the analysis of systems of partial differential equations, asymptotic analysis, as well as methods of dynamical systems. These techniques will be used to formulate the models and to study the spatio-temporal behaviour of solutions, especially stability and dependence on characteristic scales, geometry, initial data and key parameters. Advanced numerical methods will be applied to simulate the models. This comprehensive methodology goes beyond the state-of-the-art, since usually the analyses are limited to a single aspect of model behaviour. Groundbreaking impacts envisioned are threefold: (i) The project will contribute to the understanding of mechanisms of structure formation in the developmental process, in the context of recently discovered signalling pathways. In addition, some of the factors and mechanisms playing a role in developmental processes, such as Wnt signalling, are implicated in carcinogenesis, for instance colon and lung cancer. (ii) Accurate quantitative and predictive mathematical models of cell proliferation and differentiation are important for the control of tumour growth and tissue egeneration; (iii) Qualitative analysis of multiscale mathematical models of biological phenomena generates challenging mathematical problems and, therefore, the project will lead to the development of new mathematical theories and tools.
Max ERC Funding
750 000 €
Duration
Start date: 2008-09-01, End date: 2013-08-31
Project acronym BirNonArchGeom
Project Birational and non-archimedean geometries
Researcher (PI) Michael TEMKIN
Host Institution (HI) THE HEBREW UNIVERSITY OF JERUSALEM
Call Details Consolidator Grant (CoG), PE1, ERC-2017-COG
Summary Resolution of singularities is one of classical, central and difficult areas of algebraic geometry, with a centennial history of intensive research and contributions of such great names as Zariski, Hironaka and Abhyankar. Nowadays, desingularization of schemes of characteristic zero is very well understood, while semistable reduction of morphisms and desingularization in positive characteristic are still waiting for major breakthroughs. In addition to the classical techniques with their triumph in characteristic zero, modern resolution of singularities includes de Jong's method of alterations, toroidal methods, formal analytic and non-archimedean methods, etc.
The aim of the proposed research is to study nearly all directions in resolution of singularities and semistable reduction, as well as the wild ramification phenomena, which are probably the main obstacle to transfer methods from characteristic zero to positive characteristic.
The methods of algebraic and non-archimedean geometries are intertwined in the proposal, though algebraic geometry is somewhat dominating, especially due to the new stack-theoretic techniques. It seems very probable that increasing the symbiosis between birational and non-archimedean geometries will be one of by-products of this research.
Summary
Resolution of singularities is one of classical, central and difficult areas of algebraic geometry, with a centennial history of intensive research and contributions of such great names as Zariski, Hironaka and Abhyankar. Nowadays, desingularization of schemes of characteristic zero is very well understood, while semistable reduction of morphisms and desingularization in positive characteristic are still waiting for major breakthroughs. In addition to the classical techniques with their triumph in characteristic zero, modern resolution of singularities includes de Jong's method of alterations, toroidal methods, formal analytic and non-archimedean methods, etc.
The aim of the proposed research is to study nearly all directions in resolution of singularities and semistable reduction, as well as the wild ramification phenomena, which are probably the main obstacle to transfer methods from characteristic zero to positive characteristic.
The methods of algebraic and non-archimedean geometries are intertwined in the proposal, though algebraic geometry is somewhat dominating, especially due to the new stack-theoretic techniques. It seems very probable that increasing the symbiosis between birational and non-archimedean geometries will be one of by-products of this research.
Max ERC Funding
1 365 600 €
Duration
Start date: 2018-05-01, End date: 2023-04-30
Project acronym BLOC
Project Mathematical study of Boundary Layers in Oceanic Motions
Researcher (PI) Anne-Laure Perrine Dalibard
Host Institution (HI) SORBONNE UNIVERSITE
Call Details Starting Grant (StG), PE1, ERC-2014-STG
Summary Boundary layer theory is a large component of fluid dynamics. It is ubiquitous in Oceanography, where boundary layer currents, such as the Gulf Stream, play an important role in the global circulation. Comprehending the underlying mechanisms in the formation of boundary layers is therefore crucial for applications. However, the treatment of boundary layers in ocean dynamics remains poorly understood at a theoretical level, due to the variety and complexity of the forces at stake.
The goal of this project is to develop several tools to bridge the gap between the mathematical state of the art and the physical reality of oceanic motion. There are four points on which we will mainly focus: degeneracy issues, including the treatment Stewartson boundary layers near the equator; rough boundaries (meaning boundaries with small amplitude and high frequency variations); the inclusion of the advection term in the construction of stationary boundary layers; and the linear and nonlinear stability of the boundary layers. We will address separately Ekman layers and western boundary layers, since they are ruled by equations whose mathematical behaviour is very different.
This project will allow us to have a better understanding of small scale phenomena in fluid mechanics, and in particular of the inviscid limit of incompressible fluids.
The team will be composed of the PI, two PhD students and three two-year postdocs over the whole period. We will also rely on the historical expertise of the host institution on fluid mechanics and asymptotic methods.
Summary
Boundary layer theory is a large component of fluid dynamics. It is ubiquitous in Oceanography, where boundary layer currents, such as the Gulf Stream, play an important role in the global circulation. Comprehending the underlying mechanisms in the formation of boundary layers is therefore crucial for applications. However, the treatment of boundary layers in ocean dynamics remains poorly understood at a theoretical level, due to the variety and complexity of the forces at stake.
The goal of this project is to develop several tools to bridge the gap between the mathematical state of the art and the physical reality of oceanic motion. There are four points on which we will mainly focus: degeneracy issues, including the treatment Stewartson boundary layers near the equator; rough boundaries (meaning boundaries with small amplitude and high frequency variations); the inclusion of the advection term in the construction of stationary boundary layers; and the linear and nonlinear stability of the boundary layers. We will address separately Ekman layers and western boundary layers, since they are ruled by equations whose mathematical behaviour is very different.
This project will allow us to have a better understanding of small scale phenomena in fluid mechanics, and in particular of the inviscid limit of incompressible fluids.
The team will be composed of the PI, two PhD students and three two-year postdocs over the whole period. We will also rely on the historical expertise of the host institution on fluid mechanics and asymptotic methods.
Max ERC Funding
1 267 500 €
Duration
Start date: 2015-09-01, End date: 2020-08-31
Project acronym BLOWDISOL
Project "BLOW UP, DISPERSION AND SOLITONS"
Researcher (PI) Franck Merle
Host Institution (HI) UNIVERSITE DE CERGY-PONTOISE
Call Details Advanced Grant (AdG), PE1, ERC-2011-ADG_20110209
Summary "Many physical models involve nonlinear dispersive problems, like wave
or laser propagation, plasmas, ferromagnetism, etc. So far, the mathematical under-
standing of these equations is rather poor. In particular, we know little about the
detailed qualitative behavior of their solutions. Our point is that an apparent com-
plexity hides universal properties of these models; investigating and uncovering such
properties has started only recently. More than the equations themselves, these univer-
sal properties are essential for physical modelisation.
By considering several standard models such as the nonlinear Schrodinger, nonlinear
wave, generalized KdV equations and related geometric problems, the goal of this pro-
posal is to describe the generic global behavior of the solutions and the profiles which
emerge either for large time or by concentration due to strong nonlinear effects, if pos-
sible through a few relevant solutions (sometimes explicit solutions, like solitons). In
order to do this, we have to elaborate different mathematical tools depending on the
context and the specificity of the problems. Particular emphasis will be placed on
- large time asymptotics for global solutions, decomposition of generic solutions into
sums of decoupled solitons in non integrable situations,
- description of critical phenomenon for blow up in the Hamiltonian situation, stable
or generic behavior for blow up on critical dynamics, various relevant regularisations of
the problem,
- global existence for defocusing supercritical problems and blow up dynamics in the
focusing cases.
We believe that the PI and his team have the ability to tackle these problems at present.
The proposal will open whole fields of investigation in Partial Differential Equations in
the future, clarify and simplify our knowledge on the dynamical behavior of solutions
of these problems and provide Physicists some new insight on these models."
Summary
"Many physical models involve nonlinear dispersive problems, like wave
or laser propagation, plasmas, ferromagnetism, etc. So far, the mathematical under-
standing of these equations is rather poor. In particular, we know little about the
detailed qualitative behavior of their solutions. Our point is that an apparent com-
plexity hides universal properties of these models; investigating and uncovering such
properties has started only recently. More than the equations themselves, these univer-
sal properties are essential for physical modelisation.
By considering several standard models such as the nonlinear Schrodinger, nonlinear
wave, generalized KdV equations and related geometric problems, the goal of this pro-
posal is to describe the generic global behavior of the solutions and the profiles which
emerge either for large time or by concentration due to strong nonlinear effects, if pos-
sible through a few relevant solutions (sometimes explicit solutions, like solitons). In
order to do this, we have to elaborate different mathematical tools depending on the
context and the specificity of the problems. Particular emphasis will be placed on
- large time asymptotics for global solutions, decomposition of generic solutions into
sums of decoupled solitons in non integrable situations,
- description of critical phenomenon for blow up in the Hamiltonian situation, stable
or generic behavior for blow up on critical dynamics, various relevant regularisations of
the problem,
- global existence for defocusing supercritical problems and blow up dynamics in the
focusing cases.
We believe that the PI and his team have the ability to tackle these problems at present.
The proposal will open whole fields of investigation in Partial Differential Equations in
the future, clarify and simplify our knowledge on the dynamical behavior of solutions
of these problems and provide Physicists some new insight on these models."
Max ERC Funding
2 079 798 €
Duration
Start date: 2012-04-01, End date: 2017-03-31
Project acronym BOPNIE
Project Boundary value problems for nonlinear integrable equations
Researcher (PI) Jonatan Carl Anders Lenells
Host Institution (HI) KUNGLIGA TEKNISKA HOEGSKOLAN
Call Details Consolidator Grant (CoG), PE1, ERC-2015-CoG
Summary The purpose of this project is to develop new methods for solving boundary value problems (BVPs) for nonlinear integrable partial differential equations (PDEs). Integrable PDEs can be analyzed by means of the Inverse Scattering Transform, whose introduction was one of the most important developments in the theory of nonlinear PDEs in the 20th century. Until the 1990s the inverse scattering methodology was pursued almost entirely for pure initial-value problems. However, in many laboratory and field situations, the solution is generated by what corresponds to the imposition of boundary conditions rather than initial conditions. Thus, an understanding of BVPs is crucial.
In an exciting sequence of events taking place in the last two decades, new tools have become available to deal with BVPs for integrable PDEs. Although some important issues have already been resolved, several major problems remain open.
The aim of this project is to solve a number of these open problems and to find solutions of BVPs which were heretofore not solvable. More precisely, the proposal has eight objectives:
1. Develop methods for solving problems with time-periodic boundary conditions.
2. Answer some long-standing open questions raised by series of wave-tank experiments 35 years ago.
3. Develop a new approach for the study of space-periodic solutions.
4. Develop new approaches for the analysis of BVPs for equations with 3 x 3-matrix Lax pairs.
5. Derive new asymptotic formulas by using a nonlinear version of the steepest descent method.
6. Construct disk and disk/black-hole solutions of the stationary axisymmetric Einstein equations.
7. Solve a BVP in Einstein's theory of relativity describing two colliding gravitational waves.
8. Extend the above methods to BVPs in higher dimensions.
Summary
The purpose of this project is to develop new methods for solving boundary value problems (BVPs) for nonlinear integrable partial differential equations (PDEs). Integrable PDEs can be analyzed by means of the Inverse Scattering Transform, whose introduction was one of the most important developments in the theory of nonlinear PDEs in the 20th century. Until the 1990s the inverse scattering methodology was pursued almost entirely for pure initial-value problems. However, in many laboratory and field situations, the solution is generated by what corresponds to the imposition of boundary conditions rather than initial conditions. Thus, an understanding of BVPs is crucial.
In an exciting sequence of events taking place in the last two decades, new tools have become available to deal with BVPs for integrable PDEs. Although some important issues have already been resolved, several major problems remain open.
The aim of this project is to solve a number of these open problems and to find solutions of BVPs which were heretofore not solvable. More precisely, the proposal has eight objectives:
1. Develop methods for solving problems with time-periodic boundary conditions.
2. Answer some long-standing open questions raised by series of wave-tank experiments 35 years ago.
3. Develop a new approach for the study of space-periodic solutions.
4. Develop new approaches for the analysis of BVPs for equations with 3 x 3-matrix Lax pairs.
5. Derive new asymptotic formulas by using a nonlinear version of the steepest descent method.
6. Construct disk and disk/black-hole solutions of the stationary axisymmetric Einstein equations.
7. Solve a BVP in Einstein's theory of relativity describing two colliding gravitational waves.
8. Extend the above methods to BVPs in higher dimensions.
Max ERC Funding
2 000 000 €
Duration
Start date: 2016-05-01, End date: 2021-04-30
Project acronym BREAD
Project Breaking the curse of dimensionality: numerical challenges in high dimensional analysis and simulation
Researcher (PI) Albert Cohen
Host Institution (HI) UNIVERSITE PIERRE ET MARIE CURIE - PARIS 6
Call Details Advanced Grant (AdG), PE1, ERC-2013-ADG
Summary "This project is concerned with problems that involve a very large number of variables, and whose efficient numerical treatment is challenged by the so-called curse of dimensionality, meaning that computational complexity increases exponentially in the variable dimension.
The PI intend to establish in his host institution a scientific leadership on the mathematical understanding and numerical treatment of these problems, and to contribute to the development of this area of research through international collaborations, organization of workshops and research schools, and training of postdocs and PhD students.
High dimensional problems are ubiquitous in an increasing number of areas of scientific computing, among which statistical or active learning theory, parametric and stochastic partial differential equations, parameter optimization in numerical codes. There is a high demand from the industrial world of efficient numerical methods for treating such problems.
The practical success of various numerical algorithms, that have been developed in recent years in these application areas, is often limited to moderate dimensional setting.
In addition, these developments tend to be, as a rule, rather problem specific and not always founded on a solid mathematical analysis.
The central scientific objectives of this project are therefore: (i) to identify fundamental mathematical principles behind overcoming the curse of dimensionality, (ii) to understand how these principles enter in relevant instances of the above applications, and (iii) based on the these principles beyond particular problem classes, to develop broadly applicable numerical strategies that benefit from such mechanisms.
The performances of these strategies should be provably independent of the variable dimension, and in that sense break the curse of dimensionality. They will be tested on both synthetic benchmark tests and real world problems coming from the afore-mentioned applications."
Summary
"This project is concerned with problems that involve a very large number of variables, and whose efficient numerical treatment is challenged by the so-called curse of dimensionality, meaning that computational complexity increases exponentially in the variable dimension.
The PI intend to establish in his host institution a scientific leadership on the mathematical understanding and numerical treatment of these problems, and to contribute to the development of this area of research through international collaborations, organization of workshops and research schools, and training of postdocs and PhD students.
High dimensional problems are ubiquitous in an increasing number of areas of scientific computing, among which statistical or active learning theory, parametric and stochastic partial differential equations, parameter optimization in numerical codes. There is a high demand from the industrial world of efficient numerical methods for treating such problems.
The practical success of various numerical algorithms, that have been developed in recent years in these application areas, is often limited to moderate dimensional setting.
In addition, these developments tend to be, as a rule, rather problem specific and not always founded on a solid mathematical analysis.
The central scientific objectives of this project are therefore: (i) to identify fundamental mathematical principles behind overcoming the curse of dimensionality, (ii) to understand how these principles enter in relevant instances of the above applications, and (iii) based on the these principles beyond particular problem classes, to develop broadly applicable numerical strategies that benefit from such mechanisms.
The performances of these strategies should be provably independent of the variable dimension, and in that sense break the curse of dimensionality. They will be tested on both synthetic benchmark tests and real world problems coming from the afore-mentioned applications."
Max ERC Funding
1 848 000 €
Duration
Start date: 2014-01-01, End date: 2018-12-31
Project acronym BRIDGES
Project Bridging Non-Equilibrium Problems: From the Fourier Law to Gene Expression
Researcher (PI) Jean-Pierre Eckmann
Host Institution (HI) UNIVERSITE DE GENEVE
Call Details Advanced Grant (AdG), PE1, ERC-2011-ADG_20110209
Summary My goal is to study several important open mathematical problems in non-equilibrium (NEQ) systems and to build a bridge between these problems and NEQ aspects of soft sciences, in particular biological questions. Traffic on this bridge is going to be two-way, the mathematics carrying a long history as a language of science towards the soft sciences, and the soft sciences fruitfully asking new questions and building new paradigms for mathematical research.
Out-of-equilibrium systems pose several fascinating problems: The Fourier law which says that resistance of a wire is proportional to its length is still presenting hard problems for research, and even the existence and the convergence to a NEQ steady state are continuously posing new puzzles, as do questions of smoothness and correlations of such states. These will be addressed with stochastic differential equations, and with particlescatterer systems, both canonical and grand-canonical. The latter are extensions of the well-known Lorentz gas and the study of hyperbolic billiards.
Another field where NEQ plays an important role is the study of glassy systems. They were studied with molecular dynamics (MD) but I have used a topological variant, which mimics astonishingly well what happens in MD simulations. The aim is to extend this to 3 dimensions, where new problems appear.
Finally, I will apply the NEQ studies to biological systems: How a system copes with the varying environment,adapting in this way to a novel type of NEQ. I will study networks of communication among neurons,which are like random graphs with the additional property of being embedded, and the arrangement of genes on chromosomes in such a way as to optimize the adaptation to the different cell types which must be produced using the same genetic information.
I will answer such questions with students and collaborators, who will specialize in the subprojects but will interact with my help across the common bridge.
Summary
My goal is to study several important open mathematical problems in non-equilibrium (NEQ) systems and to build a bridge between these problems and NEQ aspects of soft sciences, in particular biological questions. Traffic on this bridge is going to be two-way, the mathematics carrying a long history as a language of science towards the soft sciences, and the soft sciences fruitfully asking new questions and building new paradigms for mathematical research.
Out-of-equilibrium systems pose several fascinating problems: The Fourier law which says that resistance of a wire is proportional to its length is still presenting hard problems for research, and even the existence and the convergence to a NEQ steady state are continuously posing new puzzles, as do questions of smoothness and correlations of such states. These will be addressed with stochastic differential equations, and with particlescatterer systems, both canonical and grand-canonical. The latter are extensions of the well-known Lorentz gas and the study of hyperbolic billiards.
Another field where NEQ plays an important role is the study of glassy systems. They were studied with molecular dynamics (MD) but I have used a topological variant, which mimics astonishingly well what happens in MD simulations. The aim is to extend this to 3 dimensions, where new problems appear.
Finally, I will apply the NEQ studies to biological systems: How a system copes with the varying environment,adapting in this way to a novel type of NEQ. I will study networks of communication among neurons,which are like random graphs with the additional property of being embedded, and the arrangement of genes on chromosomes in such a way as to optimize the adaptation to the different cell types which must be produced using the same genetic information.
I will answer such questions with students and collaborators, who will specialize in the subprojects but will interact with my help across the common bridge.
Max ERC Funding
2 135 385 €
Duration
Start date: 2012-04-01, End date: 2017-07-31
Project acronym BSD
Project Euler systems and the conjectures of Birch and Swinnerton-Dyer, Bloch and Kato
Researcher (PI) Victor Rotger cerdà
Host Institution (HI) UNIVERSITAT POLITECNICA DE CATALUNYA
Call Details Consolidator Grant (CoG), PE1, ERC-2015-CoG
Summary In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $1.000.000 Prize Problems. One of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960's. The main object of this proposal is developing innovative and unconventional strategies for proving groundbreaking results towards the resolution of this problem and their generalizations by Bloch and Kato (BK).
Breakthroughs on BSD were achieved by Coates-Wiles, Gross, Zagier and Kolyvagin, and Kato. Since then, there have been nearly no new ideas on how to tackle BSD. Only very recently, three independent revolutionary approaches have seen the light: the works of (1) the Fields medalist Bhargava, (2) Skinner and Urban, and (3) myself and my collaborators. In spite of that, our knowledge of BSD is rather poor. In my proposal I suggest innovating strategies for approaching new horizons in BSD and BK that I aim to develop with the team of PhD and postdoctoral researchers that the CoG may allow me to consolidate. The results I plan to prove represent a departure from the achievements obtained with my coauthors during the past years:
I. BSD over totally real number fields. I plan to prove new ground-breaking instances of BSD in rank 0 for elliptic curves over totally real number fields, generalizing the theorem of Kato (by providing a new proof) and covering many new scenarios that have never been considered before.
II. BSD in rank r=2. Most of the literature on BSD applies when r=0 or 1. I expect to prove p-adic versions of the theorems of Gross-Zagier and Kolyvagin in rank 2.
III. Darmon's 2000 conjecture on Stark-Heegner points. I plan to prove Darmon’s striking conjecture announced at the ICM2000 by recasting it in terms of special values of p-adic L-functions.
Summary
In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $1.000.000 Prize Problems. One of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960's. The main object of this proposal is developing innovative and unconventional strategies for proving groundbreaking results towards the resolution of this problem and their generalizations by Bloch and Kato (BK).
Breakthroughs on BSD were achieved by Coates-Wiles, Gross, Zagier and Kolyvagin, and Kato. Since then, there have been nearly no new ideas on how to tackle BSD. Only very recently, three independent revolutionary approaches have seen the light: the works of (1) the Fields medalist Bhargava, (2) Skinner and Urban, and (3) myself and my collaborators. In spite of that, our knowledge of BSD is rather poor. In my proposal I suggest innovating strategies for approaching new horizons in BSD and BK that I aim to develop with the team of PhD and postdoctoral researchers that the CoG may allow me to consolidate. The results I plan to prove represent a departure from the achievements obtained with my coauthors during the past years:
I. BSD over totally real number fields. I plan to prove new ground-breaking instances of BSD in rank 0 for elliptic curves over totally real number fields, generalizing the theorem of Kato (by providing a new proof) and covering many new scenarios that have never been considered before.
II. BSD in rank r=2. Most of the literature on BSD applies when r=0 or 1. I expect to prove p-adic versions of the theorems of Gross-Zagier and Kolyvagin in rank 2.
III. Darmon's 2000 conjecture on Stark-Heegner points. I plan to prove Darmon’s striking conjecture announced at the ICM2000 by recasting it in terms of special values of p-adic L-functions.
Max ERC Funding
1 428 588 €
Duration
Start date: 2016-09-01, End date: 2021-08-31
Project acronym CASe
Project Combinatorics with an analytic structure
Researcher (PI) Karim ADIPRASITO
Host Institution (HI) THE HEBREW UNIVERSITY OF JERUSALEM
Call Details Starting Grant (StG), PE1, ERC-2016-STG
Summary "Combinatorics, and its interplay with geometry, has fascinated our ancestors as shown by early stone carvings in the Neolithic period. Modern combinatorics is motivated by the ubiquity of its structures in both pure and applied mathematics.
The work of Hochster and Stanley, who realized the relation of enumerative questions to commutative algebra and toric geometry made a vital contribution to the development of this subject. Their work was a central contribution to the classification of face numbers of simple polytopes, and the initial success lead to a wealth of research in which combinatorial problems were translated to algebra and geometry and then solved using deep results such as Saito's hard Lefschetz theorem. As a caveat, this also made branches of combinatorics reliant on algebra and geometry to provide new ideas.
In this proposal, I want to reverse this approach and extend our understanding of geometry and algebra guided by combinatorial methods. In this spirit I propose new combinatorial approaches to the interplay of curvature and topology, to isoperimetry, geometric analysis, and intersection theory, to name a few. In addition, while these subjects are interesting by themselves, they are also designed to advance classical topics, for example, the diameter of polyhedra (as in the Hirsch conjecture), arrangement theory (and the study of arrangement complements), Hodge theory (as in Grothendieck's standard conjectures), and realization problems of discrete objects (as in Connes embedding problem for type II factors).
This proposal is supported by the review of some already developed tools, such as relative Stanley--Reisner theory (which is equipped to deal with combinatorial isoperimetries), combinatorial Hodge theory (which extends the ``K\""ahler package'' to purely combinatorial settings), and discrete PDEs (which were used to construct counterexamples to old problems in discrete geometry)."
Summary
"Combinatorics, and its interplay with geometry, has fascinated our ancestors as shown by early stone carvings in the Neolithic period. Modern combinatorics is motivated by the ubiquity of its structures in both pure and applied mathematics.
The work of Hochster and Stanley, who realized the relation of enumerative questions to commutative algebra and toric geometry made a vital contribution to the development of this subject. Their work was a central contribution to the classification of face numbers of simple polytopes, and the initial success lead to a wealth of research in which combinatorial problems were translated to algebra and geometry and then solved using deep results such as Saito's hard Lefschetz theorem. As a caveat, this also made branches of combinatorics reliant on algebra and geometry to provide new ideas.
In this proposal, I want to reverse this approach and extend our understanding of geometry and algebra guided by combinatorial methods. In this spirit I propose new combinatorial approaches to the interplay of curvature and topology, to isoperimetry, geometric analysis, and intersection theory, to name a few. In addition, while these subjects are interesting by themselves, they are also designed to advance classical topics, for example, the diameter of polyhedra (as in the Hirsch conjecture), arrangement theory (and the study of arrangement complements), Hodge theory (as in Grothendieck's standard conjectures), and realization problems of discrete objects (as in Connes embedding problem for type II factors).
This proposal is supported by the review of some already developed tools, such as relative Stanley--Reisner theory (which is equipped to deal with combinatorial isoperimetries), combinatorial Hodge theory (which extends the ``K\""ahler package'' to purely combinatorial settings), and discrete PDEs (which were used to construct counterexamples to old problems in discrete geometry)."
Max ERC Funding
1 337 200 €
Duration
Start date: 2016-12-01, End date: 2021-11-30
Project acronym CatDT
Project Categorified Donaldson-Thomas Theory
Researcher (PI) Nicholas David James (Ben) DAVISON
Host Institution (HI) THE UNIVERSITY OF EDINBURGH
Call Details Starting Grant (StG), PE1, ERC-2017-STG
Summary According to string theory, coherent sheaves on three-dimensional Calabi-Yau spaces encode fundamental properties of the universe. On the other hand, they have a purely mathematical definition. We will develop and use the new field of categorified Donaldson-Thomas (DT) theory, which counts these objects. Via the powerful perspective of noncommutative algebraic geometry, this theory has found application in recent years in a wide variety of contexts, far from classical algebraic geometry.
Categorification has proved tremendously powerful across mathematics, for example the entire subject of algebraic topology was started by the categorification of Betti numbers. The categorification of DT theory leads to the replacement of the numbers of DT theory by vector spaces, of which these numbers are the dimensions. In the area of categorified DT theory we have been able to prove fundamental conjectures upgrading the famous wall crossing formula and integrality conjecture in noncommutative algebraic geometry. The first three projects involve applications of the resulting new subject:
1. Complete the categorification of quantum cluster algebras, proving the strong positivity conjecture.
2. Use cohomological DT theory to prove the outstanding conjectures in the nonabelian Hodge theory of Riemann surfaces, and the subject of Higgs bundles.
3. Prove the comparison conjecture, realising the study of Yangian quantum groups and the geometric representation theory around them as a special case of DT theory.
The final objective involves coming full circle, and applying our recent advances in noncommutative DT theory to the original theory that united string theory with algebraic geometry:
4. Develop a generalised theory of categorified DT theory extending our results in noncommutative DT theory, proving the integrality conjecture for categories of coherent sheaves on Calabi-Yau 3-folds.
Summary
According to string theory, coherent sheaves on three-dimensional Calabi-Yau spaces encode fundamental properties of the universe. On the other hand, they have a purely mathematical definition. We will develop and use the new field of categorified Donaldson-Thomas (DT) theory, which counts these objects. Via the powerful perspective of noncommutative algebraic geometry, this theory has found application in recent years in a wide variety of contexts, far from classical algebraic geometry.
Categorification has proved tremendously powerful across mathematics, for example the entire subject of algebraic topology was started by the categorification of Betti numbers. The categorification of DT theory leads to the replacement of the numbers of DT theory by vector spaces, of which these numbers are the dimensions. In the area of categorified DT theory we have been able to prove fundamental conjectures upgrading the famous wall crossing formula and integrality conjecture in noncommutative algebraic geometry. The first three projects involve applications of the resulting new subject:
1. Complete the categorification of quantum cluster algebras, proving the strong positivity conjecture.
2. Use cohomological DT theory to prove the outstanding conjectures in the nonabelian Hodge theory of Riemann surfaces, and the subject of Higgs bundles.
3. Prove the comparison conjecture, realising the study of Yangian quantum groups and the geometric representation theory around them as a special case of DT theory.
The final objective involves coming full circle, and applying our recent advances in noncommutative DT theory to the original theory that united string theory with algebraic geometry:
4. Develop a generalised theory of categorified DT theory extending our results in noncommutative DT theory, proving the integrality conjecture for categories of coherent sheaves on Calabi-Yau 3-folds.
Max ERC Funding
1 239 435 €
Duration
Start date: 2017-11-01, End date: 2022-10-31
Project acronym CausalStats
Project Statistics, Prediction and Causality for Large-Scale Data
Researcher (PI) Peter Lukas Bühlmann
Host Institution (HI) EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH
Call Details Advanced Grant (AdG), PE1, ERC-2017-ADG
Summary Understanding cause-effect relationships between variables is of great interest in many fields of science. However, causal inference from data is much more ambitious and difficult than inferring (undirected) measures of association such as correlations, partial correlations or multivariate regression coefficients, mainly because of fundamental identifiability
problems. A main objective of the proposal is to exploit advantages from large-scale heterogeneous data for causal inference where heterogeneity arises from different experimental conditions or different unknown sub-populations. A key idea is to consider invariance or stability across different experimental conditions of certain conditional probability distributions: the invariants correspond on the one hand to (properly defined) causal variables which are of main interest in causality; andon the other hand, they correspond to the features for constructing powerful predictions for new scenarios which are unobserved in the data (new probability distributions). This opens novel perspectives: causal inference
can be phrased as a prediction problem of a certain kind, and vice versa, new prediction methods which work well across different scenarios (unobserved in the data) should be based on or regularized towards causal variables. Fundamental identifiability limits will become weaker with increased degree of heterogeneity, as we expect in large-scale data. The topic is essentially unexplored, yet it opens new avenues for causal inference, structural equation and graphical modeling, and robust prediction based on large-scale complex data. We will develop mathematical theory, statistical methodology and efficient algorithms; and we will also work and collaborate on major application problems such as inferring causal effects (i.e., total intervention effects) from gene knock-out or RNA interference perturbation experiments, genome-wide association studies and novel prediction tasks in economics.
Summary
Understanding cause-effect relationships between variables is of great interest in many fields of science. However, causal inference from data is much more ambitious and difficult than inferring (undirected) measures of association such as correlations, partial correlations or multivariate regression coefficients, mainly because of fundamental identifiability
problems. A main objective of the proposal is to exploit advantages from large-scale heterogeneous data for causal inference where heterogeneity arises from different experimental conditions or different unknown sub-populations. A key idea is to consider invariance or stability across different experimental conditions of certain conditional probability distributions: the invariants correspond on the one hand to (properly defined) causal variables which are of main interest in causality; andon the other hand, they correspond to the features for constructing powerful predictions for new scenarios which are unobserved in the data (new probability distributions). This opens novel perspectives: causal inference
can be phrased as a prediction problem of a certain kind, and vice versa, new prediction methods which work well across different scenarios (unobserved in the data) should be based on or regularized towards causal variables. Fundamental identifiability limits will become weaker with increased degree of heterogeneity, as we expect in large-scale data. The topic is essentially unexplored, yet it opens new avenues for causal inference, structural equation and graphical modeling, and robust prediction based on large-scale complex data. We will develop mathematical theory, statistical methodology and efficient algorithms; and we will also work and collaborate on major application problems such as inferring causal effects (i.e., total intervention effects) from gene knock-out or RNA interference perturbation experiments, genome-wide association studies and novel prediction tasks in economics.
Max ERC Funding
2 184 375 €
Duration
Start date: 2018-10-01, End date: 2023-09-30
Project acronym CAVE
Project Challenges and Advancements in Virtual Elements
Researcher (PI) Lourenco Beirao da veiga
Host Institution (HI) UNIVERSITA' DEGLI STUDI DI MILANO-BICOCCA
Call Details Consolidator Grant (CoG), PE1, ERC-2015-CoG
Summary The Virtual Element Method (VEM) is a novel technology for the discretization of partial differential equations (PDEs), that shares the same variational background as the Finite Element Method. First but not only, the VEM responds to the strongly increasing interest in using general polyhedral and polygonal meshes in the approximation of PDEs without the limit of using tetrahedral or hexahedral grids. By avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the stiffness matrixes, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for polygonal/polyhedral meshes (even non-conforming) with non-convex elements and possibly with curved faces; it allows for discrete spaces of arbitrary C^k regularity on unstructured meshes.
The main scope of the project is to address the recent theoretical challenges posed by VEM and to assess whether this promising technology can achieve a breakthrough in applications. First, the theoretical and computational foundations of VEM will be made stronger. A deeper theoretical insight, supported by a wider numerical experience on benchmark problems, will be developed to gain a better understanding of the method's potentials and set the foundations for more applicative purposes. Second, we will focus our attention on two tough and up-to-date problems of practical interest: large deformation elasticity (where VEM can yield a dramatically more efficient handling of material inclusions, meshing of the domain and grid adaptivity, plus a much stronger robustness with respect to large grid distortions) and the cardiac bidomain model (where VEM can lead to a more accurate domain approximation through MRI data, a flexible refinement/de-refinement procedure along the propagation front, to an exact satisfaction of conservation laws).
Summary
The Virtual Element Method (VEM) is a novel technology for the discretization of partial differential equations (PDEs), that shares the same variational background as the Finite Element Method. First but not only, the VEM responds to the strongly increasing interest in using general polyhedral and polygonal meshes in the approximation of PDEs without the limit of using tetrahedral or hexahedral grids. By avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the stiffness matrixes, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for polygonal/polyhedral meshes (even non-conforming) with non-convex elements and possibly with curved faces; it allows for discrete spaces of arbitrary C^k regularity on unstructured meshes.
The main scope of the project is to address the recent theoretical challenges posed by VEM and to assess whether this promising technology can achieve a breakthrough in applications. First, the theoretical and computational foundations of VEM will be made stronger. A deeper theoretical insight, supported by a wider numerical experience on benchmark problems, will be developed to gain a better understanding of the method's potentials and set the foundations for more applicative purposes. Second, we will focus our attention on two tough and up-to-date problems of practical interest: large deformation elasticity (where VEM can yield a dramatically more efficient handling of material inclusions, meshing of the domain and grid adaptivity, plus a much stronger robustness with respect to large grid distortions) and the cardiac bidomain model (where VEM can lead to a more accurate domain approximation through MRI data, a flexible refinement/de-refinement procedure along the propagation front, to an exact satisfaction of conservation laws).
Max ERC Funding
980 634 €
Duration
Start date: 2016-07-01, End date: 2021-06-30
Project acronym CC
Project Combinatorial Construction
Researcher (PI) Peter Keevash
Host Institution (HI) THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD
Call Details Consolidator Grant (CoG), PE1, ERC-2014-CoG
Summary Combinatorial Construction is a mathematical challenge with many applications. Examples include the construction of networks that are very sparse but highly connected, or codes that can correct many transmission errors with little overhead in communication costs. For a general class of combinatorial objects, and some desirable property, the fundamental question in Combinatorial Construction is to demonstrate the existence of an object with the property, preferably via an explicit algorithmic construction. Thus it is ubiquitous in Computer Science, including applications to expanders, sorting networks, distributed communication, data storage, codes, cryptography and derandomisation. In popular culture it appears as the unsolved `lottery problem' of determining the minimum number of tickets that guarantee a prize. In a recent preprint I prove the Existence Conjecture for combinatorial designs, via a new method of Randomised Algebraic Constructions; this result has already attracted considerable attention in the mathematical community. The significance is not only in the solution of a problem posed by Steiner in 1852, but also in the discovery of a powerful new method, that promises to have many further applications in Combinatorics, and more widely in Mathematics and Theoretical Computer Science. I am now poised to resolve many other problems of combinatorial construction.
Summary
Combinatorial Construction is a mathematical challenge with many applications. Examples include the construction of networks that are very sparse but highly connected, or codes that can correct many transmission errors with little overhead in communication costs. For a general class of combinatorial objects, and some desirable property, the fundamental question in Combinatorial Construction is to demonstrate the existence of an object with the property, preferably via an explicit algorithmic construction. Thus it is ubiquitous in Computer Science, including applications to expanders, sorting networks, distributed communication, data storage, codes, cryptography and derandomisation. In popular culture it appears as the unsolved `lottery problem' of determining the minimum number of tickets that guarantee a prize. In a recent preprint I prove the Existence Conjecture for combinatorial designs, via a new method of Randomised Algebraic Constructions; this result has already attracted considerable attention in the mathematical community. The significance is not only in the solution of a problem posed by Steiner in 1852, but also in the discovery of a powerful new method, that promises to have many further applications in Combinatorics, and more widely in Mathematics and Theoretical Computer Science. I am now poised to resolve many other problems of combinatorial construction.
Max ERC Funding
1 706 729 €
Duration
Start date: 2016-01-01, End date: 2020-12-31
Project acronym CCOSA
Project Classes of combinatorial objects: from structure to algorithms
Researcher (PI) Daniel Kral
Host Institution (HI) THE UNIVERSITY OF WARWICK
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The proposed project aims at analyzing fundamental problems from combinatorics using the most current methods available and at providing new structural and algorithmic insights to such problems. The problems considered will be treated on a general level of classes of combinatorial objects of the same kind and the developed general methods will also be applied to specific open problems. Classes of dense and sparse objects will be treated using different techniques. Dense combinatorial objects appear in extremal combinatorics and tools developed to handle them found their applications in different
areas of mathematics and computer science. The project will focus on extending known methods to new classes of combinatorial objects, in particular those from algebra, and applying the most current techniques including Razborov flag algebras to problems from extremal combinatorics. Applications of the obtained results in property testing will also be considered. On the other hand, algorithmic applications often include manipulating with sparse objects. Examples of sparse objects are graphs embeddable in a fixed surface and more general minor-closed classes of graphs. The project objectives include providing new structural results and algorithmic metatheorems for classes of sparse objects using both classical tools based on the theory of graph minors as well as new tools based on the framework of classes of nowhere-dense structures.
Summary
The proposed project aims at analyzing fundamental problems from combinatorics using the most current methods available and at providing new structural and algorithmic insights to such problems. The problems considered will be treated on a general level of classes of combinatorial objects of the same kind and the developed general methods will also be applied to specific open problems. Classes of dense and sparse objects will be treated using different techniques. Dense combinatorial objects appear in extremal combinatorics and tools developed to handle them found their applications in different
areas of mathematics and computer science. The project will focus on extending known methods to new classes of combinatorial objects, in particular those from algebra, and applying the most current techniques including Razborov flag algebras to problems from extremal combinatorics. Applications of the obtained results in property testing will also be considered. On the other hand, algorithmic applications often include manipulating with sparse objects. Examples of sparse objects are graphs embeddable in a fixed surface and more general minor-closed classes of graphs. The project objectives include providing new structural results and algorithmic metatheorems for classes of sparse objects using both classical tools based on the theory of graph minors as well as new tools based on the framework of classes of nowhere-dense structures.
Max ERC Funding
849 000 €
Duration
Start date: 2010-12-01, End date: 2015-11-30