Project acronym BETLIV
Project Returning to a Better Place: The (Re)assessment of the ‘Good Life’ in Times of Crisis
Researcher (PI) Valerio SIMONI RIBA
Host Institution (HI) FONDATION POUR L INSTITUT DE HAUTES ETUDES INTERNATIONALES ET DU DEVELOPPEMENT
Call Details Starting Grant (StG), SH5, ERC-2017-STG
Summary What makes for a valuable and good life is a question that many people in the contemporary world ask themselves, yet it is one that social science research has seldom addressed. Only recently have scholars started undertaking inductive comparative research on different notions of the ‘good life’, highlighting socio-cultural variations and calling for a better understanding of the different imaginaries, aspirations and values that guide people in their quest for better living conditions. Research is still lacking, however, on how people themselves evaluate, compare, and put into perspective different visions of good living and their socio-cultural anchorage. This project addresses such questions from an anthropological perspective, proposing an innovative study of how ideals of the good life are articulated, (re)assessed, and related to specific places and contexts as a result of the experience of crisis and migration. The case studies chosen to operationalize these lines of enquiry focus on the phenomenon of return migration, and consist in an analysis of the imaginaries and experience of return by Ecuadorian and Cuban men and women who migrated to Spain, are dissatisfied with their life there, and envisage/carry out the project of going back to their countries of origin (Ecuador and Cuba respectively). The project’s ambition is to bring together and contribute to three main scholarly areas of enquiry: 1) the study of morality, ethics and what counts as ‘good life’, 2) the study of the field of economic practice, its definition, value regimes, and ‘crises’, and 3) the study of migratory aspirations, projects, and trajectories. A multi-sited endeavour, the research is designed in three subprojects carried out in Spain (PhD student), Ecuador (Post-Doc), and Cuba (PI), in which ethnographic methods will be used to provide the first empirically grounded study of the links between notions and experiences of crisis, return migration, and the (re)assessment of good living.
Summary
What makes for a valuable and good life is a question that many people in the contemporary world ask themselves, yet it is one that social science research has seldom addressed. Only recently have scholars started undertaking inductive comparative research on different notions of the ‘good life’, highlighting socio-cultural variations and calling for a better understanding of the different imaginaries, aspirations and values that guide people in their quest for better living conditions. Research is still lacking, however, on how people themselves evaluate, compare, and put into perspective different visions of good living and their socio-cultural anchorage. This project addresses such questions from an anthropological perspective, proposing an innovative study of how ideals of the good life are articulated, (re)assessed, and related to specific places and contexts as a result of the experience of crisis and migration. The case studies chosen to operationalize these lines of enquiry focus on the phenomenon of return migration, and consist in an analysis of the imaginaries and experience of return by Ecuadorian and Cuban men and women who migrated to Spain, are dissatisfied with their life there, and envisage/carry out the project of going back to their countries of origin (Ecuador and Cuba respectively). The project’s ambition is to bring together and contribute to three main scholarly areas of enquiry: 1) the study of morality, ethics and what counts as ‘good life’, 2) the study of the field of economic practice, its definition, value regimes, and ‘crises’, and 3) the study of migratory aspirations, projects, and trajectories. A multi-sited endeavour, the research is designed in three subprojects carried out in Spain (PhD student), Ecuador (Post-Doc), and Cuba (PI), in which ethnographic methods will be used to provide the first empirically grounded study of the links between notions and experiences of crisis, return migration, and the (re)assessment of good living.
Max ERC Funding
1 500 000 €
Duration
Start date: 2018-02-01, End date: 2023-01-31
Project acronym COMPLEXDATA
Project Statistics for Complex Data: Understanding Randomness, Geometry and Complexity with a view Towards Biophysics
Researcher (PI) Victor Michael Panaretos
Host Institution (HI) ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The ComplexData project aims at advancing our understanding of the statistical treatment of varied types of complex data by generating new theory and methods, and to obtain progress in concrete current biophysical problems through the implementation of the new tools developed. Complex Data constitute data where the basic object of observation cannot be described in the standard Euclidean context of statistics, but rather needs to be thought of as an element of an abstract mathematical space with special properties. Scientific progress has, in recent years, begun to generate an increasing number of new and complex types of data that require statistical understanding and analysis. Four such types of data that are arising in the context of current scientific research and that the project will be focusing on are: random integral transforms, random unlabelled shapes, random flows of functions, and random tensor fields. In these unconventional contexts for statistics, the strategy of the project will be to carefully exploit the special aspects involved due to geometry, dimension and randomness in order to be able to either adapt and synthesize existing statistical methods, or to generate new statistical ideas altogether. However, the project will not restrict itself to merely studying the theoretical aspects of complex data, but will be truly interdisciplinary. The connecting thread among all the above data types is that their study is motivated by, and will be applied to concrete practical problems arising in the study of biological structure, dynamics, and function: biophysics. For this reason, the programme will be in interaction with local and international contacts from this field. In particular, the theoretical/methodological output of the four programme research foci will be applied to gain insights in the following corresponding four application areas: electron microscopy, protein homology, DNA molecular dynamics, brain imaging.
Summary
The ComplexData project aims at advancing our understanding of the statistical treatment of varied types of complex data by generating new theory and methods, and to obtain progress in concrete current biophysical problems through the implementation of the new tools developed. Complex Data constitute data where the basic object of observation cannot be described in the standard Euclidean context of statistics, but rather needs to be thought of as an element of an abstract mathematical space with special properties. Scientific progress has, in recent years, begun to generate an increasing number of new and complex types of data that require statistical understanding and analysis. Four such types of data that are arising in the context of current scientific research and that the project will be focusing on are: random integral transforms, random unlabelled shapes, random flows of functions, and random tensor fields. In these unconventional contexts for statistics, the strategy of the project will be to carefully exploit the special aspects involved due to geometry, dimension and randomness in order to be able to either adapt and synthesize existing statistical methods, or to generate new statistical ideas altogether. However, the project will not restrict itself to merely studying the theoretical aspects of complex data, but will be truly interdisciplinary. The connecting thread among all the above data types is that their study is motivated by, and will be applied to concrete practical problems arising in the study of biological structure, dynamics, and function: biophysics. For this reason, the programme will be in interaction with local and international contacts from this field. In particular, the theoretical/methodological output of the four programme research foci will be applied to gain insights in the following corresponding four application areas: electron microscopy, protein homology, DNA molecular dynamics, brain imaging.
Max ERC Funding
681 146 €
Duration
Start date: 2011-06-01, End date: 2016-05-31
Project acronym CONSTAMIS
Project Connecting Statistical Mechanics and Conformal Field Theory: an Ising Model Perspective
Researcher (PI) CLEMENT HONGLER
Host Institution (HI) ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Call Details Starting Grant (StG), PE1, ERC-2016-STG
Summary The developments of Statistical Mechanics and Quantum Field Theory are among the major achievements of the 20th century's science. During the second half of the century, these two subjects started to converge. In two dimensions, this resulted in a most remarkable chapter of mathematical physics: Conformal Field Theory (CFT) reveals deep structures allowing for extremely precise investigations, making such theories powerful building blocks of many subjects of mathematics and physics. Unfortunately, this convergence has remained non-rigorous, leaving most of the spectacular field-theoretic applications to Statistical Mechanics conjectural.
About 15 years ago, several mathematical breakthroughs shed new light on this picture. The development of SLE curves and discrete complex analysis has enabled one to connect various statistical mechanics models with conformally symmetric processes. Recently, major progress was made on a key statistical mechanics model, the Ising model: the connection with SLE was established, and many formulae predicted by CFT were proven.
Important advances towards connecting Statistical Mechanics and CFT now appear possible. This is the goal of this proposal, which is organized in three objectives:
(I) Build a deep correspondence between the Ising model and CFT: reveal clear links between the objects and structures arising in the Ising and CFT frameworks.
(II) Gather the insights of (I) to study new connections to CFT, particularly for minimal models, current algebras and parafermions.
(III) Combine (I) and (II) to go beyond conformal symmetry: link the Ising model with massive integrable field theories.
The aim is to build one of the first rigorous bridges between Statistical Mechanics and CFT. It will help to close the gap between physical derivations and mathematical theorems. By linking the deep structures of CFT to concrete models that are applicable in many subjects, it will be potentially useful to theoretical and applied scientists.
Summary
The developments of Statistical Mechanics and Quantum Field Theory are among the major achievements of the 20th century's science. During the second half of the century, these two subjects started to converge. In two dimensions, this resulted in a most remarkable chapter of mathematical physics: Conformal Field Theory (CFT) reveals deep structures allowing for extremely precise investigations, making such theories powerful building blocks of many subjects of mathematics and physics. Unfortunately, this convergence has remained non-rigorous, leaving most of the spectacular field-theoretic applications to Statistical Mechanics conjectural.
About 15 years ago, several mathematical breakthroughs shed new light on this picture. The development of SLE curves and discrete complex analysis has enabled one to connect various statistical mechanics models with conformally symmetric processes. Recently, major progress was made on a key statistical mechanics model, the Ising model: the connection with SLE was established, and many formulae predicted by CFT were proven.
Important advances towards connecting Statistical Mechanics and CFT now appear possible. This is the goal of this proposal, which is organized in three objectives:
(I) Build a deep correspondence between the Ising model and CFT: reveal clear links between the objects and structures arising in the Ising and CFT frameworks.
(II) Gather the insights of (I) to study new connections to CFT, particularly for minimal models, current algebras and parafermions.
(III) Combine (I) and (II) to go beyond conformal symmetry: link the Ising model with massive integrable field theories.
The aim is to build one of the first rigorous bridges between Statistical Mechanics and CFT. It will help to close the gap between physical derivations and mathematical theorems. By linking the deep structures of CFT to concrete models that are applicable in many subjects, it will be potentially useful to theoretical and applied scientists.
Max ERC Funding
998 005 €
Duration
Start date: 2017-03-01, End date: 2022-02-28
Project acronym EllipticPDE
Project Regularity and singularities in elliptic PDE's: beyond monotonicity formulas
Researcher (PI) Xavier ROS-OTON
Host Institution (HI) UNIVERSITAT ZURICH
Call Details Starting Grant (StG), PE1, ERC-2018-STG
Summary One of the oldest and most important questions in PDE theory is that of regularity. A classical example is Hilbert's XIXth problem (1900), solved by De Giorgi and Nash in 1956. During the second half of the XXth century, the regularity theory for elliptic and parabolic PDE's experienced a huge development, and many fundamental questions were answered by Caffarelli, Nirenberg, Krylov, Evans, Nadirashvili, Friedman, and many others. Still, there are problems of crucial importance that remain open.
The aim of this project is to go significantly beyond the state of the art in some of the most important open questions in this context. In particular, three key objectives of the project are the following. First, to introduce new techniques to obtain fine description of singularities in nonlinear elliptic PDE's. Aside from its intrinsic interest, a good regularity theory for singular points is likely to provide insightful applications in other contexts. A second aim of the project is to establish generic regularity results for free boundaries and other PDE problems. The development of methods which would allow one to prove generic regularity results may be viewed as one of the greatest challenges not only for free boundary problems, but for PDE problems in general. Finally, the third main objective is to achieve a complete regularity theory for nonlinear elliptic PDE's that does not rely on monotonicity formulas. These three objectives, while seemingly different, are in fact deeply interrelated.
Summary
One of the oldest and most important questions in PDE theory is that of regularity. A classical example is Hilbert's XIXth problem (1900), solved by De Giorgi and Nash in 1956. During the second half of the XXth century, the regularity theory for elliptic and parabolic PDE's experienced a huge development, and many fundamental questions were answered by Caffarelli, Nirenberg, Krylov, Evans, Nadirashvili, Friedman, and many others. Still, there are problems of crucial importance that remain open.
The aim of this project is to go significantly beyond the state of the art in some of the most important open questions in this context. In particular, three key objectives of the project are the following. First, to introduce new techniques to obtain fine description of singularities in nonlinear elliptic PDE's. Aside from its intrinsic interest, a good regularity theory for singular points is likely to provide insightful applications in other contexts. A second aim of the project is to establish generic regularity results for free boundaries and other PDE problems. The development of methods which would allow one to prove generic regularity results may be viewed as one of the greatest challenges not only for free boundary problems, but for PDE problems in general. Finally, the third main objective is to achieve a complete regularity theory for nonlinear elliptic PDE's that does not rely on monotonicity formulas. These three objectives, while seemingly different, are in fact deeply interrelated.
Max ERC Funding
1 335 250 €
Duration
Start date: 2019-01-01, End date: 2023-12-31
Project acronym FLIRT
Project Fluid Flows and Irregular Transport
Researcher (PI) Gianluca Crippa
Host Institution (HI) UNIVERSITAT BASEL
Call Details Starting Grant (StG), PE1, ERC-2015-STG
Summary "Several important partial differential equations (PDEs) arising in the mathematical description of physical phenomena exhibit transport features: physical quantities are advected by velocity fields that drive the dynamics of the system. This is the case for instance for the Euler equation of fluid dynamics, for conservation laws, and for kinetic equations.
An ubiquitous feature of these phenomena is their intrinsic lack of regularity. From the mathematical point of view this stems from the nonlinearity and/or nonlocality of the PDEs. Moreover, the lack of regularity also encodes actual properties of the underlying physical systems: conservation laws develop shocks (discontinuities that propagate in time), solutions to the Euler equation exhibit rough and ""disordered"" behaviors. This irregularity is the major difficulty in the mathematical analysis of such problems, since it prevents the use of many standard methods, foremost the classical (and powerful) theory of characteristics.
For these reasons, the study in a non smooth setting of transport and continuity equations, and of flows of ordinary differential equations, is a fundamental tool to approach challenging important questions concerning these PDEs.
This project aims at establishing:
(1) deep insight into the structure of solutions of nonlinear PDEs, in particular the Euler equation and multidimensional systems of conservation laws,
(2) rigorous bounds for mixing phenomena in fluid flows, phenomena for which giving a precise mathematical formulation is extremely challenging.
The unifying factor of this proposal is that the analysis will rely on major advances in the theory of flows of ordinary differential equations in a non smooth setting, thus providing a robust formulation via characteristics for the PDEs under consideration. The guiding thread is the crucial role of geometric measure theory techniques, which are extremely efficient to describe and investigate irregular phenomena."
Summary
"Several important partial differential equations (PDEs) arising in the mathematical description of physical phenomena exhibit transport features: physical quantities are advected by velocity fields that drive the dynamics of the system. This is the case for instance for the Euler equation of fluid dynamics, for conservation laws, and for kinetic equations.
An ubiquitous feature of these phenomena is their intrinsic lack of regularity. From the mathematical point of view this stems from the nonlinearity and/or nonlocality of the PDEs. Moreover, the lack of regularity also encodes actual properties of the underlying physical systems: conservation laws develop shocks (discontinuities that propagate in time), solutions to the Euler equation exhibit rough and ""disordered"" behaviors. This irregularity is the major difficulty in the mathematical analysis of such problems, since it prevents the use of many standard methods, foremost the classical (and powerful) theory of characteristics.
For these reasons, the study in a non smooth setting of transport and continuity equations, and of flows of ordinary differential equations, is a fundamental tool to approach challenging important questions concerning these PDEs.
This project aims at establishing:
(1) deep insight into the structure of solutions of nonlinear PDEs, in particular the Euler equation and multidimensional systems of conservation laws,
(2) rigorous bounds for mixing phenomena in fluid flows, phenomena for which giving a precise mathematical formulation is extremely challenging.
The unifying factor of this proposal is that the analysis will rely on major advances in the theory of flows of ordinary differential equations in a non smooth setting, thus providing a robust formulation via characteristics for the PDEs under consideration. The guiding thread is the crucial role of geometric measure theory techniques, which are extremely efficient to describe and investigate irregular phenomena."
Max ERC Funding
1 009 351 €
Duration
Start date: 2016-06-01, End date: 2021-05-31
Project acronym GRAPHCPX
Project A graph complex valued field theory
Researcher (PI) Thomas Hans Willwacher
Host Institution (HI) EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH
Call Details Starting Grant (StG), PE1, ERC-2015-STG
Summary The goal of the proposed project is to create a universal (AKSZ type) topological field theory with values in graph complexes, capturing the rational homotopy types of manifolds, configuration and embedding spaces.
If successful, such a theory will unite certain areas of mathematical physics, topology, homological algebra and algebraic geometry. More concretely, from the physical viewpoint it would give a precise topological interpretation of a class of well studied topological field theories, as opposed to the current state of the art, in which these theories are defined by giving formulae without guarantees on the non-triviality of the produced invariants.
From the topological viewpoint such a theory will provide new tools to study much sought after objects like configuration and embedding spaces, and tentatively also diffeomorphism groups, through small combinatorial models given by Feynman diagrams. In particular, this will unite and extend existing graphical models of configuration and embedding spaces due to Kontsevich, Lambrechts, Volic, Arone, Turchin and others.
From the homological algebra viewpoint a field theory as above provides a wealth of additional algebraic structures on the graph complexes, which are some of the most central and most mysterious objects in the field.
Such algebraic structures are expected to yield constraints on the graph cohomology, as well as ways to construct series of previously unknown classes.
Summary
The goal of the proposed project is to create a universal (AKSZ type) topological field theory with values in graph complexes, capturing the rational homotopy types of manifolds, configuration and embedding spaces.
If successful, such a theory will unite certain areas of mathematical physics, topology, homological algebra and algebraic geometry. More concretely, from the physical viewpoint it would give a precise topological interpretation of a class of well studied topological field theories, as opposed to the current state of the art, in which these theories are defined by giving formulae without guarantees on the non-triviality of the produced invariants.
From the topological viewpoint such a theory will provide new tools to study much sought after objects like configuration and embedding spaces, and tentatively also diffeomorphism groups, through small combinatorial models given by Feynman diagrams. In particular, this will unite and extend existing graphical models of configuration and embedding spaces due to Kontsevich, Lambrechts, Volic, Arone, Turchin and others.
From the homological algebra viewpoint a field theory as above provides a wealth of additional algebraic structures on the graph complexes, which are some of the most central and most mysterious objects in the field.
Such algebraic structures are expected to yield constraints on the graph cohomology, as well as ways to construct series of previously unknown classes.
Max ERC Funding
1 162 500 €
Duration
Start date: 2016-07-01, End date: 2021-06-30
Project acronym MAQD
Project Mathematical Aspects of Quantum Dynamics
Researcher (PI) Benjamin Schlein
Host Institution (HI) UNIVERSITAT ZURICH
Call Details Starting Grant (StG), PE1, ERC-2009-StG
Summary The main goal of this proposal is to reach
a better mathematical understanding of
the dynamics of quantum mechanical
systems. In particular I plan to work
on the following three projects along
this direction. A. Effective Evolution
Equations for Macroscopic Systems.
The derivation of effective evolution
equations from first principle microscopic
theories is a fundamental task of statistical
mechanics. I have been involved in
several projects related to the derivation
of the Hartree and the Gross-Piteavskii
equation from many body quantum
dynamics. I plan to continue to work on
these problems and to use these results
to obtain new information on the many
body dynamics. B. Spectral Properties
of Random Matrices. The correlations
among eigenvalues of large random
matrices are expected to be independent
of the distribution of the entries. This
conjecture, known as universality, is
of great importance for random matrix
theory. In collaboration with L. Erdos and
H.-T. Yau, we established the validity of
Wigner's semicircle law on
microscopic scales, and we proved the
emergence of eigenvalue repulsion. In
the future, we plan to continue to study
Wigner matrices to prove, on the longer
term, universality. C. Locality Estimates in
Quantum Dynamics. Anharmonic lattice
systems are very important models in
non-equilibrium statistical mechanics.
With B. Nachtergaele, H. Raz, and R.
Sims, we proved Lieb-Robinson type
inequalities (giving an upper bound on
the speed of propagation of signals), for
a certain class of anharmonicity. Next, we
plan to extend these results to a larger
class of anharmonic potentials, and to
apply these bounds to establish other
fundamental properties of the dynamics
of anharmonic systems, such as the
existence of its thermodynamical limit.
Summary
The main goal of this proposal is to reach
a better mathematical understanding of
the dynamics of quantum mechanical
systems. In particular I plan to work
on the following three projects along
this direction. A. Effective Evolution
Equations for Macroscopic Systems.
The derivation of effective evolution
equations from first principle microscopic
theories is a fundamental task of statistical
mechanics. I have been involved in
several projects related to the derivation
of the Hartree and the Gross-Piteavskii
equation from many body quantum
dynamics. I plan to continue to work on
these problems and to use these results
to obtain new information on the many
body dynamics. B. Spectral Properties
of Random Matrices. The correlations
among eigenvalues of large random
matrices are expected to be independent
of the distribution of the entries. This
conjecture, known as universality, is
of great importance for random matrix
theory. In collaboration with L. Erdos and
H.-T. Yau, we established the validity of
Wigner's semicircle law on
microscopic scales, and we proved the
emergence of eigenvalue repulsion. In
the future, we plan to continue to study
Wigner matrices to prove, on the longer
term, universality. C. Locality Estimates in
Quantum Dynamics. Anharmonic lattice
systems are very important models in
non-equilibrium statistical mechanics.
With B. Nachtergaele, H. Raz, and R.
Sims, we proved Lieb-Robinson type
inequalities (giving an upper bound on
the speed of propagation of signals), for
a certain class of anharmonicity. Next, we
plan to extend these results to a larger
class of anharmonic potentials, and to
apply these bounds to establish other
fundamental properties of the dynamics
of anharmonic systems, such as the
existence of its thermodynamical limit.
Max ERC Funding
750 000 €
Duration
Start date: 2009-12-01, End date: 2014-11-30
Project acronym POLYTE
Project Polynomial term structure models
Researcher (PI) Damir Filipovic
Host Institution (HI) ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Call Details Starting Grant (StG), PE1, ERC-2012-StG_20111012
Summary "The term structure of interest rates plays a central role in the functioning of the interbank market. It also represents a key factor for the valuation and management of long term liabilities, such as pensions. The financial crisis has revealed the multivariate risk nature of the term structure, which includes inflation, credit and liquidity risk, resulting in multiple spread adjusted discount curves. This has generated a strong interest in tractable stochastic models for the movements of the term structure that can match all determining risk factors.
We propose a new class of term structure models based on polynomial factor processes which are defined as jump-diffusions whose generator leaves the space of polynomials of any fixed degree invariant. The moments of their transition distributions are polynomials in the initial state. The coefficients defining this relationship are given as solutions of a system of nested linear ordinary differential equations. As a consequence polynomial processes yield closed form polynomial-rational expressions for the term structure of interest rates. Polynomial processes include affine processes, whose transition functions admit an exponential-affine characteristic function. Affine processes are among the most widely used models in finance to date, but come along with some severe specification limitations. We propose to overcome these shortcomings by studying polynomial processes and polynomial expansion methods achieving a comparable efficiency as Fourier methods in the affine case.
In sum, the objectives of this project are threefold. First, we plan to develop a theory for polynomial processes and entirely explore their statistical properties. This fills a gap in the literature on affine processes in particular. Second, we aim to develop polynomial-rational term structure models addressing the new paradigm of multiple spread adjusted discount curves. Third, we plan to implement and estimate these models using real market data."
Summary
"The term structure of interest rates plays a central role in the functioning of the interbank market. It also represents a key factor for the valuation and management of long term liabilities, such as pensions. The financial crisis has revealed the multivariate risk nature of the term structure, which includes inflation, credit and liquidity risk, resulting in multiple spread adjusted discount curves. This has generated a strong interest in tractable stochastic models for the movements of the term structure that can match all determining risk factors.
We propose a new class of term structure models based on polynomial factor processes which are defined as jump-diffusions whose generator leaves the space of polynomials of any fixed degree invariant. The moments of their transition distributions are polynomials in the initial state. The coefficients defining this relationship are given as solutions of a system of nested linear ordinary differential equations. As a consequence polynomial processes yield closed form polynomial-rational expressions for the term structure of interest rates. Polynomial processes include affine processes, whose transition functions admit an exponential-affine characteristic function. Affine processes are among the most widely used models in finance to date, but come along with some severe specification limitations. We propose to overcome these shortcomings by studying polynomial processes and polynomial expansion methods achieving a comparable efficiency as Fourier methods in the affine case.
In sum, the objectives of this project are threefold. First, we plan to develop a theory for polynomial processes and entirely explore their statistical properties. This fills a gap in the literature on affine processes in particular. Second, we aim to develop polynomial-rational term structure models addressing the new paradigm of multiple spread adjusted discount curves. Third, we plan to implement and estimate these models using real market data."
Max ERC Funding
995 155 €
Duration
Start date: 2012-12-01, End date: 2017-11-30
Project acronym RAM
Project Regularity theory for area minimizing currents
Researcher (PI) Camillo De Lellis
Host Institution (HI) UNIVERSITAT ZURICH
Call Details Starting Grant (StG), PE1, ERC-2012-StG_20111012
Summary "The Plateau's problem consists in finding the surface of least area spanning a given contour. This question has attracted the attention of many mathematicians in the last two centuries, providing a prototypical problem for several fields of research in mathematics. For hypersurfaces a lot is known about the existence and regularity thanks to the classical works of De Giorgi, Almgren, Fleming, Federer, Simons, Allard, Simon, Schoen and several other authors.
In higher codimension a quite powerful existence theory, the ``theory of currents'', was developed by Federer and Fleming in 1960. The success of this theory relies on its homological flavor and indeed it has found several applications to problems in differential geometry. Many geometric objects which are widely studied in the modern literature are naturally area-minimizing currents: two examples among many are special lagrangians and holomorphic subvarieties. However the understanding of the regularity issues is, compared to the case of hypersurfaces, much poorer. Aside from its intrinsic interest, a good regularity theory is likely to provide more insightful geometric applications. A quite striking example is Taubes' proof of the equivalence between the Gromov and Seiberg-Witten invariants.
A very complicated and far reaching regularity theory has been developed by Almgren thirty years ago in a monumental work of almost 1000 pages. The first part of this project aims at reaching the same conclusions of Almgren with a more flexible and accessible theory. In the second part I wish to go beyond Almgren's work and attack some of the many open questions which still remain in the field."
Summary
"The Plateau's problem consists in finding the surface of least area spanning a given contour. This question has attracted the attention of many mathematicians in the last two centuries, providing a prototypical problem for several fields of research in mathematics. For hypersurfaces a lot is known about the existence and regularity thanks to the classical works of De Giorgi, Almgren, Fleming, Federer, Simons, Allard, Simon, Schoen and several other authors.
In higher codimension a quite powerful existence theory, the ``theory of currents'', was developed by Federer and Fleming in 1960. The success of this theory relies on its homological flavor and indeed it has found several applications to problems in differential geometry. Many geometric objects which are widely studied in the modern literature are naturally area-minimizing currents: two examples among many are special lagrangians and holomorphic subvarieties. However the understanding of the regularity issues is, compared to the case of hypersurfaces, much poorer. Aside from its intrinsic interest, a good regularity theory is likely to provide more insightful geometric applications. A quite striking example is Taubes' proof of the equivalence between the Gromov and Seiberg-Witten invariants.
A very complicated and far reaching regularity theory has been developed by Almgren thirty years ago in a monumental work of almost 1000 pages. The first part of this project aims at reaching the same conclusions of Almgren with a more flexible and accessible theory. In the second part I wish to go beyond Almgren's work and attack some of the many open questions which still remain in the field."
Max ERC Funding
919 500 €
Duration
Start date: 2012-09-01, End date: 2017-08-31
Project acronym RandMat
Project Spectral Statistics of Structured Random Matrices
Researcher (PI) Antti Kenneth Viktor KNOWLES
Host Institution (HI) UNIVERSITE DE GENEVE
Call Details Starting Grant (StG), PE1, ERC-2016-STG
Summary The purpose of this proposal is a better mathematical understanding of certain classes of large random matrices. Up to very recently, random matrix theory has been mainly focused on mean-field models with independent entries. In this proposal I instead consider random matrices that incorporate some nontrivial structure. I focus on two types of structured random matrices that arise naturally in important applications and lead to a rich mathematical behaviour: (1) random graphs with fixed degrees, such as random regular graphs, and (2) random band matrices, which constitute a good model of disordered quantum Hamiltonians.
The goals are strongly motivated by the applications to spectral graph theory and quantum chaos for (1) and to the physics of conductance in disordered media for (2). Specifically, I will work in the following directions. First, derive precise bounds on the locations of the extremal eigenvalues and the spectral gap, ultimately obtaining their limiting distributions. Second, characterize the spectral statistics in the bulk of the spectrum, using both eigenvalue correlation functions on small scales and linear eigenvalue statistics on intermediate mesoscopic scales. Third, prove the delocalization of eigenvectors and derive the distribution of their components. These results will address several of the most important questions about the structured random matrices (1) and (2), such as expansion properties of random graphs, hallmarks of quantum chaos in random regular graphs, crossovers in the eigenvalue statistics of disordered conductors, and quantum diffusion.
To achieve these goals I will combine tools introduced in my previous work, such as local resampling of graphs and subdiagram resummation techniques, and in addition develop novel, robust techniques to address the more challenging goals. I expect the output of this proposal to contribute significantly to the understanding of structured random matrices.
Summary
The purpose of this proposal is a better mathematical understanding of certain classes of large random matrices. Up to very recently, random matrix theory has been mainly focused on mean-field models with independent entries. In this proposal I instead consider random matrices that incorporate some nontrivial structure. I focus on two types of structured random matrices that arise naturally in important applications and lead to a rich mathematical behaviour: (1) random graphs with fixed degrees, such as random regular graphs, and (2) random band matrices, which constitute a good model of disordered quantum Hamiltonians.
The goals are strongly motivated by the applications to spectral graph theory and quantum chaos for (1) and to the physics of conductance in disordered media for (2). Specifically, I will work in the following directions. First, derive precise bounds on the locations of the extremal eigenvalues and the spectral gap, ultimately obtaining their limiting distributions. Second, characterize the spectral statistics in the bulk of the spectrum, using both eigenvalue correlation functions on small scales and linear eigenvalue statistics on intermediate mesoscopic scales. Third, prove the delocalization of eigenvectors and derive the distribution of their components. These results will address several of the most important questions about the structured random matrices (1) and (2), such as expansion properties of random graphs, hallmarks of quantum chaos in random regular graphs, crossovers in the eigenvalue statistics of disordered conductors, and quantum diffusion.
To achieve these goals I will combine tools introduced in my previous work, such as local resampling of graphs and subdiagram resummation techniques, and in addition develop novel, robust techniques to address the more challenging goals. I expect the output of this proposal to contribute significantly to the understanding of structured random matrices.
Max ERC Funding
1 257 442 €
Duration
Start date: 2017-01-01, End date: 2021-12-31
