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 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 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 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 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 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
