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 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 CRITIQUEUE
Project Critical queues and reflected stochastic processes
Researcher (PI) Johannes S.H. Van Leeuwaarden
Host Institution (HI) TECHNISCHE UNIVERSITEIT EINDHOVEN
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary Our primary motivation stems from queueing theory, the branch of applied probability that deals with congestion phenomena. Congestion levels are typically nonnegative, which is why reflected stochastic processes arise naturally in queueing theory. Other applications of reflected stochastic processes are in the fields of branching processes and random graphs.
We are particularly interested in critically-loaded queueing systems (close to 100% utilization), also referred to as queues in heavy traffic. Heavy-traffic analysis typically reduces complicated queueing processes to much simpler (reflected) limit processes or scaling limits. This makes the analysis of complex systems tractable, and from a mathematical point of view, these results are appealing since they can be made rigorous. Within the large
body of literature on heavy-traffic theory and critical stochastic processes, we launch two new research lines:
(i) Time-dependent analysis through scaling limits.
(ii) Dimensioning stochastic systems via refined scaling limits and optimization.
Both research lines involve mathematical techniques that combine stochastic theory with asymptotic theory, complex analysis, functional analysis, and modern probabilistic methods. It will provide a platform enabling collaborations between researchers in pure and applied probability and researchers in performance analysis of queueing systems. This will particularly be the case at TU/e, the host institution, and at
the affiliated institution EURANDOM.
Summary
Our primary motivation stems from queueing theory, the branch of applied probability that deals with congestion phenomena. Congestion levels are typically nonnegative, which is why reflected stochastic processes arise naturally in queueing theory. Other applications of reflected stochastic processes are in the fields of branching processes and random graphs.
We are particularly interested in critically-loaded queueing systems (close to 100% utilization), also referred to as queues in heavy traffic. Heavy-traffic analysis typically reduces complicated queueing processes to much simpler (reflected) limit processes or scaling limits. This makes the analysis of complex systems tractable, and from a mathematical point of view, these results are appealing since they can be made rigorous. Within the large
body of literature on heavy-traffic theory and critical stochastic processes, we launch two new research lines:
(i) Time-dependent analysis through scaling limits.
(ii) Dimensioning stochastic systems via refined scaling limits and optimization.
Both research lines involve mathematical techniques that combine stochastic theory with asymptotic theory, complex analysis, functional analysis, and modern probabilistic methods. It will provide a platform enabling collaborations between researchers in pure and applied probability and researchers in performance analysis of queueing systems. This will particularly be the case at TU/e, the host institution, and at
the affiliated institution EURANDOM.
Max ERC Funding
970 800 €
Duration
Start date: 2010-08-01, End date: 2016-07-31
Project acronym DIFFINCL
Project Differential Inclusions and Fluid Mechanics
Researcher (PI) Laszlo SZEKELYHIDI
Host Institution (HI) UNIVERSITAET LEIPZIG
Call Details Consolidator Grant (CoG), PE1, ERC-2016-COG
Summary Important problems in science often involve structures on several distinct length scales. Two typical examples are fine phase mixtures in solid-solid phase transitions and the complex mixing patterns in turbulent or multiphase flows. The microstructures in such situations influence in a crucial way the macroscopic behavior of the system, and understanding the formation, interaction and overall effect of these structures is a great scientific challenge. Although there is a large variety of models and descriptions for such phenomena, a recurring issue in the mathematical analysis is that one has to deal with very complex and highly non-smooth structures in solutions of the associated partial differential equations.
A common ground is provided by the analysis of differential inclusions, a theory whose development was strongly influenced by the influx of ideas from the work of Gromov on partial differential relations, building on celebrated constructions of Nash for isometric immersions, and the work of Tartar in the study of oscillation phenomena in nonlinear partial differential equations. A recent success of this approach is provided by my work on the h-principle in fluid mechanics and Onsager's conjecture. Against this background my aim in this project is to go significantly beyond the state of the art, both in terms of the methods and in terms of applications of differential inclusions. One part of the project is to continue my work on fluid mechanics with the ultimate goal to address important challenges in the field: providing an analytic foundation for the K41 statistical theory of turbulence and for the behavior of turbulent flows near instabilities and boundaries. A further aim is to explore rigidity phenomena and to attack several outstanding open problems in the context of differential inclusions, most prominently Morrey's conjecture on quasiconvexity and rank-one convexity.
Summary
Important problems in science often involve structures on several distinct length scales. Two typical examples are fine phase mixtures in solid-solid phase transitions and the complex mixing patterns in turbulent or multiphase flows. The microstructures in such situations influence in a crucial way the macroscopic behavior of the system, and understanding the formation, interaction and overall effect of these structures is a great scientific challenge. Although there is a large variety of models and descriptions for such phenomena, a recurring issue in the mathematical analysis is that one has to deal with very complex and highly non-smooth structures in solutions of the associated partial differential equations.
A common ground is provided by the analysis of differential inclusions, a theory whose development was strongly influenced by the influx of ideas from the work of Gromov on partial differential relations, building on celebrated constructions of Nash for isometric immersions, and the work of Tartar in the study of oscillation phenomena in nonlinear partial differential equations. A recent success of this approach is provided by my work on the h-principle in fluid mechanics and Onsager's conjecture. Against this background my aim in this project is to go significantly beyond the state of the art, both in terms of the methods and in terms of applications of differential inclusions. One part of the project is to continue my work on fluid mechanics with the ultimate goal to address important challenges in the field: providing an analytic foundation for the K41 statistical theory of turbulence and for the behavior of turbulent flows near instabilities and boundaries. A further aim is to explore rigidity phenomena and to attack several outstanding open problems in the context of differential inclusions, most prominently Morrey's conjecture on quasiconvexity and rank-one convexity.
Max ERC Funding
1 860 875 €
Duration
Start date: 2017-04-01, End date: 2022-03-31
Project acronym FLAT SURFACES
Project SL(2,R)-action on flat surfaces and geometry of extremal subvarieties of moduli spaces
Researcher (PI) Martin Moeller
Host Institution (HI) JOHANN WOLFGANG GOETHE-UNIVERSITATFRANKFURT AM MAIN
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary Dynamics on polygonal billiard tables is best understood by unfolding the table and studying the resulting flat surface. The moduli space of flat surfaces carries a natural action of SL(2,R) and all the questions about Lie group actions on homogeneous spaces reappear in this
non-homogeneous setting in an even more interesting way.
Closed SL(2,R)-orbits give rise to totally geodesic
subvarieties of the moduli space of curves, called
Teichmueller curves. Their classifcation is a major goal over the coming years. The applicant's algebraic characterization of Teichmueller curves plus the comprehension of the Deligne-Mumford compactification of Hilbert modular varities
make this goal feasible.
on polygonal billiard tables is best understood
unfolding the table and studying the resulting
surface. The moduli space of flat surfaces carries
action of SL(2,R) and all the questions about
group actions on homogeneous spaces reappear in this homogeneous setting in an even more interesting way.
SL(2,R)-orbits give rise to totally geodesic
of the moduli space of curves, called
curves. Their classifcation is a major goal
the coming years. The applicant's algebraic characterization Teichmueller curves plus the comprehension of the Mumford compactification of Hilbert modular varities this goal feasible.
Summary
Dynamics on polygonal billiard tables is best understood by unfolding the table and studying the resulting flat surface. The moduli space of flat surfaces carries a natural action of SL(2,R) and all the questions about Lie group actions on homogeneous spaces reappear in this
non-homogeneous setting in an even more interesting way.
Closed SL(2,R)-orbits give rise to totally geodesic
subvarieties of the moduli space of curves, called
Teichmueller curves. Their classifcation is a major goal over the coming years. The applicant's algebraic characterization of Teichmueller curves plus the comprehension of the Deligne-Mumford compactification of Hilbert modular varities
make this goal feasible.
on polygonal billiard tables is best understood
unfolding the table and studying the resulting
surface. The moduli space of flat surfaces carries
action of SL(2,R) and all the questions about
group actions on homogeneous spaces reappear in this homogeneous setting in an even more interesting way.
SL(2,R)-orbits give rise to totally geodesic
of the moduli space of curves, called
curves. Their classifcation is a major goal
the coming years. The applicant's algebraic characterization Teichmueller curves plus the comprehension of the Mumford compactification of Hilbert modular varities this goal feasible.
Max ERC Funding
1 005 600 €
Duration
Start date: 2010-10-01, End date: 2015-09-30
Project acronym HARG
Project Harmonic analysis on reductive groups
Researcher (PI) Eric Marcus Opdam
Host Institution (HI) UNIVERSITEIT VAN AMSTERDAM
Call Details Advanced Grant (AdG), PE1, ERC-2010-AdG_20100224
Summary We propose to attack a variety of fundamental open problems in
harmonic analysis on $p$-adic and real reductive groups.
Specifically we seek solutions to the local Langlands conjectures
and various normalization problems of discrete series representations.
For $p$-adic groups, affine Hecke algebras are a major technical tool.
Our understanding of these algebras with unequal parameters has
advanced recently and allows us to address these problems.
We will compute the Plancherel measure on the Bernstein components
explicitly. Using a new transfer principle of Plancherel measures
between Hecke algebras we will combine Bernstein components to form
$L$-packets, following earlier work of Reeder in small rank.
We start with the tamely ramified case, building on work of
Reeder-Debacker. We will also explore these methods for $L$-packets
of positive depth, using recent progress due to Yu and others.
Furthermore we intend to study non-tempered
unitary representations via affine Hecke algebras, extending the
work of Barbasch-Moy on the Iwahori spherical unitary dual.
As for real reductive groups we intend to address essential
questions on the convergence of the Fourier-transform. This theory
is widely developed for functions which transform finitely under a
maximal compact subgroup. We wish to drop this condition in order
to obtain global final statements for various classes of rapidly
decreasing functions. We intend to extend our results to certain types of
homogeneous spaces, e.g symmetric and multiplicity one spaces. For doing
so we will embark to develop a suitable spherical character theory for
discrete series representations and solve the corresponding normalization
problems.
The analytic nature of the Plancherel measure and the correct interpretation
thereof is the underlying theme which connects the various parts of
this proposal.
Summary
We propose to attack a variety of fundamental open problems in
harmonic analysis on $p$-adic and real reductive groups.
Specifically we seek solutions to the local Langlands conjectures
and various normalization problems of discrete series representations.
For $p$-adic groups, affine Hecke algebras are a major technical tool.
Our understanding of these algebras with unequal parameters has
advanced recently and allows us to address these problems.
We will compute the Plancherel measure on the Bernstein components
explicitly. Using a new transfer principle of Plancherel measures
between Hecke algebras we will combine Bernstein components to form
$L$-packets, following earlier work of Reeder in small rank.
We start with the tamely ramified case, building on work of
Reeder-Debacker. We will also explore these methods for $L$-packets
of positive depth, using recent progress due to Yu and others.
Furthermore we intend to study non-tempered
unitary representations via affine Hecke algebras, extending the
work of Barbasch-Moy on the Iwahori spherical unitary dual.
As for real reductive groups we intend to address essential
questions on the convergence of the Fourier-transform. This theory
is widely developed for functions which transform finitely under a
maximal compact subgroup. We wish to drop this condition in order
to obtain global final statements for various classes of rapidly
decreasing functions. We intend to extend our results to certain types of
homogeneous spaces, e.g symmetric and multiplicity one spaces. For doing
so we will embark to develop a suitable spherical character theory for
discrete series representations and solve the corresponding normalization
problems.
The analytic nature of the Plancherel measure and the correct interpretation
thereof is the underlying theme which connects the various parts of
this proposal.
Max ERC Funding
1 769 000 €
Duration
Start date: 2011-03-01, End date: 2016-02-29
Project acronym MODSIMCONMP
Project Modeling, Simulation and Control of Multi-Physics Systems
Researcher (PI) Volker Mehrmann
Host Institution (HI) TECHNISCHE UNIVERSITAT BERLIN
Call Details Advanced Grant (AdG), PE1, ERC-2010-AdG_20100224
Summary This proposal is aimed at developing and analyzing a fundamentally new interdisciplinary approach for the modeling, simulation, control and optimization of multi-physics, multi-scale dynamical systems. The new innovative feature is to generate models via a network of modularized uni-physics components, where each component incorporates a mathematical model for the dynamical behavior as well as a model for the uncertainties, e.g. due to modeling, discretization or finite precision computation errors. Based on this new modeling concept also new numerical simulation, control, and optimization techniques will be developed and incorporated, that allow a systematic adaptive error control (including the appropriate treatment of different scales,and the uncertainties) for the components as well as for the whole multi-physics model. In order to cope with the differential-algebraic and multi-scale character of the systems we will develop and analyze remodeling techniques for the components as well as for the whole network including the uncertainties and special structures. The new remodeled systems will be designed such that they allow an efficient and accurate dynamical simulation with high order numerical integration techniques as well as efficient methods for model reduction and open and closed loop control.
In an interdisciplinary corporation with colleagues from computer science and engineering we will extend the modeling language MODELICA to be able to incorporate the new features (in particular the uncertainties and modeling errors) and we also plan to implement the complete approach as a new software platform.
Summary
This proposal is aimed at developing and analyzing a fundamentally new interdisciplinary approach for the modeling, simulation, control and optimization of multi-physics, multi-scale dynamical systems. The new innovative feature is to generate models via a network of modularized uni-physics components, where each component incorporates a mathematical model for the dynamical behavior as well as a model for the uncertainties, e.g. due to modeling, discretization or finite precision computation errors. Based on this new modeling concept also new numerical simulation, control, and optimization techniques will be developed and incorporated, that allow a systematic adaptive error control (including the appropriate treatment of different scales,and the uncertainties) for the components as well as for the whole multi-physics model. In order to cope with the differential-algebraic and multi-scale character of the systems we will develop and analyze remodeling techniques for the components as well as for the whole network including the uncertainties and special structures. The new remodeled systems will be designed such that they allow an efficient and accurate dynamical simulation with high order numerical integration techniques as well as efficient methods for model reduction and open and closed loop control.
In an interdisciplinary corporation with colleagues from computer science and engineering we will extend the modeling language MODELICA to be able to incorporate the new features (in particular the uncertainties and modeling errors) and we also plan to implement the complete approach as a new software platform.
Max ERC Funding
1 899 924 €
Duration
Start date: 2011-04-01, End date: 2016-03-31
Project acronym PEPCo
Project Problems in Extremal and Probabilistic Combinatorics
Researcher (PI) Mathias DR. SCHACHT
Host Institution (HI) UNIVERSITAET HAMBURG
Call Details Consolidator Grant (CoG), PE1, ERC-2016-COG
Summary Extremal and probabilistic combinatorics is a central and currently maybe the most active and fastest growing area in discrete mathematics. The field can be traced back to the work of Turán and it was established by Erdős through his fundamental contributions and his uncounted guiding questions. Since then it has grown into an important discipline with strong ties to other mathematical areas such as theoretical computer science, number theory, and ergodic theory.
The PI proposes a variety of extremal problems for hypergraphs and for sparse random and pseudorandom graphs. The work for hypergraphs is motivated by Turán’s problem, maybe the most prominent open problem in the area. After solving an analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of three-element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem seems to be out of reach by our current methods and despite a great deal of effort over the last 70 years, our knowledge is still very limited.
We suggest a variant of the problem by imposing additional restrictions on the distribution of the three-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and hopefully more manageable subproblems, some of which were already considered by Erdős and Sós. We suggest a unifying framework for these problems and one of the main goals would be the development of new techniques for this type of problems. These additional assumptions on the hyperedge distribution are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extension by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research.
Summary
Extremal and probabilistic combinatorics is a central and currently maybe the most active and fastest growing area in discrete mathematics. The field can be traced back to the work of Turán and it was established by Erdős through his fundamental contributions and his uncounted guiding questions. Since then it has grown into an important discipline with strong ties to other mathematical areas such as theoretical computer science, number theory, and ergodic theory.
The PI proposes a variety of extremal problems for hypergraphs and for sparse random and pseudorandom graphs. The work for hypergraphs is motivated by Turán’s problem, maybe the most prominent open problem in the area. After solving an analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of three-element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem seems to be out of reach by our current methods and despite a great deal of effort over the last 70 years, our knowledge is still very limited.
We suggest a variant of the problem by imposing additional restrictions on the distribution of the three-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and hopefully more manageable subproblems, some of which were already considered by Erdős and Sós. We suggest a unifying framework for these problems and one of the main goals would be the development of new techniques for this type of problems. These additional assumptions on the hyperedge distribution are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extension by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research.
Max ERC Funding
1 800 000 €
Duration
Start date: 2017-10-01, End date: 2022-09-30
Project acronym QC&C
Project Quantum fields and Curvature--Novel Constructive Approach via Operator Product Expansion
Researcher (PI) Stefan Hollands
Host Institution (HI) UNIVERSITAET LEIPZIG
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary It was realized already from the beginning that the theory of quantized fields (QFT) does not easily fit into known mathematical structures, and the quest for a satisfactory mathematical foundation continues to-day. Parts of this theory have already been tremendously successful, e.g. in the quantitative description of elementary particles, and ideas from QFT have revolutionized entire fields of mathematics. But the non-perturbative construction of the most important QFT s, namely renormalizable theories in 4d, remains unsolved. The aim of this project is to make a substantial contribution to this quest for the mathematical construction of such QFT s (on curved manifolds), and the exploration of their mathematical structure. We want to pursue a novel ansatz to achieve this goal. The essence of our novel approach is to focus attention on the algebraic backbone of the theory, which manifests itself in the so-called operator-product-expansion. The study of such algebraic structures related to operator products has already been tremendously useful in the study of conformal field theories in low dimensions, but we here propose that a suitable version of it also has great potential to be used as a constructive tool for the much more complicated quantum gauge theories in four dimensions. It is not expected that an explicit solution can be obtained for such models-especially so in curved space-but the idea is instead to analyze powerful consistency conditions on the quantum field theory arising from the OPE ( associativity conditions ) and to use them to prove that the theory exists in a mathematically rigorous sense. Our approach will be complemented by other powerful and deep mathematical tools that have been developed over the past decades, such as the sophisticated non-perturbative expansions uncovered in the school of constructive quantum fields theory , Hochschild cohomology, RG-flow equation techniques, microlocal analysis, curvature expansions, and many more.
Summary
It was realized already from the beginning that the theory of quantized fields (QFT) does not easily fit into known mathematical structures, and the quest for a satisfactory mathematical foundation continues to-day. Parts of this theory have already been tremendously successful, e.g. in the quantitative description of elementary particles, and ideas from QFT have revolutionized entire fields of mathematics. But the non-perturbative construction of the most important QFT s, namely renormalizable theories in 4d, remains unsolved. The aim of this project is to make a substantial contribution to this quest for the mathematical construction of such QFT s (on curved manifolds), and the exploration of their mathematical structure. We want to pursue a novel ansatz to achieve this goal. The essence of our novel approach is to focus attention on the algebraic backbone of the theory, which manifests itself in the so-called operator-product-expansion. The study of such algebraic structures related to operator products has already been tremendously useful in the study of conformal field theories in low dimensions, but we here propose that a suitable version of it also has great potential to be used as a constructive tool for the much more complicated quantum gauge theories in four dimensions. It is not expected that an explicit solution can be obtained for such models-especially so in curved space-but the idea is instead to analyze powerful consistency conditions on the quantum field theory arising from the OPE ( associativity conditions ) and to use them to prove that the theory exists in a mathematically rigorous sense. Our approach will be complemented by other powerful and deep mathematical tools that have been developed over the past decades, such as the sophisticated non-perturbative expansions uncovered in the school of constructive quantum fields theory , Hochschild cohomology, RG-flow equation techniques, microlocal analysis, curvature expansions, and many more.
Max ERC Funding
818 099 €
Duration
Start date: 2011-04-01, End date: 2016-03-31
