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 INDEX
Project Rigidity of groups and higher index theory
Researcher (PI) Piotr Wojciech Nowak
Host Institution (HI) INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Call Details Starting Grant (StG), PE1, ERC-2015-STG
Summary The Atiyah-Singer index theorem was one of the most spectacular achievements of mathematics in the XXth century, connecting the analytic and topological properties of manifolds. The Baum-Connes conjecture is a hugely successful approach to generalizing the index theorem to a much broader setting. It has remarkable applications in topology and analysis. For instance, it implies the Novikov conjecture on the homotopy invariance of higher signatures of a closed manifold and the Kaplansky-Kadison conjecture on the existence of non-trivial idempotents in the reduced group C*-algebra of a torsion-free group. At present, the Baum-Connes conjecture is known to hold for a large class of groups, including groups admitting metrically proper isometric actions on Hilbert spaces and Gromov hyperbolic groups.
The Baum-Connes conjecture with certain coefficients is known to fail for a class of groups, whose Cayley graphs contain coarsely embedded expander graphs. Nevertheless, the conjecture in full generality remains open and there is a growing need for new examples of groups and group actions, that would be counterexamples to the Baum-Connes conjecture. The main objective of this project is to exhibit such examples.
Our approach relies on strengthening Kazhdan’s property (T), a prominent cohomological rigidity property, from its original setting of Hilbert spaces to much larger classes of Banach spaces. Such properties are an emerging direction in the study of cohomological rigidity and are not yet well-understood. They lie at the intersection of geometric group theory, non-commutative geometry and index theory. In their study we will implement novel approaches, combining geometric and analytic techniques with variety of new cohomological constructions.
Summary
The Atiyah-Singer index theorem was one of the most spectacular achievements of mathematics in the XXth century, connecting the analytic and topological properties of manifolds. The Baum-Connes conjecture is a hugely successful approach to generalizing the index theorem to a much broader setting. It has remarkable applications in topology and analysis. For instance, it implies the Novikov conjecture on the homotopy invariance of higher signatures of a closed manifold and the Kaplansky-Kadison conjecture on the existence of non-trivial idempotents in the reduced group C*-algebra of a torsion-free group. At present, the Baum-Connes conjecture is known to hold for a large class of groups, including groups admitting metrically proper isometric actions on Hilbert spaces and Gromov hyperbolic groups.
The Baum-Connes conjecture with certain coefficients is known to fail for a class of groups, whose Cayley graphs contain coarsely embedded expander graphs. Nevertheless, the conjecture in full generality remains open and there is a growing need for new examples of groups and group actions, that would be counterexamples to the Baum-Connes conjecture. The main objective of this project is to exhibit such examples.
Our approach relies on strengthening Kazhdan’s property (T), a prominent cohomological rigidity property, from its original setting of Hilbert spaces to much larger classes of Banach spaces. Such properties are an emerging direction in the study of cohomological rigidity and are not yet well-understood. They lie at the intersection of geometric group theory, non-commutative geometry and index theory. In their study we will implement novel approaches, combining geometric and analytic techniques with variety of new cohomological constructions.
Max ERC Funding
880 625 €
Duration
Start date: 2016-08-01, End date: 2021-07-31
