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Herbstsemester 2019

Datum / Zeit Referent:in Titel Ort
25. September 2019
10:15-11:45 		Pavel Safronov
University of Zurich
Details

Talks in Mathematical Physics

Titel Finite-dimensionality of skein modules
Referent:in, Affiliation Pavel Safronov, University of Zurich
Datum, Zeit 25. September 2019, 10:15-11:45
Ort Y27 H 25
Abstract In this talk I will explain what skein modules of 3-manifolds are and how they fit in the framework of topological field theory and factorization homology. I will also outline a proof that skein modules of closed 3-manifolds are finite-dimensional, proving a conjecture of Witten. This is joint work with Sam Gunningham and David Jordan
Finite-dimensionality of skein modulesread_more
Y27 H 25
25. September 2019
13:00-14:30 		Paul Wedrich
Australian National University
Details

Talks in Mathematical Physics

Titel Invariants of 4-manifolds from Khovanov-Rozansky link homology
Referent:in, Affiliation Paul Wedrich, Australian National University
Datum, Zeit 25. September 2019, 13:00-14:30
Ort Y35 F 08a
Abstract Ribbon categories are 3-dimensional algebraic structures that control quantum link polynomials and that give rise to 3-manifold invariants known as skein modules. I will describe how to use Khovanov-Rozansky link homology, a categorification of the gl(N) quantum link polynomial, to obtain a 4-dimensional algebraic structure that gives rise to vector space-valued invariants of smooth 4-manifolds. The technical heart of this construction is the newly established functoriality of Khovanov-Rozansky homology in the 3-sphere.
Invariants of 4-manifolds from Khovanov-Rozansky link homologyread_more
Y35 F 08a
26. September 2019
14:00-15:00 		Ivan Cherednik
ETH-ITS and UNC Chapel Hill
Details

Talks in Mathematical Physics

Titel Torus iterated links via double affine Hecke algebras I
Referent:in, Affiliation Ivan Cherednik, ETH-ITS and UNC Chapel Hill
Datum, Zeit 26. September 2019, 14:00-15:00
Ort HG G 43
Abstract This mini-course (3 lectures) is devoted to the recent construction of DAHA invariants of torus iterated links (including all algebraic links). Presumably they coincide with reduced stable Khovanov-Rozansky polynomials for such links, though the theory of the latter mostly exists by now in the uncolored, non-reduced setting and for knots (not for links); also these polynomials are generally quite difficult to calculate topologically. The DAHA approach works for any colored iterated links (and for any root systems, not only in type A); also they are known to coincide with the corresponding HOMFLY-PT polynomials as t=q. The explicit (colored) formulas are calculated well beyond torus knots. The DAHA construction connects invariants of algebraic links with the physics approaches, based on Verlinde algebras (for torus knots) and via M-theory in the refined case (for any q,t). Importantly, DAHA invariants are connected (conjecturally) with zeta-functions of plane curve singularities. However the focus will be on the DAHA approach, with all necessary tools and definitions, some proofs, and various examples. No prior knowledge of DAHA is assumed. The HOMFLY-PT polynomials will appear only to state the connection results; they are not actually needed in this course. Motivic zeta-functions and physics links will not be discussed. The course is designed to be understandable for (motivated) students. We will begin with the definition of (uncolored) HOMFLY-PT polynomials and the key facts concerning DAHA for sl(2) and in type A.
Torus iterated links via double affine Hecke algebras Iread_more
HG G 43
26. September 2019
15:15-16:15 		Mikhail Isachenkov
IHES
Details

Talks in Mathematical Physics

Titel Calogero-Sutherland-type models in d>2 conformal field theory
Referent:in, Affiliation Mikhail Isachenkov, IHES
Datum, Zeit 26. September 2019, 15:15-16:15
Ort HG G 43
Abstract Conformal bootstrap programme in the spacetime dimensions d>2 has seen a new upsurge of activity in the last decade. I will attempt at briefly reviewing its current state and then switch to the connection between important constituents of the bootstrap method, conformal blocks, and the Calogero-Sutherland-type integrable models. It goes via harmonic analysis on the conformal group and paves the way for using double affine Hecke algebras and the related theory of special functions in the study of conformal field theories.
Calogero-Sutherland-type models in d>2 conformal field theoryread_more
HG G 43
3. Oktober 2019
14:00-15:00 		Ivan Cherednik
ETH-ITS and UNC Chapel Hill
Details

Talks in Mathematical Physics

Titel Torus iterated links via double affine Hecke algebras II
Referent:in, Affiliation Ivan Cherednik, ETH-ITS and UNC Chapel Hill
Datum, Zeit 3. Oktober 2019, 14:00-15:00
Ort HG G 43
Abstract This mini-course (3 lectures) is devoted to the recent construction of DAHA invariants of torus iterated links (including all algebraic links). Presumably they coincide with reduced stable Khovanov-Rozansky polynomials for such links, though the theory of the latter mostly exists by now in the uncolored, non-reduced setting and for knots (not for links); also these polynomials are generally quite difficult to calculate topologically. The DAHA approach works for any colored iterated links (and for any root systems, not only in type A); also they are known to coincide with the corresponding HOMFLY-PT polynomials as t=q. The explicit (colored) formulas are calculated well beyond torus knots. The DAHA construction connects invariants of algebraic links with the physics approaches, based on Verlinde algebras (for torus knots) and via M-theory in the refined case (for any q,t). Importantly, DAHA invariants are connected (conjecturally) with zeta-functions of plane curve singularities. However the focus will be on the DAHA approach, with all necessary tools and definitions, some proofs, and various examples. No prior knowledge of DAHA is assumed. The HOMFLY-PT polynomials will appear only to state the connection results; they are not actually needed in this course. Motivic zeta-functions and physics links will not be discussed. The course is designed to be understandable for (motivated) students. We will begin with the definition of (uncolored) HOMFLY-PT polynomials and the key facts concerning DAHA for sl(2) and in type A.
Torus iterated links via double affine Hecke algebras IIread_more
HG G 43
10. Oktober 2019
14:00-15:00 		Ivan Cherednik
ETH-ITS and UNC Chapel Hill
Details

Talks in Mathematical Physics

Titel Torus iterated links via double affine Hecke algebras III
Referent:in, Affiliation Ivan Cherednik, ETH-ITS and UNC Chapel Hill
Datum, Zeit 10. Oktober 2019, 14:00-15:00
Ort HG G 43
Abstract This mini-course (3 lectures) is devoted to the recent construction of DAHA invariants of torus iterated links (including all algebraic links). Presumably they coincide with reduced stable Khovanov-Rozansky polynomials for such links, though the theory of the latter mostly exists by now in the uncolored, non-reduced setting and for knots (not for links); also these polynomials are generally quite difficult to calculate topologically. The DAHA approach works for any colored iterated links (and for any root systems, not only in type A); also they are known to coincide with the corresponding HOMFLY-PT polynomials as t=q. The explicit (colored) formulas are calculated well beyond torus knots. The DAHA construction connects invariants of algebraic links with the physics approaches, based on Verlinde algebras (for torus knots) and via M-theory in the refined case (for any q,t). Importantly, DAHA invariants are connected (conjecturally) with zeta-functions of plane curve singularities. However the focus will be on the DAHA approach, with all necessary tools and definitions, some proofs, and various examples. No prior knowledge of DAHA is assumed. The HOMFLY-PT polynomials will appear only to state the connection results; they are not actually needed in this course. Motivic zeta-functions and physics links will not be discussed. The course is designed to be understandable for (motivated) students. We will begin with the definition of (uncolored) HOMFLY-PT polynomials and the key facts concerning DAHA for sl(2) and in type A.
Torus iterated links via double affine Hecke algebras IIIread_more
HG G 43
24. Oktober 2019
15:15-16:15 		Xiaomeng Xu
ETH Zurich
Details

Talks in Mathematical Physics

Titel Stokes phenomenon and its applications in mathematical physics
Referent:in, Affiliation Xiaomeng Xu, ETH Zurich
Datum, Zeit 24. Oktober 2019, 15:15-16:15
Ort HG G 43
Abstract This talk includes an introduction to the Stokes phenomenon of meromorphic ODEs with singularities, and then explores its relations with Yang-Baxter equations and integrable systems.
Stokes phenomenon and its applications in mathematical physicsread_more
HG G 43
31. Oktober 2019
15:15-16:15 		Donald Youmans
Université de Genève
Details

Talks in Mathematical Physics

Titel Two-dimensional BF theory as a CFT
Referent:in, Affiliation Donald Youmans, Université de Genève
Datum, Zeit 31. Oktober 2019, 15:15-16:15
Ort HG G 43
Abstract Two-dimensional BF theory is an example of a topological gauge theory. Imposing the Lorenz gauge-fixing condition introduces an auxiliary geometric datum in terms of a metric. We will show that the theory is topological conformal, i.e. it depends only on the conformal structure of the introduced metric. Moreover, the stress-energy tensor is Q-exact (hence vanishes in Q-cohomology and therefore on physical states). We will then study the consequences of Q-exactness and possible constructions involving the Q-primitive of the stress-energy tensor, such as topological correlation functions, a BV algebra structure on the Q-cohomology (space of physical observables) and deformations in the space of topological conformal field theories. This is a joint work with Andrey Losev (University Higher School of Economic, Moscow) and Pavel Mnev (University of Notre Dame, USA)
Two-dimensional BF theory as a CFTread_more
HG G 43
7. November 2019
14:00-15:00 		Nicolai Reshetikhin
ETH-ITS and University of California, Berkeley
Details

Talks in Mathematical Physics

Titel Integrable and superintegrable systems on moduli spaces of flat connections I
Referent:in, Affiliation Nicolai Reshetikhin, ETH-ITS and University of California, Berkeley
Datum, Zeit 7. November 2019, 14:00-15:00
Ort HG G 43
Abstract Let G be a simple Lie group. For a compact topological surface the moduli space of flat connections has a natural symplectic structure when the surface is closed and a Poisson structure when the surface has boundary. Symplectic leaves of this Poisson variety are parametrized by conjugacy classes of monodromies along connected components of the boundary. The goal of these lectures is to explain that central functions on holonomies along any simple collection of curves on a surface define a superintegrable system and to describe these systems in some special cases. The first lecture will focus on the definition and properties of superintegrable systems of moduli spaces of flat connections. In the second lecture superintegrable systems corresponding to simple curves on a surface will be introduced. In the third lecture the relation between these systems and various versions of spin Calogero-Moser systems will be explained.
Integrable and superintegrable systems on moduli spaces of flat connections Iread_more
HG G 43
7. November 2019
15:15-16:15 		Batu Güneysu
Universität Bonn
Details

Talks in Mathematical Physics

Titel The Chern Character of theta-summable Fredholm Modules over dg Algebras and Localization on Loop Space
Referent:in, Affiliation Batu Güneysu, Universität Bonn
Datum, Zeit 7. November 2019, 15:15-16:15
Ort HG G 43
Abstract I am going to explain the notion of a theta-summable Fredholm module over a locally convex dg algebra $\Omega$ and how one constructs its Chern character as a cocycle on the entire cyclic complex of $\Omega$. This extends the construction of Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this Chern character, we prove an index theorem involving an abstract version of a Bismut-Chern character, first constructed by Getzler, Jones and Petrack in the context of loop spaces. This theory leads to a rigorous construction of the path integral for N=1/2 supersymmetry which satisfies a Duistermaat-Heckman type localization formula on loop space, solving a long standing open problem.
The Chern Character of theta-summable Fredholm Modules over dg Algebras and Localization on Loop Spaceread_more
HG G 43
8. November 2019
14:15-15:15 		Jasper Stokman
University of Amsterdam
Details

Talks in Mathematical Physics

Titel Formal spherical and boundary correlation functions
Referent:in, Affiliation Jasper Stokman, University of Amsterdam
Datum, Zeit 8. November 2019, 14:15-15:15
Ort CLV B 4
Clausiusstrasse 47
Formal spherical and boundary correlation functions
CLV B 4
Clausiusstrasse 47
14. November 2019
14:00-15:00 		Nicolai Reshetikhin
ETH-ITS and University of California, Berkeley
Details

Talks in Mathematical Physics

Titel Integrable and superintegrable systems on moduli spaces of flat connections II
Referent:in, Affiliation Nicolai Reshetikhin, ETH-ITS and University of California, Berkeley
Datum, Zeit 14. November 2019, 14:00-15:00
Ort HG G 43
Abstract Let G be a simple Lie group. For a compact topological surface the moduli space of flat connections has a natural symplectic structure when the surface is closed and a Poisson structure when the surface has boundary. Symplectic leaves of this Poisson variety are parametrized by conjugacy classes of monodromies along connected components of the boundary. The goal of these lectures is to explain that central functions on holonomies along any simple collection of curves on a surface define a superintegrable system and to describe these systems in some special cases. The first lecture will focus on the definition and properties of superintegrable systems of moduli spaces of flat connections. In the second lecture superintegrable systems corresponding to simple curves on a surface will be introduced. In the third lecture the relation between these systems and various versions of spin Calogero-Moser systems will be explained.
Integrable and superintegrable systems on moduli spaces of flat connections IIread_more
HG G 43
14. November 2019
15:15-16:15 		Rinat Kedem
University of Illinois
Details

Talks in Mathematical Physics

Titel Quantum Q-system algebras and Macdonald theory
Referent:in, Affiliation Rinat Kedem, University of Illinois
Datum, Zeit 14. November 2019, 15:15-16:15
Ort HG G 43
Abstract Quantum Q-systems are quantizations of the Q-system cluster algebras, and are responsible for the graded tensor products (“Feigin-Loktev fusion products’’) of current algebra modules. They can be viewed as discrete integrable systems whose conserved quantities are difference Toda hamiltonians. There is a representation of the generators of the algebra via q-difference operators, a generalization of degenerate Macdonald operators. This gives a relation between the quantum Q-system algebra and spherical nil DAHAs. Joint work with Philippe Di Francesco.
Quantum Q-system algebras and Macdonald theoryread_more
HG G 43
21. November 2019
14:00-15:00 		Nicolai Reshetikhin
ETH-ITS and University of California, Berkeley
Details

Talks in Mathematical Physics

Titel Integrable and superintegrable systems on moduli spaces of flat connections III
Referent:in, Affiliation Nicolai Reshetikhin, ETH-ITS and University of California, Berkeley
Datum, Zeit 21. November 2019, 14:00-15:00
Ort HG G 43
Abstract Let G be a simple Lie group. For a compact topological surface the moduli space of flat connections has a natural symplectic structure when the surface is closed and a Poisson structure when the surface has boundary. Symplectic leaves of this Poisson variety are parametrized by conjugacy classes of monodromies along connected components of the boundary. The goal of these lectures is to explain that central functions on holonomies along any simple collection of curves on a surface define a superintegrable system and to describe these systems in some special cases. The first lecture will focus on the definition and properties of superintegrable systems of moduli spaces of flat connections. In the second lecture superintegrable systems corresponding to simple curves on a surface will be introduced. In the third lecture the relation between these systems and various versions of spin Calogero-Moser systems will be explained.
Integrable and superintegrable systems on moduli spaces of flat connections IIIread_more
HG G 43
22. November 2019
15:15-16:15 		Theodore Voronov
University of Manchester
Details

Talks in Mathematical Physics

Titel Thick morphisms and spinor representation
Referent:in, Affiliation Theodore Voronov, University of Manchester
Datum, Zeit 22. November 2019, 15:15-16:15
Ort Y27 H 28
Abstract "Thick" morphisms (also called microformal morphisms) between manifolds or supermanifolds are a generalization of smooth maps --- and which are not usual maps, but rather relations between the corresponding cotangent bundles equipped with some extra data. They were discovered for the purpose of constructing L-infinity morphisms of higher (homotopy) brackets, when manifolds in question have an S-infinity ("homotopy Schouten") or P-infinity ("homotopy Poisson") structure. The key feature of thick morphisms is that they induce NONLINEAR, in general, pullbacks on functions. (This nonlinearity is exactly the feature making them useful for homotopy brackets purposes.) It was also found that there is a "quantum version" of thick morphisms in the form of integral operators of special type. Such "quantum pullbacks" can be seen as a generalization of spinor representation, as it became clear recently. I will try to explain all that. (See arXiv:1409.6475,arXiv:1411.6720,arXiv:1903.02884, also arXiv:1506.02417, arXiv:1512.04163, arXiv:1710.04335; and arXiv:1909.00290 , the latter joint with H. Khudaverdian.)
Thick morphisms and spinor representationread_more
Y27 H 28
28. November 2019
14:00-15:00 		Yuri Berest
Cornell University
Details

Talks in Mathematical Physics

Titel Representation homology and contact homology of topological spaces I
Referent:in, Affiliation Yuri Berest, Cornell University
Datum, Zeit 28. November 2019, 14:00-15:00
Ort HG G 43
Abstract Representation homology is an algebraic homology theory associated with derived representation schemes, which are natural homological extensions of classical representation varieties. The subject may be plainly viewed as part of derived algebraic geometry; however, somewhat surprisingly, there are more elementary constructions. In the first half of this mini-course, we will give two constructions of representation homology of topological spaces: one in terms of simplicial groups and the other in terms of classical homological algebra in functor categories. The first construction is inspired by our earlier work on representation homology of algebras and the recent work of Galatius and Venkatesh on derived Galois deformation rings; the second - by the classical work of Connes on cyclic homology and the work of Loday and Pirashvili on higher Hochschild homology. Both constructions are quite simple and accessible to calculations: we demonstrate this by computing representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces and some 3-dimensional manifolds, such as link complements in R3 and some lens spaces. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in R3. In the second part of the course, we will discuss knot contact homology in the sense of L. Ng. This is an interesting geometric invariant of knots in R3 defined by Floer-theoretic counting of pseudoholomorphic disks in the cosphere bundle of a knot K in ST*R3. In recent years, this invariant has been extensively studied by means of symplectic geometry and symplectic topology in the works of Ng, Ekholm, Etnyre, Shende, Sullivan and others. We will give a new, purely algebraic construction of knot contact homology based on the homotopy theory of (small) dg categories. This algebraic construction is remarkably parallel to the construction of representation homology of link complements: based on this analogy, we will propose a natural extension of contact homology to arbitrary spaces.
Representation homology and contact homology of topological spaces Iread_more
HG G 43
28. November 2019
15:15-16:15 		Mikaela Iacobelli
ETH Zurich
Details

Talks in Mathematical Physics

Titel Well-posedness and singular limits for the VPME system
Referent:in, Affiliation Mikaela Iacobelli, ETH Zurich
Datum, Zeit 28. November 2019, 15:15-16:15
Ort HG G 43
Abstract The Vlasov-Poisson system with massless electrons (VPME) is widely used in plasma physics to model the evolution of ions in a plasma. It differs from the classical Vlasov-Poisson system (VP) in that the Poisson coupling has an exponential nonlinearity that creates several mathematical difficulties. We will discuss a recent result proving uniqueness for VPME in the class of solutions with bounded density, and global existence of solutions with bounded density for a general class of initial data, generalising to this setting all the previous results known for VP. Moreover we will talk about a mean field derivation of the VPME and a rigorous quasi neutral limit for initial data that are close to analytic data deriving the Kinetic Isothermal Euler (KIE) system from the VPME in dimensions d=1,2,3. Lastly, we combine these two singular limits in order to show how to obtain the KIE system from an underlying particle system.
Well-posedness and singular limits for the VPME systemread_more
HG G 43
5. Dezember 2019
14:00-15:00 		Yuri Berest
Cornell University
Details

Talks in Mathematical Physics

Titel Representation homology and contact homology of topological spaces II
Referent:in, Affiliation Yuri Berest, Cornell University
Datum, Zeit 5. Dezember 2019, 14:00-15:00
Ort HG G 43
Abstract Representation homology is an algebraic homology theory associated with derived representation schemes, which are natural homological extensions of classical representation varieties. The subject may be plainly viewed as part of derived algebraic geometry; however, somewhat surprisingly, there are more elementary constructions. In the first half of this mini-course, we will give two constructions of representation homology of topological spaces: one in terms of simplicial groups and the other in terms of classical homological algebra in functor categories. The first construction is inspired by our earlier work on representation homology of algebras and the recent work of Galatius and Venkatesh on derived Galois deformation rings; the second - by the classical work of Connes on cyclic homology and the work of Loday and Pirashvili on higher Hochschild homology. Both constructions are quite simple and accessible to calculations: we demonstrate this by computing representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces and some 3-dimensional manifolds, such as link complements in R3 and some lens spaces. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in R3. In the second part of the course, we will discuss knot contact homology in the sense of L. Ng. This is an interesting geometric invariant of knots in R3 defined by Floer-theoretic counting of pseudoholomorphic disks in the cosphere bundle of a knot K in ST*R3. In recent years, this invariant has been extensively studied by means of symplectic geometry and symplectic topology in the works of Ng, Ekholm, Etnyre, Shende, Sullivan and others. We will give a new, purely algebraic construction of knot contact homology based on the homotopy theory of (small) dg categories. This algebraic construction is remarkably parallel to the construction of representation homology of link complements: based on this analogy, we will propose a natural extension of contact homology to arbitrary spaces.
Representation homology and contact homology of topological spaces IIread_more
HG G 43
* 11. Dezember 2019
10:15-11:45 		Michael Ehrig
Beijing Institute of Technology
Details

Talks in Mathematical Physics

Titel Lie Superalgebras via Schur-Weyl duality and Categorification
Referent:in, Affiliation Michael Ehrig, Beijing Institute of Technology
Datum, Zeit 11. Dezember 2019, 10:15-11:45
Ort Y27 H 28
Abstract In this talk, I will outline an approach to understand and describe the category of finite dimensional representations of a classical Lie superalgebra. Due to the non semi simplicity, methods different from the ones for semi-simple Lie algebras need to be applied to describe this category. Using variations of Schur-Weyl duality, respectively the fundamental theorems of invariant theory, we formulate the problem of understanding it in terms of centralizer algebras. These centralizer algebras are then described via methods from categorification of quantum groups and link invariants, yielding the description of the category of finite dimensional representations for some of the classical Lie superalgebras. This is joint work with Catharina Stroppel.
Lie Superalgebras via Schur-Weyl duality and Categorificationread_more
Y27 H 28
* 11. Dezember 2019
13:00-14:30 		Andrew Mathas
University of Sydney
Details

Talks in Mathematical Physics

Titel An overview of the representation theory of the symmetric groups
Referent:in, Affiliation Andrew Mathas, University of Sydney
Datum, Zeit 11. Dezember 2019, 13:00-14:30
Ort Y23 G 04
Abstract A representation of the symmetric group is a vector space together with an action of the group. Studying these representations is a rich beautiful subject, with its roots in combinatorial and with connections to areas ranging from general linear groups, to Kac-Moody algebras, 2-category theory, knot theory, statistical mechanics and Markov traces. This talk will start with results from the classical representation theory and a summary of the main open problems in the field. We then slowly build towards some recent exciting developments with the aim of describing the Ariki-Brundan-Kleshchev categorification theorem. Roughly speaking, this result shows that the representation theory of the symmetric groups is intimately connected with the highest weight representation theory of the affine special linear groups.
An overview of the representation theory of the symmetric groupsread_more
Y23 G 04
12. Dezember 2019
14:00-15:00 		Yuri Berest
Cornell University
Details

Talks in Mathematical Physics

Titel Representation homology and contact homology of topological spaces III
Referent:in, Affiliation Yuri Berest, Cornell University
Datum, Zeit 12. Dezember 2019, 14:00-15:00
Ort HG G 43
Abstract Representation homology is an algebraic homology theory associated with derived representation schemes, which are natural homological extensions of classical representation varieties. The subject may be plainly viewed as part of derived algebraic geometry; however, somewhat surprisingly, there are more elementary constructions. In the first half of this mini-course, we will give two constructions of representation homology of topological spaces: one in terms of simplicial groups and the other in terms of classical homological algebra in functor categories. The first construction is inspired by our earlier work on representation homology of algebras and the recent work of Galatius and Venkatesh on derived Galois deformation rings; the second - by the classical work of Connes on cyclic homology and the work of Loday and Pirashvili on higher Hochschild homology. Both constructions are quite simple and accessible to calculations: we demonstrate this by computing representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces and some 3-dimensional manifolds, such as link complements in R3 and some lens spaces. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in R3. In the second part of the course, we will discuss knot contact homology in the sense of L. Ng. This is an interesting geometric invariant of knots in R3 defined by Floer-theoretic counting of pseudoholomorphic disks in the cosphere bundle of a knot K in ST*R3. In recent years, this invariant has been extensively studied by means of symplectic geometry and symplectic topology in the works of Ng, Ekholm, Etnyre, Shende, Sullivan and others. We will give a new, purely algebraic construction of knot contact homology based on the homotopy theory of (small) dg categories. This algebraic construction is remarkably parallel to the construction of representation homology of link complements: based on this analogy, we will propose a natural extension of contact homology to arbitrary spaces.
Representation homology and contact homology of topological spaces IIIread_more
HG G 43
* 13. Dezember 2019
11:15-12:15 		Kenji Iohara
Université de Lyon
Details

Talks in Mathematical Physics

Titel Around the Temperley-Lieb algebras
Referent:in, Affiliation Kenji Iohara, Université de Lyon
Datum, Zeit 13. Dezember 2019, 11:15-12:15
Ort HG G 43
Abstract We first recall some facts about the Temperley-Lieb algebras of type A and B, together with their representations. In particular, the Schur-Weyl type duality and its application will be treated.
Around the Temperley-Lieb algebrasread_more
HG G 43
18. Dezember 2019
13:00-15:00 		Jürg Fröhlich
ETH Zurich
Details

Talks in Mathematical Physics

Titel What does Quantum Mechanics mean, after all?
Referent:in, Affiliation Jürg Fröhlich, ETH Zurich
Datum, Zeit 18. Dezember 2019, 13:00-15:00
Ort Y13 M 12
What does Quantum Mechanics mean, after all?
Y13 M 12
19. Dezember 2019
15:15-16:15 		Yuri Berest
Cornell University
Details

Talks in Mathematical Physics

Titel Perverse sheaves, contact homology and cubical approximations
Referent:in, Affiliation Yuri Berest, Cornell University
Datum, Zeit 19. Dezember 2019, 15:15-16:15
Ort HG G 43
Abstract Knot contact homology is an interesting geometric invariant of a knot K in R3 defined by Floer-theoretic counting of pseudoholomorphic disks in the sphere conormal bundle of K in T∗R3. In its simplest form, this invariant was introduced by L. Ng and has been extensively studied in recent years by means of symplectic geometry and topology. In this talk, we will give a purely algebraic construction of knot contact homology based on the homotopy theory of (small) dg categories. For a link L in R3, we define a dg k-category AL with a distinguished object, whose quasi-equivalence class is a topological invariant of L. In the case when L is a knot, the endomorphism algebra of the distinguished object of AL coincides with a geometric dg algebra model of the knot contact homology of L constructed by Ekholm, Etnyre, Ng and Sullivan (2013). The input of our construction is a natural action of the Artin braid group Bn on the category of perverse sheaves on a two-dimensional disk with singularities at n marked points, studied by Gelfand, MacPherson and Vilonen (1996). Time permitting, we will also discuss a possible generalization of contact homology to arbitrary spaces using the homotopy theory of cubical diagrams of simplicial sets.
Perverse sheaves, contact homology and cubical approximationsread_more
HG G 43
9. Januar 2020
15:15-16:45 		Pavel Mnev
University of Notre Dame
Details

Talks in Mathematical Physics

Titel Two-dimensional perturbative scalar field theory with polynomial potential and cutting-gluing
Referent:in, Affiliation Pavel Mnev, University of Notre Dame
Datum, Zeit 9. Januar 2020, 15:15-16:45
Ort HG G 43
Abstract We study perturbative path integral quantization of the two-dimensional scalar field theory with a polynomial (or power series) interaction potential on manifolds with boundary. The perturbative partition function defined in terms of configuration space integrals on the surface satisfies an Atiyah-Segal type gluing formula. Moreover, partition functions (modified by an interesting nonlocal boundary term) do organize into a functor (in the sense of Segal's axiomatics), from Riemannian cobordism category to the category of Hilbert spaces. A crucial role in the result is played by the tadpoles (short loops). We will discuss functorial assignments of tadpoles and the relation to RG flow in the space of potentials. This is a report on a joint work with Santosh Kandel and Konstantin Wernli, arXiv:1912.11202 [math-ph].
Two-dimensional perturbative scalar field theory with polynomial potential and cutting-gluingread_more
HG G 43
16. Januar 2020
15:15-16:15 		Nicola Pinamonti
University of Genova
Details

Talks in Mathematical Physics

Titel Bose Einstein condensation and spontaneous symmetry breaking for relativistic quantum field theory at finite temperature
Referent:in, Affiliation Nicola Pinamonti, University of Genova
Datum, Zeit 16. Januar 2020, 15:15-16:15
Ort HG G 43
Abstract During this talk we shall discuss the construction of states describing a Bose Einstein condensate at finite temperature for a relativistic massive complex scalar field with a phi^4 interaction. The starting point is the construction of the linear theory around a classical background which represent the condensate. Afterwards, the interacting field will be obtained using perturbation theory. We shall build equilibrium states adapting methods discussed by Araki in the context of rigorous statistical mechanics to perturbation theory, following a similar analysis performed recently by Fredenhagen and Lindner. We shall show that the limit where the interaction Lagrangian is supported everywhere in the spacetime exists for the correlation functions. In particular, infrared divergences will be avoided thanks to the thermal masses present in the linearized theory. The presence of this effective masses is not in conflict with Goldstone theorem because in the linearized theory the internal U(1)-symmetry is explicitly broken. The U(1)-symmetry and thus the validity of Goldstone theorem is restored only if the full perturbation series is considered.
Bose Einstein condensation and spontaneous symmetry breaking for relativistic quantum field theory at finite temperatureread_more
HG G 43

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