### ICMS Summer School and Workshop: Geometric representation theory and low-dimensional topology.June 3rd-7th, 10th-14th 2019

Below is an archive of the summer school and workshop which was held at the ICMS in June 2019. Below you can find the complete program for both events, notes for the summer school talks, and videos for talks both weeks.

### Scientific Organisers

• Dan Freed, University of Texas, Austin
• David Jordan, University of Edinburgh
• Peter Samuelson, University of California, Riverside
• Olivier Schiffmann, University of Paris-Sud, Orsay

The aim of this workshop is to bring together researchers in topological field theory, geometric representation theory, mathematical physics, and algebraic combinatorics to report on recent progress and open questions at the intersection of these fields. In recent years, several independent research groups have raised the possibility of new 3- and 4-dimensional topological field theories tying together Khovanov-Rozansky knot homology, double affine Hecke algebras and elliptic Hall algebras, super Yang-Mills theory, Betti quantum geometric Langlands, and Hilbert schemes. As each of these topics constitutes a specialist field in its own right, it is essential that we bring researchers together to exchange ideas and insights.

This workshop is organised by the International Centre for Mathematical Sciences in partnership with the European Research Council and the National Science Foundation. It is partially supported by the NSF, Division of Mathematical Sciences.

### Workshop programme:

The programme is shown below. Skip to the bottom for the summer school programme.

Monday 10 June 2019

 09:00-09:45 Registration & coffee 09:45-10:00 Welcome & opening 10:00-11:00 Catharina Stroppel (University of Bonn) DAHA actions on fusion rings In this talk I will describe fusion algebras arising from quantum groups at roots of unity. After a short overview of the general theory we study the rings in more details. The goal is to construct actions of certain double affine Hecke algebras on these algebras. 11:10-12:10 Mina Aganagic (University of California, Berkeley) Knot categorification from geometry and mirror symmetry, via string theory I will describe how two geometric approaches to categorification of RTW invariants of knots emerge from string theory. The first approach is based on a category of B-type branes on resolutions of slices in affine Grassmannians. The second is based on a category of A-branes in a Landau-Ginzburg theory. The relation between them is two dimensional (equivariant) mirror symmetry. String theory also predicts that a third approach to categorification, based on counting solutions to five dimensional Haydys-Witten equations, is equivalent to the first two. This talk is mostly based on joint work with Andrei Okounkov. 12:10-14:00 Lunch 14:00-15:00 Sergei Gukov (California Institute of Technology) New TQFTs from DAHANo abstract provided. 15:00-16:00 Tea & coffee 16:00-17:00 Claudia Scheimbauer (Norwegian University of Science and Technology) En-algebras, extended topological field theories and dualizability E_n -algebras, extended topological field theories and dualizability The Cobordism Hypothesis provides a beautiful interplay between extended topological field theories and dualizability conditions, allowing for a conceptual explanation of certain finiteness conditions appearing in representation theory, and, vice versa, a geometric understanding of algebraic objects. In turn, E_n -algebras, which are algebras for the little disks operad, (e.g. associative algebras and quantum groups) provide important examples. I will explain why every E_n -algebra leads to a categorified topological field theory using dualizability arguments. Furthermore, I will explore extensions and further directions. 17:00-19:00 Wine & whisky reception

Tuesday 11 June 2019

 09:00-10:00 Sam Gunningham (University of Edinburgh) q-Character sheaves and Springer theory The category of equivariant D_q(G)-modules sits at the interface of low-dimensional topology and geometric representation theory. It appears naturally in the context of factorization homology and skein theory, but it may also be thought of as a q-deformation of the category of equivariant modules for the ring D(G) of differential operators on a complex reductive group G. In this talk I will survey various aspects of an ongoing program to adapt the powerful tools available in the D-module setting (e.g. character sheaves, parabolic induction, Springer theory) to the realm of D_q-modules. As an example, I will present a new derivation of a recent result of Carrega and Gilmer on the 9- dimensionality of the skein module for a 3-torus. This is joint work with (combinations of) David Jordan, Monica Vazirani, and Pavel Safronov. 10:00-10:30 Tea & coffee 10:30-11:30 Adrien Brochier (IMJ-PRG, Université Paris-Diderot) Quantization of character varieties, topological field theories and the Riemann-Hilbert correspondence I will give a gentle exposition of some recent work joint with D. Ben-Zvi, D. Jordan and N. Snyder on quantization of character varieties of surfaces, using the language of topological field theories. I will first recall various points of view on the representation category of quantum groups and the associated topological invariants. I will then explain how, using the formalism of factorization homology, one can construct and compute explicitly a certain category-valued invariant of surfaces. This provides a canonical quantization of the Atiyah--Bott-Goldman Poisson structure on character varieties, unifies and generalizes a number of previously known constructions and has interesting applications in representation theory and low-dimensional topology. Time permitting, I will mention a work in progress on a higher genus analog of the Kohno-Drinfeld theorem, using this formalism to compute explicitly the monodromy of certain differential equations arising in conformal field theory. This can be thought of as a quantization of the Riemann-Hilbert correspondence. 11:40-12:40 Harold Williams (University of California, Davis) Canonical bases for Coulomb branches Following work of Kapustin-Saulina and Gaiotto-Moore-Neitzke it is anticipated that the expectation values of irreducible half-BPS line defects define a canonical basis in the quantized Coulomb branch of a 4d N=2 field theory. In this talk we propose a mathematical definition of the category of such defects, hence of the associated canonical basis, in the case of N=2 gauge theories of cotangent type. The form of the definition is a finite length t-structure on the DG category of coherent sheaves on a derived enhancement of the space of triples introduced by Braverman-Finkelberg-Nakajima. This heart of this t-structure is a non-Noetherian extension of the category of perverse coherent sheaves on the affine Grassmannian, to which it specializes in the case of pure N=2 gauge theory. It is expected that these categories categorify the cluster algebra associated to the BPS quiver of the theory, which we confirm in various examples. This is work in progress with Sabin Cautis. 12:40-14:30 Lunch 14:30-15:30 Anton Mellit (University of Vienna) $A_{q,t}$ algebra as a partially symmetrized DAHA and computations of knot invariants $A_{q,t}$ algebra as a partially symmetrized DAHA and computations of knot invariants $A_{q,t}$ algebra arose from my work with Erik Carlsson on the shuffle conjecture. I will explain how this algebra can be constructed naturally from a topological (skein-theoretic) point of view. Similarly to the way representation theory of quantum groups $gl_N$ can be interpolated in the limit $N\to\infty$, the $A_{q,t}$ algebra arises if one attempts to interpolate the symmetric power of the standard representation $Sym_M$ in the limit $M\to\infty$. Then I will show how the algebra was used to compute the triply graded homology (superpolynomials) of torus knots. 15:30-16:00 Tea & coffee 16:00-17:00 Du Pei (Caltech, Aarhus University) New TQFTs from DAHA (Part 2) We will continue to study DAHA and its representations via quantum physics, focusing on new connections with mirror symmetry and skein theory.

Wednesday 12 June 2019

 09:00-10:00 John Francis (Northwestern University) Moduli of stratifications and factorization homology The Ran space Ran(X) is the space of finite subsets of X, topologized so that points can collide. Ran spaces have been studied in diverse works from Borsuk–Ulam and Bott, to Beilinson–Drinfeld, Gaitsgory–Lurie and others. The alpha form of factorization homology takes as input a manifold or variety X together with a suitable algebraic coefficient system A, and it outputs the sheaf homology of Ran(X) with coefficients defined by A. Factorization homology simultaneously generalizes singular homology, Hochschild homology, and conformal blocks or observables in conformal field theory. This alpha form of factorization homology has applications to the study of mapping spaces in algebraic topology, bundles on algebraic curves, and perturbative quantum field theory. There is also a beta form of factorization homology, where one replaces the Ran space with a moduli space of stratifications, designed to overcome certain strict limitations of the alpha form. The key notion is that of a constructible bundle: in terms of its functor of points, a K-point of this moduli space is a constructible bundle over K. One application is to proving the Cobordism Hypothesis, after Baez– Dolan, Costello, Hopkins–Lurie, and Lurie. This is joint work with David Ayala. 10:00-10:30 Tea & coffee 10:30-11:30 Constantin Teleman (University of California, Berkeley) Coulomb branches, passée topology and non-polarizable matter No abstract provided. 11:40-12:40 Alexei Oblomkov (University of Massachusetts) Dualisable link homology and 3D TQFT This talk is based on the joint work with Lev Rozansky. It is elementary to see that the HOMFLYPT polynomial of a knot K $P_K(a,q)$ is palindromic: $P_K(a,q)=P_K(a,1/q)$. It was conjectured by Dunfield-Gukov-Rasmussen that the Poincare polynomial $\mathcal{P}_L(a,q,t)$ of the HOMFLYPT homology has similar property. In my talk I explain a proof of the conjecture that relies on the construction 3D TQFT that provides a mathematical model for 3D sigma model of Kapustin-Rozansky- Saulina. As by-product of our construction we find an interpretation of the knot homology as v space of sections of a particular sheaf on the Hilbert scheme of points on the plane as well as another explanation why the skein algebra of torus is an elliptic Hall algebra. 12:40-12:50 Group photo 12:50 onwards Free afternoon 19:00 Workshop dinner at Playfair Library, Old College, South Bridge

Thursday 13 June 2019

 09:00-10:00 Fabian Haiden (University of Miami) Legendrian skein algebras and Hall algebras This is a report on work in progress in understanding the relation between skein algebras and Hall algebras of Fukaya categories, motivated by a recent paper of Cooper-Samuelson. Another motivation is the problem of defining Hall algebras for Z/2-graded categories, and the limiting case q=1. The main novelty is to assign to a Lagrangian submanifold a positive linear combination of isomorphism classes of objects in the Fukaya category (i.e. a “random object”). 10:00-10:30 Tea & coffee 10:30-11:30 Yuri Berest (Cornell University) Perverse sheaves, contact homology and cubical approximations 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. 11:40-12:40 Noah Snyder (Indiana University) Local topological field theories with values in Morita categories The ordinary Morita symmetric monoidal 2-category has objects that are algebras over a field k, 1- morphisms from A to B are A-B bimodules, and 2-morphisms are bimodule maps, and the monoidal structure is giving by tensoring over k. More generally one can consider an Morita (r+m)-category whose objects are E_r-algebras in some m-category, 1-morphisms are E_{r-1}-bimodules, etc. Some notable specific examples are the Morita 3-category of monoidal categories and the Morita 4- category of braided monoidal categories. An n-dimensional local topological field theory is a symmetric monoidal functor from a bordism n-category to some target n-category. The Hopkins- Lurie-Baez-Dolan cobordism hypothesis says that such TFTs are classified in terms of n-dualizable objects in the target. The goal of this talk is to give a survey of what is known about n-dualizability of objects in Morita categories. This will touch on results of Calaque-Scheimbauer, Haugseng, Johnson--Freyd-Scheimbauer, Gwilliam-Scheimbauer, Brandenburg-Chirvasitu-Johnson--Freyd, and Morrison-Walker, and focus on my joint work with Douglas-Schommer--Pries and Brochier-Jordan on dualizability of tensor categories and braided tensor categories. 12:40-14:30 Lunch 14:30-15:30 Dimitri Wyss (Sorbonne Université) Non-archimedean integrals on the Hitchin fibration Non-archimedean or p-adic integration is an analytic tool to study rational points of algebraic varieties over finite fields. Dener-Loser and Batyrev have realized that this can be used in some cases to study the topology of complex algebraic varieties. We apply this idea to the moduli spaces of G- Higgs bundles M(G) and show in particular, that for a pair of Langlands dual groups the corresponding moduli spaces have the same non-archimedean volume. As a geometric application we find an agreement of (stringy) Hodge numbers of M(SLn) and M(PGLn) as predicted by a conjecture of Hausel-Thaddeus. For general G this leads to a new proof of the geometric stabilization theorem, a key ingredient in Ngô's proof of the fundamental lemma. This is joint work with Michael Groechenig and Paul Ziegler. 15:30-16:00 Tea & coffee 16:00-17:00 Pavel Safronov (University of Zurich) R-matrices with a spectral parameter via shifted Poisson structures Constant (dynamical) R-matrices have a categorical interpretation in terms of braided monoidal categories equipped with a monoidal forgetful functor to the category of vector spaces (Harish— Chandra bimodules). In the first part of the talk I will review analogous interpretations of classical r- matrices in terms of shifted Poisson structures. In the second part I will describe a conjectural interpretation of classical (dynamical) r-matrices with a spectral parameter (describing Yangians, quantum affine algebras and elliptic quantum groups) in terms of shifted Poisson structures and their quantizations.

Friday 14 June 2019

 10:00-11:00 Ivan Cherednik (University of North Carolina at Chapel Hill) DAHA superpolynomials for iterated links It will be mostly an introduction to our long paper with Ivan Danilenko on DAHA superpolynomials of (colored) torus iterated links. Connections with the HOMFLY-PT polynomials and the Khovanov- Rozansky stable reduced polynomials will be briefly discussed. The theory of the latter is mostly developed for uncolored knots, especially in the reduced setting. Also, DAHA superpolynomials satisfy the super-duality (a theorem) and have other remarkable symmetries, which are generally difficult to approach topologically. Though the most clarifying proof of the super-duality for algebraic links is via the so-called motivic superpolynomials of plane curve singularities, conjecturally coinciding with the DAHA ones. The super-duality becomes then the functional equation, a fundamental development, which may have physics implications. The motivic direction inspired the Riemann hypothesis for DAHA superpolynomials (to be touched a bit at the end), but it will be omitted in this talk. We will focus on the DAHA theory of colored iterated links, explaining the main steps and providing some examples. 11:00-11:30 Tea & coffee 11:30-12:30 Tudor Dimofte (University of California, Davis) 3d mirror symmetry and HOMFLY-PT homology Recently, Gorsky-Negut-Rasmussen and Oblomkov-Rozansky proposed constructions of HOMFLY-PT link homology related to coherent sheaves on Hilbert schemes. Oblomkov-Rozansky explained how their construction was realized in several different physical systems, including (most relevantly for this talk) 3d supersymmetric gauge theories. I will discuss how a duality of 3d gauge theories known as 3d mirror symmetry --- whose mathematical/categorical implications are only starting to be explored --- acts on the setups above. The result is a very different, "A type" construction of link homology, related to cohomology of affine Springer fibers and to Hilbert schemes of points on singular curves. The talk will attempt to be fairly broad, with no prior technical knowledge assumed. Based on work with N. Garner, J. Hilburn, A. Oblomkov, and L. Rozansky; and related work of N. Garner and O. Kivinen. 12:30-14:30 Lunch 14:30-15:30 Pavel Etingof (MIT) Short star-products for filtered quantizations Let $A$ be a filtered Poisson algebra with Poisson bracket $\lbrace{,\rbrace}$ of degree $-2$. A {\it star product} on $A$ is an associative product $*: A\otimes A\to A$ given by $$a*b=ab+\sum_{i\ge 1}C_i(a,b),$$ where $C_i$ has degree $-2i$ and $C_1(a,b)- C_1(b,a)=\lbrace{a,b\rbrace}$. We call the product * {\it even} if $C_i(a,b)=(-1)^iC_i(b,a)$ for all $i$, and call it {\it short} if $C_i(a,b)=0$ whenever $i>{\rm min}({\rm deg}(a), {\rm deg}(b))$. Motivated by three-dimensional $N=4$ superconformal field theory, In 2016 Beem, Peelaers and Rastelli considered short even star-products for homogeneous symplectic singularities (more precisely, hyperK\""ahler cones) and conjectured that that they exist and depend on finitely many parameters. We prove the dependence on finitely many parameters in general and existence for a large class of examples, using the connection of this problem with zeroth Hochschild homology of quantizations suggested by Kontsevich. Beem, Peelaers and Rastelli also computed the first few terms of short quantizations for Kleinian singularities of type A, which were later computed to all orders by Dedushenko, Pufu and Yacoby. We will discuss some generalizations of these results. This is joint work with Eric Rains and Douglas Stryker. 15:40-16:40 Andrei Negut (MIT) The PBW basis of the quantum toroidal algebra We give a presentation of U_{q,q'}(gl_n^^) via generators and relations, akin to the one developed by Burban-Schiffmann in the elliptic Hall algebra (which would correspond to n=1). One hopes that our presentation could help define a version of the elliptic Hall algebra for all n, although this is still only a dream. 16:40 End of workshop

### Summer school programme:

Monday 3 June 2019

 09:00-09:45 Registration & coffee 09:45-10:00 Welcome & opening 10:00-11:00 Hiro Lee Tanaka (Texas State University) Factorization homology, infinity-categories, and topological field theories I[Notes] These lectures will serve as an introduction to the topics in the title. Factorization homology is a method for producing field theories out of rich algebraic gadgets; for example, it simultaneously yields invariants of manifolds and of higher algebras. We will also give peeks into the language of infinity-categories, which are necessary in one guise or another to articulate the rich spaces that cohere operations. Topological field theories will also be presented, but from a mathematics-centred (rather than physics-centred) perspective. The (ambitious) goal is to give examples that illustrate how these notions help to organize phenomena in representation theory. 11:00-12:00 Monica Vazirani (University of California, Davis) Hecke algebras and representation theory I [Notes] The double affine Hecke algebra (DAHA) was invented by Cherednik in order to study (symmetric) Macdonald polynomials. This gave rise to the nonsymmetric Macdonald polynomials which gives a basis of (Laurent) polynomials in X's. The X's form a “third" of the DAHA in its triangular decomposition. They are also a weight basis for polynomials in Y's, which comprise another third. The middle third is the finite Hecke algebra, which in type A deforms the symmetric group Sn. The X's also form an irreducible and faithful representation of the DAHA for generic parameters q; t. One can generalize from here to studying other irreducible representations (irreps) that admit a Y -weight basis - so-called Y-semisimple or calibrated representations. Or one can study irreps on which the Y's act locally finitely - analogue of Category O. How can we understand, study, classify, or construct such irreps? This 4-hour course will focus on type A. We will start with some of the more classical representation theory and combinatorics for the symmetric group Sn, the (extended) affine symmetric group, Hecke algebras as their q-deformations, and possibly other ways Hecke algebras “arise in nature." We will examine the structure of affine Hecke algebra (AHA) representations, particularly using the functors of induction and restriction. Then we shall apply these tools toward DAHA representations. As time and interest permits, topics that may be covered include: Y –semisimple irreps, finite dimensional irreps, analogues of Schur-Weyl duality, monomial expansion of Macdonald polynomials, the spherical DAHA (possibly with connections to the elliptic Hall algebra or skein modules), double a affine braid group, orthogonal polynomials, action of SL2(Z) on DAHA. 12:00-13:30 Lunch 13:30-14:30 Tina Kanstrup (University of Massachusetts Amherst) Knot homologies and matrix factorizations I [Notes] One of the most famous link invariants is Khovanov-Rozansky triply graded homology. Its definition is completely algebraic and notoriously hard to compute. It has been conjectured by Gorsky, Negut and Rasmussen that it can also be computed as cohomology of certain coherent sheaves on the flag Hilbert scheme. A link invariant of a similar nature has been constructed by Oblomkov and Rozansky in terms of matrix factorizations. In this lecture series we will describe these different approaches and reformulate the work of Oblomkov and Rozansky into the setting of the conjecture using work of Arkhipov and Kanstrup. Time permitting we will relate this to the work of Ben-Zvi and Nadler et al. in derived algebraic geometry. 14:30-15:00 Tea & coffee 15:00-16:00 Pavel Safronov (University of Zurich) Shifted Poisson structures in representation theory I [Notes] These talks are devoted to the theory of symplectic and Poisson structures on stacks developed by (Calaque—)Pantev—Toen—Vaquie—Vezzosi. In the first lecture I will introduce the language of derived algebraic geometry and define shifted symplectic structures. In the next lecture I will sketch a relationship with shifted Poisson structures and will spend the rest of the time on examples relevant for representation theory. These include: the Kirillov—Kostant—Souriau symplectic structure on coadjoint orbits, the Grothendieck—Springer resolution, Poisson—Lie structures on simple groups, reflection equation algebras, Sklyanin and Feigin—Odesskii algebras. 16:00-17:00 Gregor Masbaum (CNRS) Introduction to Skein theory Skein modules were originally introduced to generalize the celebrated Jones polynomial of classical knots to links in other three-manifolds. They also appear naturally as quantizations of character varieties. The aim of this talk is to give an introduction to skein theory for non-experts and hopefully discuss some interesting open questions as well. 17:00-19:00 Wine & beer reception with pizza at 18:00

Tuesday 4 June 2019

 09:00-10:30 Hiro Lee Tanaka (Texas State University) Factorization homology, infinity-categories, and topological field theories II 10:30-11:00 Tea & coffee 11:00-12:30 Pavel Safronov (University of Zurich) Shifted Poisson structures in representation theory II 12:30-14:00 Lunch 14:00-15:00 Jose Simental Rodriguez (University of California, Davis) Rational double affine Hecke algebras [Notes] Rational Cherednik algebras, or rational DAHA, are a degeneration of DAHA introduced by Etingof and Ginzburg around the year 2000. We will explore the precise connection between RCA and DAHA, and review some of the basic structural and representation theoretical properties of the RCA. Time permitting, we will also see the connection between the type A RCA and the Hilbert scheme of points in the plane. 15:00-15:30 Tea & coffee 15:30-17:00 Tina Kanstrup (University of Massachusetts Amherst) Knot homologies and matrix factorizations II 17:30 Doors open for public lecture 18:00-19:00 Public lecture by Dan Freed (University of Texas, Austin) Geometry from afar [Photos] 19:00-20:00 Wine reception

Wednesday 5 June 2019

 09:00-10:00 Monica Vazirani (University of California, Davis) Hecke algebras and representation theory II 10:00-10:30 Tea & coffee 10:30-12:00 Tina Kanstrup (University of Massachusetts Amherst) Knot homologies and matrix factorizations III 12:00-13:30 Lunch 13:30-14:30 Gus Schrader (Columbia University)[Notes] The cluster approach to character varieties It was shown by Fock and Goncharov that certain character varieties parametrizing framed SL(n)-local systems on surfaces bear the structure of cluster Poisson varieties. This observation allowed them to quantize these character varieties, and conjecture that this quantization satisfies the locality property of TQFT with respect to cutting and gluing of surfaces. I will discuss their construction, and explain the idea of the proof of their conjecture obtained in joint work with A. Shapiro. 14:30-15:30 Kostiantyn Tolmachov (Perimeter Institute and University of Toronto) Knot homologies and geometric Hecke categories I will describe geometric categorifications of the Hecke algebras coming from representation theory, and survey how the knot homologies are visible from the point of view of these categorifications, following the work of Webster and Williamson. Time permitting, I will also talk about the recent progress in relating the coherent categorification of the affine Hecke algebra with the constructible categorification of the finite Hecke algebra in type A, joint with Bezrukavnikov. 15:30-16:00 Tea & coffee 16:00-17:30 Pavel Safronov (University of Zurich) Shifted Poisson structures in representation theory III

Thursday 6 June 2019

 09:00-10:30 Hiro Lee Tanaka (Texas State University) Factorization homology, infinity-categories, and topological field theories III 10:30-11:00 Tea & coffee 11:00-12:00 Juliet Cooke (University of Edinburgh) Skein categories[Notes] In this talk we will introduce a categorical analogue of skein algebras based on coloured ribbon tangles. We shall then see how these skein categories satisfy excision and can be treated as k-linear factorisation homology theories of surfaces. 12:00-13:00 Alexander Shapiro (University of Edinburgh) Around character varieties in 50 minutes There are four main approaches to character varieties and their quantization: via skein algebras, via Alekseev – Grosse – Schomerus quantization of Fock – Rosly Poisson structure, via cluster algebras due to Fock and Goncharov, and via factorization homology due to Ben-Zvi – Brochier – Jordan. I will discuss how the latter three approaches are related to each other. The talk will be based on a joint work in progress with D. Jordan, I. Le, and G. Schrader. 13.00-14:30 Lunch 14:30-15:30 Quoc Ho (Institute of Science and Technology Austria (IST Austria) Arithmetic applications of factorization homology In recent years, the theory of factorization homology has emerged as a powerful tool to study problems in diverse areas of mathematics, including higher category theory, representation theory, manifold topology, and number theory (over function fields), etc. The last two topics will be the main focus of the talk. More specifically, we will explain how factorization homology can help clarify various links between homological stability phenomena and problems in arithmetic statistics. A review of the basics of factorization homology will also be provided 15:30-16:00 Tea & coffee 16:00-17:00 Monica Vazirani (University of California, Davis) Hecke algebras and representation theory III

Friday 7 June 2019

 09:00-10:00 Alexandre Minets (IST Austria) Cohomological Hall algebras and sheaves on surfaces [ Notes] Cohomological Hall algebras (COHAs) where introduced by Kontsevich-Soibelman and SchiffmannVasserot about 10 years ago, and since then proved to be a useful language for unifying a number of constructions in geometric representation theory. I will start by explaining less technically challenging notion of Ringel-Hall algebra, and continue with the definition and examples of COHAs. If time permits, I will sketch some recent results about their action on cohomology of moduli of coherent sheaves on surfaces. 10:00-10:30 Tea & coffee 10:30-11:30 Nicolle Gonzalez (University of California) The higher structure of the Heisenberg and Clifford algebras An important role in the structure of CFT is played by vertex operators. Categorifying these operators is, thus, a natural step towards understanding the higher structure of CFT. As a motivating example, I will explain a possible approach via the categorification of the vertex operators which relate the actions of the Heisenberg and Clifford algebras on Fock space, known as the boson-fermion correspondence. In the process, I will briefly discuss Khovanov's Heisenberg category and some of the basics of diagrammatic categorification. 11:30-12:30 Monica Vazirani (University of California, Davis) Hecke algebras and representation theory IV 12:30-12:40 Group photo 12:40-14:30 Lunch 14:30-17:00 Write!