# SCIENTIFIC PROGRAMS AND ACTIVITIES

September 21, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR
 Geometric Representation Theory Seminar 2012-13 at the Fields Institute Tuesday, 3 p.m. Room 332 Organizing Committee: Karene Chu (karenemath.toronto.edu), Joel Kamnitzer (jkamnitzmath.toronto.edu), Oded Yacobi (oyacobi math.toronto. edu).
 Upcoming Seminars 2013 Tuesday June 25 3:00 p.m. Room 332 Peter Crooks Weyl Group Representations on the Homology of Springer Fibres Using the convolution algebra framework, we will construct Weyl group representations on the top-degree Borel-Moore homology of each Springer fibre. Some emphasis will be placed on examples arising in the case of sln. We will also consider these Weyl group representations in the context of the Springer Correspondence. Past Seminars 2012-13 Tuesday June 18 3:00 p.m. Room 332 Peter Samuelson In this talk we'll define Borel-Moore homology and give some basic properties. We'll also describe the basic setup of convolution algebras and give some simple examples. Then we'll discuss the convolution algebra associated to the Steinberg variety (at least when g = sl_n). Tuesday June 11 3:00 p.m. *Revised location Fields Institute, Room 332 Peter Crooks An Introduction to Springer Theory Let G be a connected, simply-connected complex semisimple linear algebraic group with Lie algebra g. One may construct a natural resolution of the (singular) nilpotent cone of g, called the Springer resolution. This has a realization in symplectic geometry as a moment map for a Hamiltonian action of G on the cotangent bundle of the flag variety of g. The fibres of the Springer resolution, called Springer fibres, are of particular interest. The top-degree Borel-Moore homology of each fibre carries a representation of the Weyl group, W. The construction of this representation is somewhat geometric in nature, as it involves identifying the group algebra of W with a subalgebra of the Borel-Moore homology of the Steinberg variety. It turns out that one can obtain every complex irreducible W-module as a certain summand of the top Borel-Moore homology of a Springer fibre. This is part of the Springer correspondence. This talk will begin with a survey of a few ideas that will be fundamental to our studying Springer Theory. Also, we will consider several examples arising in the case G=SLn. Thursday June 6, 4 p.m. Room 230 Chris Dodd Tuesday June 4 3 p.m. Room 230 Chris Dodd Localization for Modular Representations and Lusztig's conjecture This is an expository talk about the work of Bezrukavnikov, Mirkovic, and Ruminyin about localization for a semisimple lie algebra in positive characteristic and its application to understanding the category of representations. I'll start from scratch with modular reprsentations, talk about localization (in general), and explain how the affine hecke algebra works its way into the picture. Probably much of this will spill into the second week. Tuesday May 28 11 a.m. Room 230 Chris Dodd Localization for Modular Representations and Lusztig's conjecture This is an expository talk about the work of Bezrukavnikov, Mirkovic, and Ruminyin about localization for a semisimple lie algebra in positive characteristic and its application to understanding the category of representations. I'll start from scratch with modular reprsentations, talk about localization (in general), and explain how the affine hecke algebra works its way into the picture. Probably much of this will spill into the second week. Tuesday May 21 11 a.m. Stewart Library Chris Dodd Localization for Modular Representations and Lusztig's conjecture This is an expository talk about the work of Bezrukavnikov, Mirkovic, and Ruminyin about localization for a semisimple lie algebra in positive characteristic and its application to understanding the category of representations. I'll start from scratch with modular reprsentations, talk about localization (in general), and explain how the affine hecke algebra works its way into the picture. Probably much of this will spill into the second week. Tuesday May 14 3 p.m. Stewart Library Chris Dodd Localization for Modular Representations and Lusztig's conjecture This is an expository talk about the work of Bezrukavnikov, Mirkovic, and Ruminyin about localization for a semisimple lie algebra in positive characteristic and its application to understanding the category of representations. I'll start from scratch with modular reprsentations, talk about localization (in general), and explain how the affine hecke algebra works its way into the picture. Probably much of this will spill into the second week. Tuesday May 7 3 p.m. Stewart Library Joel Kamnitzer Categorification of Lie algebras [d'apres Rouquier, Khovanov-Lauda, ...] Given a vector space with an action of a semi-simple Lie algebra, we can try to "categorify" this representation, which means finding a category where the generators of the Lie algebra act by functors. Such categorical representations arise naturally in geometric representation theory. A framework for studying these categorical representations was introduced by Rouquier and Khovanov-Lauda. Their definitions are algebraic/combinatorial, but are connected to the topology of quiver varieties by the work of Varagnolo-Vasserot. Thursday May 2 3 p.m. Stewart Library Philippe Humbert, University of Strasbourg Beak diagrams and surface tangles The fact that the category of tangles can be "algebraicized" via its braided monoidal structure has played a fundamental role in the theory of quantum and finite-type invariants. How can this algebraization be generalized to tangles lying in a cylinder over an arbitrary surface? In this talk I will present one of the possible answers. Even though the approach may not be the most natural one, the good point is that it is based on a rather explicit description of surface tangles (by some kind of planar diagrams that I call "beak diagrams"). Thursday Apr. 11 3 p.m. Stewart Library Kiumars Kaveh, University of Pittsburgh Asymptotic behavior of multiplicities of reductive group actions We consider the action of a connected reductive algebraic group G on the graded algebra A of sections of a line bundle on a projective variety X. The asymptotic of multiplicities of irreducible representations appearing in A is related to the Duistermaat-Heckman function and Riemann-Roch theorem for multiplicities due to Guillemin-Sternberg, Meinrenken and others. For a given representation $\lambda$ let $m_{k, \lambda}$ denote the multiplicity of $\lambda$ appearing in the k-th degree piece A_k . We describe the asymptotic behavior of $m_{k, \lambda}$ as k goes to infinity. Our methods have elementary convex geometric nature and use the theory of Newton-Okounkov bodies. This work recovers and extends some previous results and of Brion and Paoletti who obtain similar results using Reimann-Roch theorem for multiplicities.This is a joint work with Takuya Murata. Thursday Apr. 4 3 p.m. Stewart Library Stephen Morgan, University of Toronto Quantum Hamiltonian reduction of W-algebras A W-algebra is an algebraic structure constructed from a universal enveloping algebra and a nilpotent element of the underlying Lie algebra; more precisely it can be constructed by reducing the universal enveloping algebra in a manner analogous to Hamiltonian reduction of Poisson varieties, known as quantum Hamiltonian reduction. In fact, W-algebras form a quantisation of the ring of functions on the appropriate Slodowy slices corresponding to the chosen nilpotent elements. More generally, we will show that W-algebras corresponding to more regular nilpotent elements can be obtained from more singular W-algebras using quantum Hamiltonian reduction, and mention some applications this has to categorification of tensor products of simple representations of sl2. Thursday Mar. 21 3 p.m. Room 332 Lucy Zhang, Perimeter Institute/ University of Toronto A classification of SET phases by G-extensions of spherical fusion categories We introduce the concept of topological phases and symmetry enriched topological (SET) phases. We augment the physical concepts with the study of G-graded fusion categories. In this talk, we will explore my work in progress with Xiao-Gang Wen on classifying the SET Thursday Mar. 14 3 p.m. Stewart Library Kirill Zaynullin, University of Ottawa Formal group laws, Hecke algebras and Oriented cohomology theories We generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel (e.g. Chow groups, Grothendieck's K_0, connective K-theory and algebraic cobordism). The resulting object, called a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings respectively. We also introduce a deformed version of the formal (affine) Demazure algebra, which we call a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We apply it to construct an algebraic model of the T-equivariant algebraic oriented cohomology of the variety of complete flags. The talk is based on two recent preprints arXiv:1208.4114, arXiv:1209.1676 and the paper [Invariants, torsion indices and cohomology of complete flags. Ann. Sci. Ecole Norm. Sup. (4) 46 (2013), no.3.]. Thursday Feb. 28 3 p.m. Stewart Library Xiao-Gang Wen, Perimeter Institute The mathematical languages for patterns of quantum entanglement Some quantum phases of matter are described different patterns of quantum entanglement. The patterns of quantum entanglement are new phenomena that happen in nature. But what kind of mathematical language do we use to describe quantum entanglement. Here I would like to explain that fusion category theory and group cohomology theory are nature languages to describe various patterns of quantum entanglement. As a result, we can use fusion category theory and group cohomology theory to classify new quantum states of matter. Thursday, Feb. 7 3 p.m. Stewart Library Michael McBreen, Princeton Quantum cohomology of hypertoric spaces, Bethe equations and mirror formulae A hypertoric variety is a hyperkahler analogue of a toric variety. I will discuss joint work with Daniel Shenfeld which computes the equivariant quantum cohomology ring of any smooth hypertoric variety, and provides integral formulas for the associated quantum differential equation. One could optimistically view these formulas as a form of mirror symmetry for the reduced genus 0 Gromov-Witten theory of hypertoric spaces. Time permitting, I will also sketch possible applications to the quantum cohomology of Nakajima quiver varieties and the Bethe equations of various finite dimensional quantum integrable systems, or alternatively the representation theory of the Yangian. Thursday, Jan 31, 3pm Stewart Library TBA Thursday, Jan 24, 3pm Stewart Library Andre Henriques (Utrecht) Extended conformal field theories A conformal field theory is a functor from the cobordism category of Riemann surfaces with boundary to the category of Hilbert spaces. Generalizing that idea, an extended conformal field theory is a 2-functor from a 2-category of Riemann surfaces with corners to an appropriate target 2-category that I'll describe. I'll explain how to (partially) construct extended conformal field theories. Thursday, Jan 17, 3pm Stewart Library Bhairav Singh (MIT) Twisted Geometric Satake Equivalence One of the fundamental results in geometric representation theory is the geometric Sa-take equivalence due to Lusztig, Ginzburg, Mirkovic-Vilonen, and Beilinson-Drinfeld, betweenequivariant perverse sheaves on the affine Grassmannian of a reductive group G and repre-sentations of of its Langlands dual group GV . Recently, Finkelberg-Lysenko gave a similar description for categories of monodromic perverse sheaves on the determinant line bundle of the affine Grassmannian. We will give a motivated introduction to the work of Finkelberg-Lysenko, and explain how some results in geometric representation theory generalize to themonodromic setting. January 10, 2013 Room 230 Chris Dodd, University of Toronto Modules over Algebraic Quantizations and representation theory Recently, there has been a great deal of interest in the theory of modules over algebraic quantizations of so-called symplectic resolutions. In this talk I'll discuss some new joint work that open the door to giving a geometric description to certain categories of such modules; generalizing classical theorems of Kashiwara and Bernstein in the case of D-modules on an algebraic variety. Nov. 30* 11:00 a.m. Room 210 *non-standard time and place Stavros Garoufalidis (Georgia Tech) The colored HOMFLY polynomial is q-holonomic We will prove that the colored HOMFLY polynomial of links colored by symmetric powers representation is q-holonomic. This fulfils some of the wishes of physics, has ramifications on the the SL(2,C) character variety of knots and motivates questions on the "web" approach of representation theory. Nov. 29 Joel Kamnitzer (Toronto) Crystals, geometric Satake, and the non-negative tropical flag variety IV Nov. 22 Joel Kamnitzer (Toronto) Crystals, geometric Satake, and the non-negative tropical flag variety III Nov.8 Adam Sikora (University of Buffalo) Character varieties as completely integrable systems We will give an introduction to character varieties. It is known that SL2-character varieties of surfaces are completely integrable. We prove an analogous statement for rank 2 Lie groups. Time permitting, we will discuss the relevance of this result to quantum invariants of Witten-Reshetikhin-Turaev. Nov.15 Oded Yacobi (Toronto) Yangians and quantizations of slices in the affine grassmannian We study slices to Schubert varieties in the affine Grassmannian, which arise naturally in the context of geometric representation theory. These slices carry a natural Poisson structure, and our main result is a quantization of these slices using subquotients of quantum groups called Yangians. We discuss also conjectural applications of these results to categorical representation theory. This is based on joint work with Joel Kamnitzer, Ben Webster, and Alex Weekes. Nov.1 Joel Kamnizter (Toronto) Crystals, geometric Satake, and the non-negative tropical flag variety II Friday Oct. 26, 2012 Room 210 3:30 p.m. *Please note non-standard date and time Allen Knutson (Cornell University) Combinatorial rules for branching to symmetric subgroups Given a pair G>K of compact connected Lie groups, and a dominant G-weightlambda, it is easy to use character theory to say how the irrep V_lambda decomposes as a K-representation. If G = K x K, this is tensor product decomposition, for which we have an enumerative formula: the constituents can be counted as a number of Littelmann paths or MV polytopes. I'll give a positive formula in the more general case that K is a symmetric subgroup of G, i.e., the (identity component of) the fixed-point set of an involution. The combinatorics is controlled by the poset of K-orbits on the flag manifold G/B, which reduces to the Bruhat order in the case G = K x K. I can prove this formula in the asymptotic (or, symplectic) situation replacing lambda by a large multiple, and nonasymptotically for certain pairs (G,K). This is a joint meeting with the Algebraic Combinatorics Seminar. Oct. 18 No Seminar David Nadler (Northwestern University) will be giving a talk 'Traces and loops' as part of the Fields Symposium at 2:45 p.m. in Room 230 of the Fields Institute. Oct. 19 Bahen Centre, Room 4010 4:00 p.m. *Please note change in time Masoud Kamgar (MPI, Bonn) Ramified Satake Isomorphisms I will explain how to associate a Satake-type isomorphism to certain characters of the compact torus of a split reductive group over a local field. I will then discuss the geometric analogue of this isomorphism and its possible applications. (Joint work with T. Schedler). Oct. 11 Joel Kamnitzer (Toronto) Crystals, geometric Satake, and non-negative tropical flag variety Crystals are a combinatorial model for studying the representation theory of reductive groups. In 2004, A. Henriques and I proved that the octahedron recurrence controls the category of GL_n-crystals. I will explain how this result can be generalized to any reductive group G, via configurations of points in the affine Grassmannian and the non-negative tropical flag variety. Oct.4 Josh Grochow (Toronto) Introduction to Geometric Complexity Theory The Geometric Complexity Theory (GCT) program was introduced by Mulmuley and Sohoni to attack fundamental lower bound problems in computational complexity theory—such as P vs NP—using algebraic geometry and representation theory. In addition to presenting the basic structure of the GCT program, I will discuss some of the intuition behind the use of representation theory in complexity, as well as how GCT relates to classical questions in representation theory such as the Littlewood-Richardson rule for the decomposition of tensor products of representations of GL_n into irreducibles. Sept. 27 Dave Penneys (Toronto) Classifying subfactors A subfactor is an inclusion of von Neumann algebras with trivial centers. We use several invariants to classify subfactors, including the index, the principal graphs, and the standard invariant. The standard invariant forms a unitary 2-category, so the typical planar calculus gives us the structure of a planar algebra. I will discuss all these notions, and I will give a brief overview of the classification program of small index subfactors. Sept. 20 Peter Samuelson (UofT) Quantizations of character varieties If \pi is a finitely generated group, the set $Hom(\pi, SL_2(C))$ has a natural scheme structure. We recall a description of its algebra of functions $O(\pi)$ and explain that if M is a 3-manifold, the Kauffman bracket skein module $K_q(M)$ gives a quantization of $O(\pi_1(M))$. If $M = S^3 \ K$ for a knot K, then $K_q(M)$ is a module over (a subalgebra of) the quantum torus $A_q$ which encodes the colored Jones polynomials of K. We give an indication of the types of modules that arise from this construction, and if time permits we'll discuss deformations of $K_q(M)$ to a module over the double affine Hecke algebra $H_{q,t}$. (In the talk I intend to define all unfamiliar words in this abstract. This is work in progress with Yuri Berest.)