Time: Central Time (CT)
Organizers: Ezra Getzler, Dogancan Karabas, Gus Schrader, Boris Tsygan, Hang Yuan, and Eric Zaslow

Thursday, September 30, 2021 at 4:00pm

• Speaker: Gus Schrader (Northwestern)
• Title: Bifundamental Baxter Operators
• Abstract: The Hamiltonians of the GL(n) open q-difference Toda lattice as well as their eigenfunctions, the q-Whittaker functions, can be neatly described by means of the system’s Baxter operator. The fact that the Baxter operator acts diagonally in the basis of Whittaker functions can be regarded as a q-deformed, continuous analog of the generating function of all Pieri rules for Schur functions. In joint work with with Alexander Shapiro we introduce a kind of GL(n) x GL(m)-generalization of this Baxter operator, such that the original Baxter operator corresponds to the case m=1. These new operators can be realized as quantum cluster transformations, and govern the cluster structure on K-theoretic Coulomb branches of 3d N=4 gauge theories with bifundamental matter. This perspective leads to a categorification of the GL(n) x GL(m) Baxter operators in which cluster mutations are promoted to exact triangles in the derived category of coherent sheaves on a convolution variety.
• Place: Lunt 107

Thursday, October 7, 2021 at 4:00pm

• Speaker: Hang Yuan (Northwestern)
• Title: SYZ mirror construction from quantum corrections and non-archimedean geometry
• Abstract: In the study of mirror symmetry, we still don’t know why a pair of varieties are mirror. This is a long-standing problem, and our best hope should be geometry: The Strominger-Yau-Zaslow conjecture suggests that the mirror symmetry for Calabi-Yau manifolds can be interpreted as a duality between special Lagrangian fibrations. In this talk, we use the counts of holomorphic disks in the A-side to build a rigid analytic space structure on the B-side dual torus fibration. It is based on the A-infinity structures and non-archimedean analysis. If time allowed, we will explain an application on the disk-counting and also a recent work in progress about the folklore conjecture that the critical values of the mirror Landau-Ginzburg superpotential are the eigenvalues of quantum product by c_1.
• Place: Lunt 107

Tuesday, October 12, 2021 at 4:00pm

• Speaker: Owen Gwilliam (University of Massachusetts (Amherst))
• Title: Quantizing the Loday-Quillen-Tsygan Theorem
• Abstract: The Loday-Quillen-Tsygan theorem relates the cyclic (co)homology of an associative algebra A to the Lie algebra (co)homology of gl_N(A) as N grows large. In rather poetic language, one can say it relates a topological closed string theory to a gauge theory in the large N limit. We will give a precise formulation of this idea when A is a cyclic A_∞-algebra, and we will use it to revisit the large N behavior of Gaussian random matrices, such as Harer-Zagier recurrence. This work is joint with Ginot, Hamilton, and Zeinalian.
• Place: Lunt 103

Thursday, October 21, 2021 at 4:00pm

• Speaker: Felix Janda (University of Notre Dame)
• Title: Higher genus mirror symmetry for complete intersection Calabi-Yau threefolds
• Abstract: In this talk, I will discuss progress (joint with Qile Chen, Shuai Guo and Yongbin Ruan) on computing higher Gromov-Witten invariants of quintic threefolds and more general complete intersection Calabi-Yau threefolds in projective space.
• Place: Zoom (email dogancan.karabas@northwestern.edu if it asks passcode)

Thursday, October 28, 2021 at 4:00pm

• Speaker: Dogancan Karabas (Northwestern)
• Title: Homotopy colimit of dg categories, wrapped Fukaya categories, and lens spaces
• Abstract: Ganatra, Pardon, and Shende introduced a way to compute wrapped Fukaya categories of Weinstein domains by taking the homotopy colimit of wrapped Fukaya categories of their sectorial coverings. However, homotopy colimits are hard to compute in general. In this talk, I will describe a practical formula for homotopy colimit when the categories are presented as semifree dg categories. As an application, I will show that the wrapped Fukaya category detects homotopy type of lens spaces. If time permits, I will talk about other applications of the formula, such as the calculation of wrapped Fukaya category of plumbing spaces. This is joint work with Sangjin Lee.
• Place: Lunt 107

Thursday, November 4, 2021 at 4:00pm

• Speaker: François Bonahon (USC)
• Title: Quantum invariants of surface diffeomorphisms and 3-dimensional hyperbolic geometry
• Abstract: This talk is motivated by surprising connections between two very different approaches to 3-dimensional topology, namely quantum topology and hyperbolic geometry. The Kashaev-Murakami-Murakami Volume Conjecture connects the growth of colored Jones polynomials of a knot to the hyperbolic volume of its complement. More precisely, for each integer n, one evaluates the n-th Jones polynomial of the knot at the n-root of unity exp(2 pi i/n). The Volume Conjecture predicts that this sequence grows exponentially as n tends to infinity, with exponential growth rate related to the hyperbolic volume of the knot complement. I will discuss a closely related conjecture for diffeomorphisms of surfaces, based on the representation theory of the Kauffman bracket skein algebra of the surface, a quantum topology object closely related to the Jones polynomial of a knot. I will describe the mathematics underlying this conjecture, which involves a certain Frobenius principle in quantum algebra. I will also present experimental evidence for the conjecture, and describe partial results obtained in work in progress with Helen Wong and Tian Yang.
• Place: Lunt 107

Thursday, November 11, 2021 at 4:00pm

• Speaker: Alexander Polishchuk (University of Oregon)
• Title: Homological mirror symmetry for chain type polynomials
• Abstract: This is a joint work with Umut Varolgunes. We outline the proof of an equivalence between the Fukaya-Seidel category of a chain type polynomial and the category of graded matrix factorizations of the dual polynomial (modulo some general statements about Fukaya-Seidel categories). The proof is based on a certain recursive construction for these categories.
• Place: Zoom (email dogancan.karabas@northwestern.edu if it asks passcode)

Thursday, November 18, 2021 at 4:00pm

• Speaker: James Pascaleff (University of Illinois at Urbana-Champaign)
• Title: Gluing Fukaya categories of surfaces and singularity categories
• Abstract: I will construct a sheaf of 2-periodic DG categories over decorated trivalent graphs. The sections of this sheaf over a given graph is a category that depends only on the number of loops g and the number of ends n, and in the case where n = 0, it also depends on a single continuous parameter. I will then show how this sheaf recovers
  A: The Fukaya category of a Riemann surface of genus g with n punctures with a particular total area,
  B: The derived category of singularities of certain normal crossings surfaces, with a particular 2-periodic structure.
  These results may be interpreted as mirror symmetry statements for Riemann surfaces.
  (This is joint work with Nicolo Sibilla. I will try to avoid too much overlap with the talk that Nicolo gave recently.)
• Place: Lunt 107

Thursday, December 2, 2021 at 8:00am

• Speaker: Chris Brav (HSE, Moscow)
  
• Title: Calabi-Yau categories and their deformation theory
• Abstract: We review the notion of relative Calabi-Yau structure on a dg functor (a non-commutative analogue of an oriented manifold with boundary introduced in joint work with Toby Dyckerhoff), explain how the Lie algebra of its automorphism group is a chain-level generalisation of the string Lie algebra of Chas-Sullivan and the necklace Lie algebra of Kontsevich. We discuss some examples and applications to the moduli space of objects in a dg category equipped with a relative Calabi-Yau structure. This is joint work Nick Rozenblyum.
• Place: Zoom
• Notes: PDF
  

Thursday, December 2, 2021 at 4:00pm

• Speaker: Camilo Arias Abad (Universidad Nacional de Colombia)
  
• Title: Singular Chains on Lie groups, the Cartan relations and Chern-Weil theory
• Abstract: For a Lie group G, we consider the space of smooth singular chains C(G), which is a differential graded Hopf algebra. We show that the category of sufficiently local modules over C(G) can be described infinitesimally, as the category of representations of a dg-Lie algebra which is universal for the Cartan relations. If G is compact and connected, the equivalence of categories can be promoted to an A-infinity equivalence of dg-categories.
  This result allows for a categorification of the Chern-Weil construction of characteristic classes. The categories mentioned above are quasi-equivalent to the category of infinity-local systems on the classifying space of G. The Chern-Weil homomorphism can be promoted to a Chern-Weil functor taking values in the dg-category of infinity-local systems. The Chern-Weil homomorphism is then recovered by applying the functor to the endomorphisms of the unit object. If time permits, I will discuss how a monoidal version of the equivalence above is related to string topology of classifying spaces and other results in Lie theory.
  The talk is based on joint works with A. Quintero and S. Pineda, and work in progress with M. Rivera.
• Place: Lunt 107