This seminar is held on Tuesdays at 4pm in 210 University Hall.

Spring Quarter, 2023

• April 11, Anne Schilling (UC Davis)
  What about type B?
  Abstract: We will discuss ten reasons of why the combinatorial theory of crystal bases is very helpful in representation theory, geometry, and beyond.
• April 25, David Nadler (UC Berkeley)
  Cocenter of affine Hecke category
  Abstract: I will discuss recent work with Penghui Li and Zhiwei Yun identifying the cocenter of the affine Hecke category with “elliptic character sheaves”, ie automorphic sheaves for a genus one curve.
• May 9, Isabel Vogt (Brown University)
  Special Time and Location: 11am in 117 Fenton Hall
  Curve classes on conic bundles threefolds and applications to rationality
  Abstract: In this talk I’ll discuss joint work with Sarah Frei, Lena Ji, Soumya Sankar and Bianca Viray on the problem of determining when a geometrically rational variety is birational to projective space over its field of definition. Hassett–Tschinkel and Benoist–Wittenberg recently refined the classical intermediate Jacobian obstruction of Clemens–Griffiths by considering torsors under the intermediate Jacobian of a geometrically rational threefold. By work of Hassett–Tschinkel, Benoist–Wittenberg and Kuznetsov–Prokhorov, this obstruction is strong enough to characterize rationality of geometrically rational Fano threefolds of geometric Picard rank 1. Moving into higher Picard rank, we compute this obstruction for conic bundles over P^2. As a consequence of our work, when the ground field is the real numbers, we show that neither the topological obstruction nor the refined intermediate Jacobian obstruction is sufficient to determine rationality.
• May 23, Brian Hepler (U Wisconsin)
  Duality between de Rham complexes with growth conditions (via enhanced ind-sheaves)
  Abstract: For any holomorphic function f on a complex manifold X, we define and study moderate growth and rapid decay objects associated to an enhanced ind-sheaf on X, which are classical sheaves on the real oriented blow-up space of X along f. We show that, in the context of the irregular Riemann–Hilbert correspondence proved by D’Agnolo—Kashiwara (’16), these objects recover the classical notions of the de Rham complexes with moderate growth and rapid decay associated to a holonomic D-module.

  In order to prove the latter, we resolve a recent conjectural duality by Claude Sabbah between these de Rham complexes of holonomic D-modules with growth conditions—initially along a normal crossing divisor–by making the connection with a classic duality result of Kashiwara–Schapira (’96) between certain topological vector spaces. Via a “standard dévissage argument”, we prove Sabbah’s conjecture for arbitrary divisors. As a fun application, we recover the well-known perfect pairing between the algebraic de Rham cohomology and rapid decay homology groups associated to integrable connections on smooth varieties due to Bloch–Esnault (’04) and Hien (’09).

• May 30, Peng Zhou (UC Berkeley)
  KLRW algebra and Fukaya category
  Abstract: Motivated by physics consideration, Mina Aganagic proposed two ways to categorify knot invariants, which are related by Homological Mirror Symmetry.
  The B-side is an additive Coulomb branch, and the A-side is an open part of a multiplicative Coulomb branch with a superpotential. The categories on both sides admit generators, whose endomorphism algebra is expected to be the Khovanov-Lauda-Rouquier-Webster (KLRW) algebra. The B-side link is proven by Webster. We provide the missing link to the A-side. This is work in progress with Aganangic-Shende-Li-Danilenko.

Winter Quarter, 2023

• March 17, Eric Larson (Brown University)
  Special Day and Time: Friday at 11am
  Interpolation for Brill–Noether Curves
  Abstract: In this talk, we determine when there is a Brill–Noether curve of given degree and given genus that passes through a given number of general points in any projective space.

Fall Quarter, 2022

• October 11, Song Yu (Columbia University)
  Open Crepant Transformation Conjecture for toric Calabi-Yau 3-orbifolds
  Abstract: The Crepant Transformation Conjecture of Ruan proposes an identification of the quantum cohomologies and Gromov-Witten theories of K-equivalent manifolds or orbifolds. In this talk, I will discuss various formulations and known cases of the conjecture, and focus on its extension to open Gromov-Witten theory. I will further present an approach to the conjecture for toric Calabi-Yau 3-orbifolds based on techniques from mirror symmetry. The talk is based on joint work with Bohan Fang, Chiu-Chu Melissa Liu, and Zhengyu Zong.
• October 20, Konstantin Aleshkin (Columbia University)
  Special Time/Location: 10am in 260 Tykeson Hall
  Central charges in abelian GLSM
  Abstract: GLSMs are enumerative theories associated to critical loci of functions in GIT quotients and generalize Gromov-Witten theory. A large part of structure of such a theory can be captured by certain generating series of genus 0 invariants called central charges which depend on a matrix factorization of the GLSM.
  These generating series remarkably have two integral representations: Euler type and Mellin-Barnes type. In this talk I will outline our construction of central charges and explain how the integral representations lead to mirror symmetry, Higgs-Coulomb correspondence and wall crossing phenomena for GLSM invariants.
• November 15, Siu-Cheong Lau (Boston University)
  Mirror symmetry for quiver algebroid stacks
  Abstract: Quiver algebroid stacks provide a unified setting for SYZ and noncommutative mirror symmetry. In this talk, I will first provide the background about SYZ and quiver mirror constructions. Then I will explain the terminologies of quiver algebroid stacks and twisted complexes. Finally, I will explain how such geometric objects arise from our mirror construction.
• November 29, Algebra course meeting

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