Welcome to the Berkeley Informal String-Math Seminar
Spring 2025
Organized by Mina Aganagic, Sujay Nair, Spencer Tamagni and Peng Zhou. Weekly on Mondays 2:10 PM (PT) at 402 Physics South. You are invited for lunch on the 4th floor of Physics South before the seminar. Subscribe to the mailing list. Youtube video archive
| Date | Speaker | Title |
|---|---|---|
| Sep 8 | Andrei Okounkov | Critical stable envelopes |
| Sep 15 | Sergei Cherkis | Towards Constructing G Monopoles |
| Sep 22 | Albert Schwarz | A new approach to superstring , slide |
| Sep 29 | Che Shen | Quasimaps to the Flag Variety and Tilting Modules in Category O |
| Oct 6 | Sujay Nair | Inverse Hamiltonian reduction for W-algebras in type A |
| Oct 13 | Peng Zhou | Fukaya Category approach to Categorification of Quantum Group (I) |
| Oct 20 | Peng Zhou | Fukaya Category approach to Categorification of Quantum Group (II) |
| Oct 27 | Qiuyu Ren | (Khovanov) skein lasagna modules of 4-manifolds |
| Nov 10 | Qiuyu Ren | Khovanov skein lasagna modules of 4-manifolds, II |
| Nov 17 | Ian Sullivan | Skein lasagna modules and Khovanov homology for $S^1 \times S^2$ |
| Dec 8 | Tom Gannon | Coulomb branches and functoriality in the geometric Langlands program |
09/08 Andrei Okounkov (Columbia)
Title: Critical stable envelopes
Abstract: This will be a report on a joint project with Yalong Cao, Yehao Zhou, and Zijun Zhou, in which we systematically explore the world of stable envelopes in critical cohomology and critical K-theory. We will compare and contrast old stable envelope to the more general critical stable envelopes, with an emphasis on the contrasting features.
09/15: Sergey Cherkis (Arizona University)
Title: Towards Constructing G Monopoles
Abstract: Nahm’s construction of magnetic monopoles produced all monopoles with gauge (structure) group G=U(n). It was generalized by Hurtubise and Murray to SO and Sp monopoles. For any compact Lie group G, in principle, Nahm’s construction can be used to obtain monopoles in any given unitary representation of G. For example, for G=E_8, the smallest such representation has dimension 248, making this construction highly impractical, if at all tractable.
We explore an alternative encoding of monopole data, based of recent work of Bielawski and Foscolo, motivating it from the D-brane configuration in Kleinian surface of ADE type. The type of the surface corresponds to the monopole structure group G. While it seems to reproduce the monopole moduli space correctly, the actual monopole construction remains an open question. This talk is an attempt to reconstruct the monopole from the string theory analysis of the concrete brane configuration.
09/22 Albert Schwarz(UC Berkeley)
We show how starting with one-string space of states in BRST formalism one can construct a large class of physical quantities containing, in particular, scattering amplitudes for bosonic string and superstring. The same techniques work for heterotic string.
Title: Quasimaps to the Flag Variety and Tilting Modules in Category O
Abstract: A quasimap from a curve to a GIT quotient is a map to the stack quotient that is generically stable. An open subset of quasimaps from P^1 to the flag variety, usually called Laumon space or handsaw quiver variety, is known to be closely related to the representation theory of gl_n. In particular, one can construct an action of gl_n on the cohomology of Laumon spaces via geometric correspondences. In this talk, I will explain how to construct an action of gl_n on the cohomology of the full moduli space of quasimaps and explore its relation to tilting modules in Category O.
10/06. Sujay Nair (UC Berkeley)
Title: Inverse Hamiltonian reduction for W-algebras in type A
Abstract: Inverse Hamiltonian reduction refers to a series of conjectural relations between W-algebras corresponding to distinct nilpotent orbits in a Lie algebra. I will outline a proof of this conjecture in type A that relies on novel geometric methods. Along the way, we shall encounter the localization of vertex algebras and, time permitting, speak briefly on the deformation theory thereof. To build intuition, I shall focus on the finite type analogue of this story, where such techniques are more commonplace. This talk is based on joint work with Dylan Butson.
10/13, 10/20. Peng Zhou (UC Berkeley)
Title: Fukaya Category approach to Categorification of Quantum Group
This is a series of two lectures on categorifying quantum group U_q(g) using Fukaya categories proposed by Mina Aganagic. In the first lecture, we will give an introduction to the theory, focusing on the geometric setup, basic calculations and connection to representation theory; in the second lecture we will discuss the TQFT structures underlying our theory, proposing the definition of Hopf(ish) category as a categorication of Kapranov-Schechtman’s factorizable perverse sheaf and primitive bialgebra. This is a joint work with Mina Aganagic, Vivek Shende, Elise LePage.
10/27 Qiuyu Ren (UC Berkeley)
Title: (Khovanov) skein lasagna modules of 4-manifolds
Abstract: For any well-behaved link homology theory for links in the 3-sphere, Morrison–Walker–Wedrich defined an invariant for smooth 4-manifolds called skein lasagna modules, which can be viewed as an upgrade of the input link homology theory. We review the definition and investigate some formal properties of the invariant. We also introduce a few variations of the invariant. In the case where the input link homology theory is Khovanov homology, the resulting skein lasagna modules are strong enough to detect exotic 4-manifolds, i.e. smooth 4-manifolds that are homeomorphic but not diffeomorphic. This is a sequence of two expository talks, partly based on joint work with I. Sullivan, P. Wedrich, M. Willis, and M. Zhang.
11/10 Qiuyu Ren (UC Berkeley)
Title: (Khovanov) skein lasagna modules of 4-manifolds, II
Abstract: This is a continuation of the previous talk about skein lasagna modules. We review some features of the Khovanov homology and its Lee deformation. We examine the resulting skein lasagna modules with these two theories as inputs, extract a lasagna version of Rasmussen’s s-invariant, and state some formal properties. We then show that Khovanov/Lee skein lasagna modules and lasagna s-invariants can distinguish exotic pairs of 4-manifolds, namely smooth 4-manifolds that are homeomorphic but not diffeomorphic. Background on smooth 4-manifolds will be explained when necessary. This is joint work with Mike Willis.
11/17 Ian Sullivan (UC Davis)
Title: Skein lasagna modules and Khovanov homology for $S^1 \times S^2$
Abstract: Skein lasagna modules are invariants of smooth 4-dimensional manifolds capable of detecting exotic phenomena. Wall-type stabilization problems ask about the behavior of exotic phenomena under various topological operations. In this talk, we will describe the invariants we use and the necessary properties. We describe, with Wall-type external stabilization problems as motivation, a method for computing the Khovanov skein lasagna module of $S^2 \times S^2$. Rozansky-Willis homology, an invariant of links in connect-sums of $S^1 \times S^2$, makes an appearance in this computational technique, and we establish a relationship between the skein lasagna module of a family of 4-manifolds and these invariants. This computational method shows that Khovanov skein lasagna modules over $\mathbb {Q}$ are annihilated by external stabilizations, and has proven useful for establishing the functoriality of Rozansky-Willis homology.
12/08 Tom Gannon (UC Riverside)
Title: Coulomb branches and functoriality in the geometric Langlands program
Abstract: In 2017, Braverman-Finkelberg-Nakajima gave a precise definition of the Coulomb branch of a 3d N = 4 supersymmetric gauge theory of cotangent type associated to a complex reductive group G and a finite dimensional complex representation N. In our first part of this talk, we will recall the definition and basic properties of such Coulomb branches, as well as give some motivation for its study. We will then discuss an alternate definition of the Coulomb branch for (G, N) due to Teleman assuming that (G, N) satisfies a certain condition that we call gluable. In the second part of this talk, we will discuss functoriality of these Coulomb branches: in other words, we will discuss how, given a map of reductive groups H –> G, one can recover the Coulomb branch associated to the pair (H, N) from the pair (G, N), assuming our map is gluable. We will also discuss the connections to functoriality in the geometric Langlands program and explain our conjecture, recently proved by Victor Ginzburg, that the Coulomb branch of (G, N) determines the S-dual (in the sense of, say, Ben-Zvi–Sakellaridis–Venkatesh) of the pair (G, N). This is joint with Ben Webster.
Seminar archive:Spring 2025, Fall 2024 Spring 2024, Fall 2023, Spring 2023, Fall 2022, Spring 2022, Fall 2021, Spring 2021, Fall 2020, Summer 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Fall 2017, Spring 2017, Fall 2016
A note to the speakers: This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas. The audience may fail to appreciate the glory of the result, otherwise.