Welcome to the Berkeley Informal String-Math Seminar
Spring 2025
Organized by Mina Aganagic, Sujay Nair, Peng Zhou and Jasper van de Kreeke. 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. Join via zoom. Youtube video archive
| Date | Speaker | Title |
|---|---|---|
| Feb 3 | Paul Wedrich | A braided monoidal (∞,2)-category from link homology |
| Feb 24 | Joerg Teschner | Schur quantization |
| Mar 3 | Cheuk-Yu Mak | Symplectic annular Khovanov homology and fixed point localisation |
| Mar 17 | Andrei Okounkov | Quasimaps to flag varieties: again |
| Mar 24 | Merlin Christ | Perverse schobers from Lefschetz fibrations |
| Apr 7 | Yalong Cao / Yehao Zhou | Critical loci and their quantum K-theory / Stable envelope for critical loci |
| Apr 14 | Vivek Shende | The skein valued mirror of the topological vertex |
| Apr 21 | Sam DeHority | Quiver folding and cohomological Hall modules |
| Apr 28 | Jasper van de Kreeke | Deformed mirror symmetry for punctured surfaces |
| May 5 | Yanki Lekili | Deformations of cyclic quotient surface singularities via mirror symmetry |
| May 12 | Elise LePage | Aganagic’s invariant is Khovanov homology |
| May 19 | Sujay Nair |
Seminar archive: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.
Feb 3rd, Paul Wedrich (Univ. Hamburg)
A braided monoidal (∞,2)-category from link homology
An early highlight of quantum topology was the observation that the Jones polynomial — and many other knot and link invariants — arise from braided monoidal categories of quantum group representations. In hindsight, this can be understood as underlying reason for the existence of associated topological quantum field theories (TQFTs) in 3 and 4 dimensions.
Not much later, Khovanov discovered a link homology theory that categorifies the Jones polynomial. It associates graded chain complexes to links, from which the Jones polynomials can be recovered. It was therefore speculated that Khovanov homology and its variants may themselves be expressible in terms of certain braided monoidal 2-categories and that there should exist associated TQFTs in 4 and 5 dimensions that may be sensitive to smooth structure.
A major challenge in fully realizing this dream is the problem of coherence: Link homology theories live in the world of homological algebra, where constructing a braided monoidal structure in principle requires an infinite amount of higher and higher homological coherence data. In this talk, I will sketch a proposed solution to this problem, joint with Leon Liu, Aaron Mazel-Gee, David Reutter, and Catharina Stroppel, and explain how we use the language of infinity-categories to build an E2-monoidal (∞,2)-category which categorifies the Hecke braided monoidal category underlying the HOMFLYPT link polynomial.
Feb 24th, Jörg Teschner (Univ. Hamburg)
Schur quantization
Schur quantization refers to a particular type of representation of the quantized algebras of functions on Coulomb branches of vacua of N=2, d=4 supersymmetric quantum field theories, providing a quantum theoretical interpretation of the Schur indices. My talk will describe how the Schur quantization encodes key aspects of the low energy physics of the underlying theory, and how it provides a new quantization of complex Chern-Simons theory related to the analytic Langlands correspondence. The talk is based on joint work with D. Gaiotto, (arXiv:2402.00494, arXiv:2406.09171) and work in progress with F. Ambrosino and D. Gaiotto.
Mar 3rd, Cheuk-Yu Mak (University of Sheffield)
Symplectic annular Khovanov homology and fixed point localisation
Khovanov homology is a powerful link invariant which has numerous applications. It is powerful not only because it is a strong invariant, but also it is functorial and has many relations with other invariants. In 2018, Stoffregen-Zhang realized that there is a spectral sequence from the Khovanov homology of a periodic link to the annular Khovanov homology of its quotient, which was later used by Baldwin-Hu-Sivek to show that Khovanov homology detects the (2,5) torus knot (i.e. any knot whose reduced Khovanov homology is the same as that of the (2,5) torus knot is the (2,5) torus knot). In 2022, Lipshitz-Sarkar extends Stoffregen-Zhang’s result to strongly invertible knot and the infinity page of the spectral sequence is, mysteriously, the cone of the annular Khovanov homology of the axis moving map between the two resolutions of the quotient. In this talk, we will explain the symplectic analogues of these spectral sequences and why they are easy consequences of fixed point localisation theorem of Lagrangian Floer theory. This is a joint work with Hendricks and Raghunath.
Mar 17th, Andrei Okounkov (Columbia University)
Quasimaps to flag varieties: again
Quasimaps to flag varieties have been studied from many, many angles using a wide variety of different techniques. I will discuss some known as well as few new features of these spaces in a way that may suggest generalizations to other targets.
Mar 24th, Merlin Christ (Institut de Mathématiques de Jussieu – Paris Rive Gauche)
Perverse schobers from Lefschetz fibrations
Perverse schobers refer to perverse sheaves of (enhanced) triangulated categories, as introduced by Kapranov-Schechtman. Though their general theory remains conjectural, there exists a robust theory of perverse schobers on surfaces with boundary. In the talk, we will discuss how such perverse schobers naturally arise from cosheaves of Fukaya categories of Lefschetz fibrations constructed by Ganatra-Pardon-Shende. For instance, in the case of the disc, the Fukaya-Seidel category arises as the category of global sections of a perverse schober. We then discuss how the formalism of perverse schobers can facilitate many computations. A key point is that by using sheaves of categories, as opposed to cosheaves, one can construct objects and morphisms in the global category via the gluing of local data. Finally, we will also indicate examples of perverse schobers in higher dimensions arising from repeated Lefschetz fibrations (obtained in joint work with Dyckerhoff and Walde).
Apr 7th, Yalong Cao (Institute of Mathematics Chinese Academy of Sciences)
Critical loci and their quantum K-theory
Critical loci are fundamental objects in geometry and representation theory. In this talk, we will introduce their quantum K-theory by counting (twisted) maps from algebraic curves.
Apr 7th, Yehao Zhou (Shanghai Institute for Mathematics and Interdisciplinary Sciences)
Stable envelope for critical loci
In this talk we will introduce a generalization of Maulik-Okounkov’s stable envelopes to equivariant critical cohomology. In the case of a tripled quiver variety with standard cubic potential, this reduces to MO’s stable envelope for the Nakajima variety of the corresponding doubled quiver along the dimensional reduction. We define non-abelian stable envelopes for quivers with potentials following a similar construction of Aganagic-Okounkov, and use them to relate critical COHAs to the abelian stable envelopes. Explicit computations are given in three examples: 1) Verma modules and higher spin representations of Yangian of sl(2); 2) oscillator representations of shifted Yangian of sl(2); 3) fundamental representation of Yangian of sl(2|1). This talk is based on joint work in progress with Yalong Cao, Andrei Okounkov, and Zijun Zhou.
Apr 14th, Vivek Shende (UC Berkeley, University of Southern Denmark)
The skein valued mirror of the topological vertex
We count holomorphic curves in complex 3-space with boundaries on three special Lagrangian solid tori. The count is valued in the HOMFLYPT skein module of the union of the tori. Using 1-parameter families of curves at infinity, we derive three skein valued operator equations which must annihilate the count, and which dequantize to a mirror of the geometry. We show algebraically that the resulting equations determine the count uniquely, and that the result agrees with the topological vertex from topological string theory.
Apr 21th, Sam DeHority (Yale University)
Quiver folding and cohomological Hall modules
The cohomological Hall algebra of a quiver can serve as model of an algebra of BPS states of a 4d N = 2 theory. We investigate modules for the CoHA which arise from (anti-)involutions of the underlying quiver, and find that the cohomology of moduli stacks of objects with classical type structure groups (e.g. for orthosymplectic quivers) gives a module which satisfies an interesting axiom (the twisted Yetter-Drinfeld condition). We also discuss examples with applications to moduli of classical type bundles on surfaces and the AGT correspondence.
Apr 28th, Jasper van de Kreeke (UC Berkeley)
Deformed Mirror Symmetry for Punctured Surfaces
Mirror symmetry aims at equivalences of Fukaya categories (A-side) and categories of coherent sheaves (B-side). Deformed mirror symmetry aims at matching deformations of A-side and B-side. In this talk, I explain how to do it in case of mirror symmetry for punctured surfaces. We start by constructing gentle algebras and matrix factorizations, which serve as A-side and B-side. Then we deform gentle algebras and show how to transfer the deformation to the B-side. (2023 PhD thesis, supervised by Raf Bocklandt)
May 5th, Yanki Lekili (Imperial College London)
Deformations of cyclic quotient surface singularities via mirror symmetry
Let X_0 be a rational surface with a cyclic quotient singularity (1,a)/r. Kawamata constructed a remarkable vector bundle K_0 on X_0 such that the finite-dimensional algebra End(K_0), called the Kalck-Karmazyn algebra, “absorbs” the singularity of X_0 in a categorical sense. If we deform over an irreducible component of the versal deformation space of X_0 (as described by Kollár and Shepherd-Barron), the bundle K_0 also deforms to a vector bundle K. These results were established using abstract methods of birational geometry, making the explicit computation of the family of algebras challenging. We will utilise homological mirror symmetry to compute End(K) explicitly in the A-model. In the case of a Q-Gorenstein smoothing, this algebra End(K) is a matrix order deforming the Kalck-Karmazyn algebra, and “absorbs” the singularity of the corresponding terminal 3-fold singularity. Time permitting, I will also discuss a conjecture describing irreducible components of the deformation space of X_0 in terms of the finite-dimensional algebra End(K_0). This is based on our joint work (on arXiv) with Jenia Tevelev.
May 12th, Elise LePage (UC Berkeley)
Aganagic’s invariant is Khovanov homology
In recent work, Aganagic proposed a categorification of quantum link invariants using Lagrangian Floer theory in multiplicative Coulomb branches equipped with a potential. The braid group action arises from monodromy of this potential. We show that in the case g=sl(2), this braid group action agrees with the braid group action constructed by Webster, which proves that Aganagic’s proposal gives a symplectic construction of Khovanov homology. This talk is based on 2505.00327 with Vivek Shende.