# Caltech/USC Algebra & Geometry Seminar

## Welcome

Welcome to the homepage of the newly-formed joint algebra and geometry seminar at California Institute of Technology and University of Southern California! We meet biweekly on Thursday afternoons, alternating between the Caltech and USC campuses. Each meeting consists of two hour-long talks followed by a seminar dinner. Topics at our seminar include but are not limited to algebra, representation theory, algebraic geometry, mirror symmetry, and mathematical physics.

## Spring 2024 Info

Organizers: Anne Dranowski and Song Yu

Time: Thursdays 2:30 - 5:00pm

Caltech location: Linde Hall (37) room 187

USC location: Kaprelian Hall room 414

## Upcoming Meeting

The seminar for the academic year 2023-24 has concluded. Thank you for the participation!

## Spring 2024 Schedule

### January 18 @ Caltech

Tariq Syed (USC): Cohomological properties of cyclic coverings

Abstract: Many examples of topologically contractible smooth affine complex varieties are given by cyclic coverings. In this talk, we discuss some new results on the cohomological properties of such cyclic coverings.

Anne Dranowski (USC): Asymptotic characters of modules for parity KLR algebras

Abstract: It has been shown that a category O associated with the affine Grassmannian is equivalent to the category of modules for a decorated KLR algebra. The affine Grassmannian gives rise to a perfect basis called the Mirkovic-Vilonen basis, while the KLR algebra gives rise to a perfect basis called Lusztig’s canonical basis. In previous work I used equivariant volumes to compare such bases. This talk will explain a relevant notion of volume for well-behaved module categories such as KLR-mod, and how it is used (in joint work in progress, with Alexis Leroux-Lapierre) to study the MV-KLR change of basis.

### February 1 @ USC

Peng Zhou (UCBerkeley): KLRW algebra from Floer theory

Abstract: Khovanov homology and its relatives are known to be governed by the representation theory of KLRW algebras (quiver Hecke algebra). Here we discuss a way to realize the KLRW algebra as the endomorphism algebra of certain Lagrangian in a wrapped Fukaya category. This is joint work with Mina Aganagic, Ivan Danilenko, Yixuan Li and Vivek Shende.

Josef Svoboda (Caltech): q-Series invariants of plumbed manifolds

Abstract: Plumbed manifolds are certain 3-manifolds which describe the topology of singularities of algebraic surfaces. I will present two results about q-series invariants of plumbed 3-manifolds defined by Gukov, Putrov, Pei and Vafa. The first result shows that a certain linear combination of these invariants coincides for two manifolds with the same universal abelian cover. The second result gives a formula for the invariants in terms of a rational function in a single variable, in the special case of Seifert manifolds. Based on joint work with Sergei Gukov and Ludmil Katzarkov.

### February 15 @ Caltech

David Nadler (UCBerkeley): Filling Legendrians with barcodes

Abstract: I'll discuss an ongoing project, joint with Alvarez-Gavela and Eliashberg, to construct fillings of Legendrians via moduli spaces of barcodes. The first step involves a closer look at various phenomena in and around rulings of Legendrian knots.

Sheel Ganatra (USC): Integrality of mirror maps and arithmetic homological mirror symmetry

Abstract: I will explain joint work with Hanlon-Hicks-Pomerleano-Sheridan which proves that the "integrality of Taylor coefficients of mirror maps" conjecture follows from an arithmetic refinement of homological mirror symmetry (HMS). Our work also establishes this latter arithmetic HMS statement - and thereby the integrality conjecture - for Greene-Plesser mirror pairs.

### February 29 @ USC

Ivan Danilenko (UCBerkeley): Mirror symmetry for Coulomb branches

Abstract: Mirror symmetry predicts equivalence between Fukaya categories of Coulomb branches (A-side) and derived categories of coherent sheaves of Coulomb branches (B-side). In the talk, I will focus on how this relation works in the decategorified setting. It will explicitly show how to identify a cohomology space associated with the A-side with the equivariant K-theory on the B-side. Based on joint work with A. Smirnov, and with M. Aganagic, Y. Li, V. Shende, P. Zhou.

Morgan Opie (UCLA): Enumerating stably trivial topological vector bundles with higher real K-theories

Abstract: The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking questions about topological vector bundles. But, in general, there are many non-equivalent vector bundles with the same K-theory class. Bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest CW complexes.

Building on work of Hu, we use Weiss calculus and a little chromatic homotopy theory to translate vector bundle enumeration questions to tractable stable homotopy theory computations. We compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. This is joint work with Hood Chatham and Yang Hu.

### March 7 @ Caltech

Tom Gannon (UCLA): Proof of the Ginzburg-Kazhdan conjecture

Abstract: The main theorem of this talk will be that the affine closure of the cotangent bundle of the basic affine space (also known as the universal hyperkahler implosion) has symplectic singularities for any reductive group, where essentially all of these terms will be defined in the course of the talk. After discussing some motivation for the theory of symplectic singularities, we will survey some of the basic facts that are known about the universal hyperkahler implosion and discuss how they are used to prove the main theorem. Time permitting, we will also discuss a recent result, joint with Harold Williams, which identifies the universal hyperkahler implosion in type A with a Coulomb branch in the sense of Braverman, Finkelberg, and Nakajima, confirming a conjectural description of Dancer, Hanany, and Kirwan.

Claire Levaillant (USC): Tangles of types En and a representation of the Birman-Murakami-Wenzl algebra of type E6

Abstract: We generalize Cohen-Gijsbers-Wales tangles of type Dn to Coxeter type En and introduce some relations on these tangles. Using the tangles, we build a representation of the parameters based Birman-Murakami-Wenzl algebra of type E6.

We show that as a representation of the Artin group, this representation is equivalent to the parameters based representation which was introduced by Cohen and Wales as a generalization of the Lawrence-Krammer representation of the braid group. The latter representation became famous as it is the only known representation of the braid group that is faithful. Likewise, the Cohen-Wales representation of the Artin group of type E6 is the only known faithful representation of the Artin group of type E6.

We use our representation to find a reducibility criterion for the faithful Cohen-Wales representation, depending on the values of its two parameters. We further derive values of the parameters for which the Birman-Murakami-Wenzl algebra of type E6 is not semisimple.

This work was achieved after the completion of my Ph.D. at Caltech under David Wales (1939-2023). This talk is given to honor his memory.

### March 21 @ Caltech

Chi Zhang (Caltech): Plücker coordinate method on the mirror symmetry of flag varieties

Abstract: We will discuss the Plücker coordinate method on the mirror symmetry of flag varieties. In the first part, we will review the current study on mirror symmetry of flag varieties. In the second part, we will introduce the construction of our Plücker coordinate superpotential. We will explain the idea of our proof on the first Chern class conjecture that the critical values of mirror superpotential are the eigenvalues of quantum multiplication of first Chern class in the cases of complex flag varieties. In the third part, we will show some examples about the images of the mirror isomorphism and briefly mention how quantum Schubert calculus plays a role in the story. The talk is based on my joint work with Changzheng Li, Konstanze Rietsch, and Mingzhi Yang.

Hiroshi Iritani (Kyoto University): Decomposition of quantum cohomology under blowups

Abstract: Quantum cohomology is a deformation of the cohomology ring defined by counting rational curves. A close relationship between quantum cohomology and birational geometry has been expected. For example, when the quantum parameter q approaches an "extremal ray", the spectrum of the quantum cohomology ring clusters in a certain way (predicted by the corresponding extremal contraction), inducing a decomposition of the quantum cohomology. In this talk, I will discuss such a decomposition for blowups: quantum cohomology of the blowup of X along a smooth center Z will decompose into QH(X) and (codim Z-1) copies of QH(Z). The proof relies on Fourier analysis and shift operators for equivariant quantum cohomology. We can describe blowups as a VGIT of a certain space W with C^* action. Then the equivariant quantum cohomology of W acts as a "global" mirror family connecting X and its blowup.

### *April 6 @ Caltech

Southern California Algebraic Geometry Seminar

### April 18 @ USC

Jessie Loucks-Tavitas (UWashington): Algebra and geometry of camera resectioning

Abstract: Algebraic vision, lying in the intersection of computer vision and projective geometry, is the study of three-dimensional objects being photographed by multiple pinhole cameras. Two natural questions arise:

Triangulation: Given multiple images as well as (relative) camera locations, can we reconstruct the scene or object being photographed?

Resectioning: Given a 3-D object or scene and multiple images of it, can we determine the (relative) positions of the cameras in the world?

We will discuss and characterize certain algebraic varieties associated with the camera resectioning problem. As an application, we will derive and re-interpret celebrated results in computer vision due to Carlsson, Weinshall, and others related to camera-point duality. This is joint work with Erin Connelly and Timothy Duff.

Vasily Krylov (MIT): From geometric realizations of affine Hecke algebras to character formulas

Abstract: I will explain an approach allowing to extract character formulas for irreducible \hat{g}-modules at the positive level from the geometry of some open subvarieties of Steinberg varieties. The approach uses Bezrukavnikov’s “exotic coherent” categorification of the canonical basis of the affine Hecke algebra for g. We will discuss the first nontrivial example and will see that it leads to explicit character formulas for all irreducible modules (with integral highest weights) in categories O for certain Vertex algebras coming from the 4D/2D correspondence. Based on joint works with Bezrukavnikov, Kac, and Suzuki.

### May 2 @ Caltech

Siddarth Kannan (UCLA): Weight zero compactly supported cohomology of moduli spaces of pointed hyperelliptic curves

Abstract: I will discuss recent joint work with Madeline Brandt and Melody Chan, studying the weight zero compactly supported cohomology of the moduli space H_g,n of n-pointed hyperelliptic curves of genus g. We find a normal crossings compactification of this space using the theory of admissible G-covers, and then study the homology of the resulting dual complex using techniques from combinatorial topology. Our main result is a graph sum formula for the weight zero Euler characteristic, considered as a virtual representation of the symmetric group.

Yefeng Shen (UOregon): GW/FJRW correspondence for quasi-homogeneous polynomials

Abstract: For a quasi-homogeneous polynomial, we study a correspondence between the genus-zero Gromov-Witten theory of the hypersurface determined by the polynomial and the genus-zero Fan-Jarvis-Ruan-Witten theory of the singularity determined by the polynomial. This generalizes the genus zero Landau-Ginzburg/Calabi-Yau correspondence studied in the work of Chiodo-Iritani-Ruan, when the hypersurface is Calabi-Yau. The Gamma structures in the GW/FJRW theory play a key role in this story. The talk is based on work (in progress) joint with Jie Zhou, and earlier work (arXiv:2309.07446) joint with Ming Zhang.

## Fall 2023 Schedule

### October 5 @ Caltech

Wenyuan Li (USC): Lagrangian cobordism functor in microlocal sheaf theory

Abstract: Consider the cosphere bundle of a manifold with the standard contact structure and Lagrangian cobordisms in its symplectization between Legendrian submanifolds. We construct a fully faithful functor between the categories of sheaves with singular support on the corresponding Legendrians, after enhancing the category at the negative end by local systems on the cobordism. In the special case where the contact manifold is the 1-jet bundle, we give a geometric model of the enhanced category as sheaves on the original manifold times the real line with microsupport on the cobordism. This gives a sheaf theoretic description of the maps between Legendrian contact homologies with coefficients enhanced by chains of the based loop spaces, defined using pseudoholomorphic curves.

Song Yu (Caltech): Open/closed correspondence and mirror symmetry

Abstract: We will discuss a mathematical formulation of the open/closed correspondence originally proposed by Mayr in physics, which is a correspondence in genus zero between the open Gromov-Witten theory of toric Calabi-Yau 3-folds and the closed Gromov-Witten theory of toric Calabi-Yau 4-folds. We will discuss different aspects of the correspondence on both the A- and B-sides of mirror symmetry. We will also discuss some applications. This is based on joint works with Chiu-Chu Melissa Liu and Zhengyu Zong.

### October 19 @ Caltech

Mohan Swaminathan (Stanford University): Constructing smoothings of stable maps

Abstract: The space of holomorphic maps from compact Riemann surfaces to a complex projective manifold X (in a fixed genus and degree) is typically non-compact, but a theorem of Gromov (refined by Kontsevich) shows that any sequence in this space always has some subsequential limit which is a "stable map" (i.e., a holomorphic map to X, defined on a compact complex curve with at worst nodal singularities, and having finite automorphism group). The compact moduli space of stable maps forms the foundation of the Gromov-Witten theory of X. Interestingly, it turns out that most stable maps which have "ghosts" (i.e., irreducible components of the domain on which the map is constant) can neverappear as the limit of any sequence described above. I will explain this background and then describe recent work (joint with Fatemeh Rezaee), where we produce a large class of stable maps (with ghosts of any genus) and give a constructive proof that these do indeed occur as limits. The proof relies on the study of explicit model cases and on general results from deformation theory.

Wern Yeong (UCLA): The algebraic Green-Griffiths-Lang conjecture for the complement of a very general hypersurface in projective space

Abstract: A complex algebraic variety is said to be Brody hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. The Green-Griffiths-Lang conjecture predicts that varieties of (log) general type are hyperbolic outside of a proper subvariety called an exceptional locus. We prove an algebraic version of this Conjecture, with respect to Demailly’s algebraic version of hyperbolicity, for the complement of a very general degree 2n hypersurface in Pn. Moreover, for the complement of a very general quartic plane curve, we completely characterize the exceptional locus as the union of the flex and bitangent lines. Based on joint work with Xi Chen and Eric Riedl.

### November 2 @ USC

Jose Yanez (UCLA): MMP for 3-dimensional Kahler generalized pairs

Abstract: In the last decade there have been several contributions toward establishing the Minimal Model Program for Kahler varieties. Even though there is not an easy generalization of the projective MMP to the Kahler MMP (no ample divisor, less subvarieties, key theorems of the projective MMP are not available), several tools have been developed in order to get similar results from the projective case. In this talk, I will introduce a notion of generalized pairs for Kahler varieties and explain how they give a more flexible notion to study the MMP in the Kahler setup. This is joint work with Omprokash Das and Christopher Hacon.

Joseph Helfer (USC): Hodge theory, modular forms, counting curves on surfaces, and maybe some other stuff

Abstract: Modular forms are certain kinds of generating functions with nice finite-dimensionality properties that allow you to prove combinatorial identities by checking finitely many cases. Famously, they have been used to solve many problems in number theory and arithmetic geometry. More recently, highly nifty results of Borcherds and Kudla-Milson have led to applications of modular forms to enumerative problems in (complex) algebraic geometry as well. In this talk, I will give an overview of some of these ideas, and describe a particular application of them to a problem about rational elliptic surfaces in joint work with François Greer and John Sheridan.

### November 16 @ Caltech

Nathan Priddis (Brigham Young University): Mirror Symmetry for Nonabelian LG models

Abstract: Almost 30 years ago, Berglund and Hübsch proposed a version of Mirror Symmetry for quasihomogeneous potentials, which was later completed by Berglund and Henningson, and then again independently by Krawitz. This has come to be known as BHK mirror symmetry. Basically it says that to a Landau-Ginzburg pair (W,G) of a potential W and a group of symmetries G, we can relate its BHK mirror (W‘,G‘) by a simple rule. Remarkably, this simple rule predicts other instances of mirror symmetry, including the very first known example of mirror symmetry for the quintic threefold. Recently, we have discovered an extension of BHK mirror symmetry that allows for nonabelian symmetries of W, namely we can allow G to be a nonabelian group. In this presentation, we will discuss BHK mirror symmetry, its relation with other forms of mirror symmetry, and the extension of BHK mirror symmetry to nonabelian groups.

Ming Zhang (UCSD): Quantum K-invariants of Grassmannian via Quot scheme

Abstract: Quantum K-theory is a K-theoretic generalization of Gromov-Witten theory, and genus-zero quantum K-invariants of a target manifold X are defined as the Euler characteristics of coherent sheaves over the Kontsevich moduli space, parametrizing stable maps from (possibly nodal) rational curves to X.

In this talk, we consider the target space to be the Grassmannian. The space of maps from the Riemann sphere to the Grassmanninan has a simpler compactification -- the Quot scheme. We define K-theoretic invariants using the Quot scheme and provide explicit bialternant-type formulas for the one-pointed K-theoretic Quot scheme invariants. We prove that these invariants determine quantum K-invariants of the Grassmannian with up to three marked points. Our approach is "stacky" in the sense that the Quot scheme parametrizes maps not to the Grassmannian but to the GIT quotient stack containing it. I will explain the advantages of our stacky approach by providing the following applications:

We present simple, closed formulas for the inverse of the quantized metric and the structure constants in quantum K-theory.

We re-prove the finiteness property of the quantum K-product using a vanishing result for the K-theoretic Quot scheme invariants.

We derive relations in the quantum K-ring using a reduction map.

We obtain a complete set of explicit formulas (for one-pointed invariants, quantized metric, structure constants, etc.) in the rank 2 case.

The talk is based on joint work with Shubham Sinha.

### November 28 @ USC

Daniele Garzoni (USC): Conjugacy classes of derangements in finite groups of Lie type

Abstract: Given a group G acting on a set, an element of G is called a derangement if it acts without fixed points. Luczak-Pyber and Fulman-Guralnick showed that if G is a finite simple group acting transitively, then the proportion of derangements is bounded away from zero absolutely. I will discuss a conjugacy-class version of this result for groups of Lie type, obtained in joint work with Sean Eberhard. I would like to discuss mainly two things: (i) why derangements are interesting, and (ii) explain some interesting connections between the proof of the result and the subject of "anatomy of polynomials", which essentially studies divisors of random polynomials.

Aravind Asok (USC): Affine grassmannians in motivic homotopy theory and applications

Abstract: I will give a brief introduction to motivic homotopy theory and discuss how the affine Grassmannian can be used to model certain loop spaces. I will conclude by discussing some joint work with Tom Bachmann and Mike Hopkins wherein we use the above facts to establish certain ``exotic periodicities" in motivic homotopy theory.