# Algebra Seminar

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

### Spring Quarter, 2022

- April 12,
**Justin Hilburn** (University of Waterloo)

**Zoom Meeting:**512 854 8536

3d mirror symmetry and 2-categories O
**Abstract**: 3d mirror symmetry predicts an equivalence between 2-categories associated to dual hyperKahler stacks. The first 2-category is of an algebro-geometric flavor and has constructions due to Kapustin/Rozansky/Saulina and Arinkin. The second category depends on symplectic topology and has a conjectural description in terms of the 3d generalized Seiberg-Witten equations. Both of these 2-categories can be thought of as categorifications of category O for symplectic resolutions.

In this talk I will explain work in progress with Ben Gammage and Aaron Mazel-Gee proving a variant 3d mirror symmetry for Gale dual hypertoric varieties. In particular, we give a combinatorial model for the symplectic 2-category in terms of perverse schobers.

- May 19,
**Tudor Padurariu** (Columbia University)

**Special Time/Location** : 11am in 260 Tykeson Hall

Categorical Hall algebras of quivers with potential
**Abstract**: Kontsevich-Soibelman defined the cohomological Hall algebra (CoHA) of a quiver with potential. By a result of Davison-Meinhardt, CoHAs are deformations of the universal enveloping algebra of the BPS Lie algebra of the quiver with potential. One can also define categorical and K-theoretic Hall algebras of a quiver with potential. Examples of such Hall algebras are (positive parts of) quantum affine algebras. I will introduce the categorical and K-theoretic replacements of the BPS spaces and explain how to prove versions of the Davison-Meinhardt theorem in these contexts. These results have applications in Donaldson-Thomas theory of local surfaces and in the study of (Porta-Sala) Hall algebras of surfaces.

### Winter Quarter, 2022

- January 11,
**Nick Addington** (UO)

Hodge number are not derived invariants in positive characteristic
**Abstract**: Derived categories of coherent sheaves behave a lot like cohomology, so it’s natural to ask which cohomological invariants are preserved by derived equivalences. After discussing the motivation and previous results, I’ll present a derived equivalence between Calabi-Yau 3-folds in characteristic 3 with different Hodge numbers; this couldn’t happen in characteristic 0. The project has a substantial computer algebra component.

- January 18,
**Nawaz Sultani** (University of Michigan)

Gromov–Witten invariants of some non-convex complete intersections
**Abstract**: For convex complete intersections, the Gromov-Witten (GW) invariants are often computed using the Quantum Lefshetz Hyperplane theorem, which relates the invariants to those of the ambient space. However, even in the genus 0 theory, the convexity condition often fails when the target is an orbifold, and so Quantum Lefshetz is no longer guaranteed. In this talk, I will showcase a method to compute these invariants, despite the failure of Quantum Lefshetz, for orbifold complete intersections in stack quotients of the form [V // G]. This talk will be based on joint work with Felix Janda (Notre Dame) and Yang Zhou (Fudan), and upcoming work with Rachel Webb (Berkeley).

- February 8,
**Anna Romanov** (University of Sydney)

**Zoom Meeting:**512 854 8536

Costandard Whittaker modules and contravariant pairingss
**Abstract**: I’ll discuss recent work with Adam Brown (IST Austria) in which we propose a new definition of costandard Whittaker modules for a complex semisimple Lie algebra using contravariant pairings between Whittaker modules and Verma modules. With these costandard objects, blocks of Milicic—Soergel’s Whittaker category have the structure of highest weight categories. This allows us to establish a BGG reciprocity theorem for Whittaker modules. Our costandard objects also give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the flag variety.

- February 15,
**Sasha Polishchuk** (UO)

Ranks of polynomials
**Abstract**: This is a report on joint work with Kazhdan. There are several natural notions of rank for homogeneous polynomials in several variables.

Perhaps, the most interesting is the Schmidt rank also known as strength. It is defined as the minimal number r such that the polynomial f can be written as the sum of r products of polynomials of smaller degree (for example, a polynomial has Schmidt rank 1 if and only if it is reducible). Our first result is that over an algebraically closed field the Schmidt rank of f is related to the codimension of the singular locus of the hypersurface f=0. We conjecture that the Schmidt rank behaves well under the extension of fields (namely, it changes by a factor bounded only in terms of the degree). We prove this conjecture for polynomials of degree at most 4. If time allows, I will also discuss the slice rank of polynomials, which corresponds to looking at the minimal codimension of a linear subspace in a projective hypersurface.

- March 8,
**Junwu Tu** (ShanghaiTech University)

**Zoom Meeting:**512 854 8536

Enumerative invariants of Calabi-Yau categories
**Abstract**: We shall review Costello’s definition of Gromov-Witten type invariants in all genus associated with Calabi-Yau A-infinity categories, following Costello (abstract definition) and Caldararu-T.(explicit form using ribbon graphs). Then we survey on known calculations of these invariants (named categorical enumerative invariants). We also report on some work in progress (joint with Lino Amorim, and Yefeng Shen) in this direction.

### Fall Quarter, 2021

- October 5, Organizational Meeting
- October 12,
**Ben Elias** (UO)

Quantum geometric Satake and K-theory
**Abstract**: There’s a certain geometric setup where, when you apply the equivariant cohomology functor, you get something called singular Soergel bimodules. The geometric Satake equivalence states that these singular Soergel bimodules are equivalent to representations of a semisimple lie algebra. I’ll explain this in easy examples – geometric Satake for dummies, if you will.

If instead you apply the equivariant K-theory functor, you get K-theoretic singular Soergel bimodules, a new concept due to myself and Geordie Williamson. Our quantum geometric Satake equivalence (conjecture) roughly states that K-theoretic singular Soergel bimodules are equivalent to representations of the corresponding quantum group. As stated, this is false, and the fix is rather surprising.

- October 26,
**Alexandra Utiralova** (MIT)

Harish-Chandra bimodules in complex rank
**Abstract**: Deligne tensor categories are defined as an interpolation of the categories of representations of groups GL_n, O_n, Sp_{2n} or S_n to the complex values of the parameter n. One can extend many classical representation-theoretic notions and constructions to this context. These complex rank analogues of classical objects provide insights on their stable behavior patterns as n goes to infinity.

I will talk about some of my results on Harish-Chandra bimodules in Deligne categories. It is known that in the classical case simple Harish-Chandra bimodules admit a classification in terms of W-orbits of certain pairs of weights. However, the notion of weight is not well-defined in the setting of Deligne categories. I will explain how in complex rank the above-mentioned classification translates to a condition on the corresponding (left and right) central characters.

Another interesting phenomenon arising in complex rank is that there are two ways to define Harish-Chandra bimodules. That is, one can either require that the center acts locally finitely on a bimodule M or that M has a finite K-type. The two conditions are known to be equivalent for a semi-simple Lie algebra in the classical setting, however, in Deligne categories that is no longer the case. I will talk about a way to construct examples of Harish-Chandra bimodules of finite K-type using the ultraproduct realization of Deligne categories.

- November 9,
**Yefeng Shen** (UO)

Gamma structures in Landau-Ginzburg models
**Abstract**: Gamma integral structures in algebraic varieties were introduced by Iritani and Katzakov-Kontsevich-Pantev a decade ago. It is a bridge to connect the bounded derived category of coherent sheaves to the quantum cohomology of the same variety. Similar structures also exist in Landau-Ginzburg models. In this talk, we will explain the Gamma structures on the Landau-Ginzburg model that consists of a quasi-homogeneous polynomial W and a finite abelian group G. The Gamma structures there will be a bridge to connect the G-equivariant matrix factorizations of W (algebraic aspect) to the Fan-Jarvis-Ruan-Witten theory (analytic aspect) of the LG model. Asymptotic analysis plays a key role in the construction. If time allows, we will also talk about the relation to the Stokes structure of a local system (topological aspect) given by the same LG model. The talk is based on work joint with Ming Zhang.

- November 16,
**Xiaohan Yan** (UC Berkeley)

Quantum K-theory of flag varieties via non-abelian localization
**Abstract**: Quantum cohomology may be generalized to K-theoretic settings by studying the “K-theoretic analogue” of Gromov-Witten invariants defined as holomorphic Euler characteristics of sheaves on the moduli space of stable maps. Generating functions of such invariants, which are called the (K-theoretic) ”big J-functions”, play a crucial role in the theory. In this talk, we provide a reconstruction theorem of the permutation-invariant big J-function of partial flag varieties (regarded as GIT quotients of vector spaces) using a family of finite-difference operators, from the quantum K-theory of their associated abelian quotients which is well-understood. Generating functions of K-theoretic quasimap invariants, e.g. the vertex functions, can be realized in this way as values of various twisted big J-functions. We also discuss properties of the level structures as applications of the method. A portion of this talk is based on a joint work with my advisor Alexander Givental.

- November 23,
**Nicholas Proudfoot** (UO)

Equivariant log concavity in the cohomology of configuration spaces
**Abstract**: Consider the space whose points consist of n-tuples of distinct complex numbers. The Betti numbers of this space are called Stirling numbers, and they form a log concave sequence by a theorem of Isaac Newton. I will state a conjectural generalization of this result that takes into account the action of the symmetric group by permuting the points. The full conjecture is open, but I will explain how to leverage the theory of representation stability to prove infinitely many cases.

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