# Algebra Seminar 2019

This seminar is held via Zoom with Meeting ID: 512 854 8536 during the Spring quarter.

### Spring Quarter, 2020

- May 7,
**Junliang Shen** (MIT)

2:00-2:30pm

Hitchin system, hyperKahler geometry and P=W conjecture
- May 21,
**Franco Rota** (Rutgers University)

1:00-1:30pm

Bridgeland sability conditions for projective Kleinian orbisurfaces
**Abstract**: In this talk, I will present a construction of Bridgeland stability conditions on the derived category of smooth quasi-projective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the general construction for smooth surfaces and Bridgeland’s work on Kleinian singularities. It hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the Toën-Hirzebruch-Riemann-Roch theorem. All this is joint work with Bronson Lim.

- May 26,
**Lei Wu** (University of Utah)

11:00-11:30am

### Winter Quarter, 2020

- January 13,
**Anthony Licata** (Australian National University)

**Special Time:**Monday at 4pm in 210 Deady

Artin groups, dynamics, and stability conditions
**Abstract**: The goal of this talk will be to describe a program to study the Artin braid groups associated to arbitrary Coxeter groups analogously to how one studies mapping class groups of surfaces. After recalling a bit about mapping class groups and Teichmuller space, I’ll explain how various important structures in Teichmuller theory have analogs in homological algebra, and how these analogs can be used to study braid groups.

- January 21,
**Eric Ramos** (UO)

The categorified graph minor theorem, and graph configuration spaces
**Abstract**: Perhaps one of the most well-known theorems in graph theory is the celebrated Graph Minor Theorem of Robertson and Seymour. This theorem states that in any infinite collection of finite graphs, there must be a pair of graphs for which one is obtained from the other by a sequence of edge contractions and deletions. In this talk, I will present work of Nick Proudfoot, Dane Miyata, and myself which proves a categorified version of the graph minor theorem. As an application, we show how configuration spaces of graphs must display some strongly uniform properties. We then show how this result can be seen as a vast generalization of a variety of classical theorems in graph configuration spaces.

- February 11,
**Patricia Hersh** (UO)

Crystal graphs and SB-labelings
**Abstract**: Crystal graphs are an important tool to study the representation theory of Kac-Moody algebras. The crystal graphs arising from highest weight representations are in fact (Hasse diagrams of) partially ordered sets. In joint work with Cristian Lenart, we study these crystal posets. We prove two types of positive results for those poset intervals which includes the highest weight vector of the representation: (1) a crystal operator analogue of the statement that any two reduced expressions for the same Coxeter group element are connected by braid moves, and (2) poset topological and Moebius function results analogous to those of weak Bruhat order. We also provide examples demonstrating that both types of results fail arbitrarily badly for arbitrary intervals in crystal posets, even in type A. The first such examples were found by computer, specifically by Moebius function computations. To our surprise, the first example we discovered with unexpectedly large Moebius function was also the first example where (1) failed. This link between (1) and (2) can be explained using the theory of SB-labelings, a theory we developed in separate joint work with Karola Meszaros. The talk will tell this story, providing background along the way.

- February 18,
**Jun Wang** (Ohio State University)

A mirror theorem for Gromov-Witten theory without convexity
**Abstract**: One central question in Gromov-Witten (GW) theory is to relate the GW invariants of a hypersurface to the GW invariants of the ambient space. In genus zero, this is usually done by the so-called quantum Lefschetz principle, which uses the twisted GW invariants of the ambient space. This approach is analogous to the classical theorem that the number of lines inside a cubic surface can be obtained by computing the Euler number of a certain vector bundle on the space of lines inside \mathbb P^3 (which is the Grassmannian Gr(2,4)). Thus, the quantum Lefschetz principle provides an effective way to calculate the GW invariants of the hypersurface when the twisted GW invariants of the ambient space are known (e,g. toric stacks). However, this approach requires a technical assumption called convexity for the line bundle over the ambient space defining the hypersurface. Hence, when convexity fails, the GW invariants of a hypersurface are much less known for a long time. In this talk, I will present a way (mirror theorem) to obtain the genus zero GW invariants of positive hypersurfaces in toric stacks for which the convexity may fail. One key ingredient in the proof is to resolve the genus zero quasimap wall-crossing conjecture proposed by Ciocan-Fontaine and Kim for these targets.

- February 25<
**Jeff Monroe** (UO)

Linking invariants and the lower central series of a free group
**Abstract**: We consider the basic question of determining whether an element of a free group is an

-fold commutator, as first considered first by Magnus and Fox. We import a technique from algebraic topology, defining (higher) linking invariants of letters in words. These are the

-analogues of Hopf invariants of higher homotopy groups.

In this talk we develop these invariants (purely algebraically – no topology or anything beyond basic group theory required) and share work in progress to prove the conjecture that these invariants span all homomorphisms from lower central series subquotients of free groups to the rational numbers. We share combinatorial algebraic questions which arise and indicate planned applications.

This work is joint with Dev Sinha.

- March 10,
**Eugene Gorsky** (UC Davis)

Derived traces for Soergel categories
**Abstract**: It is well known that Hecke algebra is spanned by braids modulo skein relations, while the span of closed braids in the annulus modulo skein relations is isomorphic to the space of symmetric functions. I will describe a categorification of these results: the category of Soergel bimodules categorifies the Hecke algebra, while the annular closure corresponds to the formalism of “derived horizontal trace”. Along the way, I will explicitly compute Hochschild homology of the category of Soergel bimodules.

All notions will be explained in the talk. This is a joint work with Matt Hogancamp and Paul Wedrich.

- March 12,
**Martin Weissman** (UC Santa Cruz)

**Special time/location** Thursday 2pm, in Deady 209

An Induction Theorem for Groups acting on Trees
**Abstract**: The irreducible representations of reductive Lie groups are typically infinite-dimensional. But we can relate them to the representations of their maximal compact subgroups, which are combinatorially classified. In particular, the discrete series representations can be constructed by a sort of analytic induction from a compact subgroup. For p-adic groups, like GL(n, Qp), a folklore conjecture states that irreducible representations can similarly be constructed via compact subgroups. I will describe a proof of this conjecture for groups of “relative rank one”. But really, I will focus on a very general theorem about groups acting on trees, from which the conjecture follows almost immediately.

### Fall Quarter, 2019

- October 8,
**Doeke Buursma** (UO)

Describing standard modules for KLR algebras
**Abstract**: KLR algebras provide a categorification for the positive part of the quantized universal enveloping algebra of (complex, finite-dimensional, semisimple) lie algebras. They are also affine quasi-hereditary and therefore come with a collection of standard modules which happen to categorify the PBW basis of the corresponding KLR algebra. We discuss standard module theory for KLR algebras, giving an explicit construction of projective resolutions of standard modules type A. We describe the Yoneda algebra of the direct sum of the standard modules in some special cases.

- October 15,
**Jacob Matherne** (UO)

Singular Hodge theory of matroids
**Abstract**: Kazhdan-Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the setting of matroids. In this talk, I will compare and contrast KL theory for Coxeter groups with KL theory for matroids.

I will also associate to any matroid a certain ring whose Hodge theory can conjecturally be used to establish the positivity of the KL polynomials of matroids as well as the “top-heavy conjecture” of Dowling and Wilson from 1974 (a statement on the shape of the poset which plays an analogous role to the Bruhat poset). Those enjoying Nick Proudfoot’s course this term are especially encouraged to attend. This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang.

- October 22,
**Alicia Lamarche** (Rice)

Derived Categories, Arithmetic, and Rationality Questions
- October 29,
**Julius Ross** (UIC)

The Hodge-Riemann Bilinear Relations for Schur classes of Ample Vector Bundles
**Abstract**: It is well known that Hodge Theory has deep consequences for the topology of algebraic varieties, in particular through the Hard-Lefschetz theorem and the Hodge-Riemann bilinear relations. Classically this involves the choice of some positive cohomology class, such as (a power of) the first Chern class of an ample vector bundle. I shall discuss a generalization of this to certain Schur classes of ample vector bundles, giving some applications such as new log-concavity results of chern numbers of ample vector bundles. This is joint work with Matei Toma.

- October 31,
**Xiaobo Liu** (BICMR, Peking University)

**Special Time/Location:** 11am, Thursday, 101 Peterson Hall

Connecting Hodge integrals to Gromov-Witten invariants by Virasoro operators
**Abstract**: Kontsevich-Witten tau function and Hodge tau functions are important tau functions for KP hierarchy which arise in geometry of moduli space of curves. Alexandrov conjectured that these two functions can be connected by Virasoro operators. In a joint work with Gehao Wang, we have proved Alexandrov’s conjecture. In a joint work with Haijiang Yu, we show that this conjecture can also be generalized to Gromov-Witten invariants and Hodge integrals over moduli spaces of stable maps to smooth projective varieties.

- November 5,
**Helen Jenne** (UO)

Combinatorics of the double-dimer model
**Abstract**: In this talk we will discuss a new result about the double-dimer model: under certain conditions, the partition function for double-dimer configurations of a planar bipartite graph satisfies an elegant recurrence, related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the number of dimer configurations (or perfect matchings) of a graph was established nearly 20 years ago by Kuo and others, and has applications to random tiling theory and the theory of cluster algebras. We expect our work will have similar applications.

- November 7,
**Michael Geline** (Northern Illinois University)

**Special Time/Location:** 1pm, Thursday, 210 Deady Hall

Local subgroups control rationality of finite group representations
**Abstract**: The local to global principle in finite group theory, introduced generally by Frobenius, asks for the extent to which questions about a finite group and a fixed prime $p$ can be reduced to questions about normalizers of nontrivial $p$-subgroups. Most of the best known open questions along these lines, attributed to Brauer, Broue, Alperin and others, seek to express numbers of irreducible representations (in both characteristic $p$ and zero) in terms of such “local subgroup.” I will show how rationality properties of irreducible representations are controlled quite definitively in this way. The method is what I believe to be a novel application of the Green correspondence.

- November 12,
**Robert Laugwitz** (Rutgers)

2-representations of dg categories
**Abstract**: Categorical representations of tensor categories and 2-categories have now been studied by several authors. Mazorchuk-Miemietz have developed a systematic framework to study ‚simple‘ categorical representations using cell combinatorics that can be applied to known categorifications of quantum groups or Hecke algebras.

In this talk, I will review some constructions of categorical representation theory and report on joint work in progress with V. Miemietz on a partial extension of the theory to pretriangulated and derived 2-categories by working with differential graded (dg) structures.

- November 19,
**Katrina Honigs** (UO)

Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic
**Abstract**: There are many results characterizing when derived categories of two complex surfaces are equivalent, including theorems of Bridgeland and Maciocia showing that derived equivalent Enriques or bielliptic surfaces must be isomorphic. The proofs of these theorems strongly use Torelli theorems and lattice-theoretic methods which are not available in positive characteristic. In this talk I will discuss how to prove these results over algebraically closed fields of positive characteristic (excluding some low characteristic cases). This work is joint with M. Lieblichand S. Tirabassi.

- November 26,
**Aaron Lauda** (USC)

Bordered Heegaard-Floer homology, category O, and higher representation theory
**Abstract**: The Alexander polynomial for knots and links can be interpreted as a quantum knot invariant associated with the quantum group of the Lie superalgebra gl(1|1). This polynomial has been famously categorified to a link homology theory, knot Floer homology, defined within the theory of Heegaard-Floer homology. Andy Manion showed that the Ozsvath-Szabo algebras used to efficiently compute knot Floer homology categorify certain tensor products of gl(1|1) representations. For representation theorists, work of Sartori provides a different categorification of these same tensors products using subquotients of BGG category O. In this talk we will explain joint work with Andy Manion establishing a direct relationship between these two constructions. Given the radically different nature of these two approaches, transporting ideas between them provides a new perspective and allows for results that would not have been apparent otherwise, including a surprising connection between Ozsvath-Szabo theory and hypertoric varieties.

- December 3, Course Planning Meeting

**Previous years**: 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005