USCAlgebra Seminar

Fall 2015/Spring 2016

Time: Monday, 3:30pm to 4:20pm     •     Location: KAP 245

Past talks

September 14: Shawn Baland (University of Washington)
Relative stable categories and their prime ideal spectra

In modular representation theory, one of the main objects of interest is the stable module category $\operatorname{stmod}(RG)$ of a group algebra $RG$. In particular, if $R$ is a field, then it is well known that $\operatorname{stmod}(RG)$ is a triangulated category. In 2012, Benson, Iyengar and Krause defined a new exact structure on the module category $\operatorname{mod}(RG)$ for any commutative ring $R$ in such a way that the resulting stable category is always triangulated. In this talk we'll discuss the support theoretic aspects of these new stable categories and some recent progress that has been made in computing the prime ideal spectrum (in the sense of Balmer) of $\operatorname{stmod}(RG)$ in the case where $G$ is the cyclic group of order $p$ and $R$ is the commutative ring $\mathbf{Z}/p^n$. This is joint work with Greg Stevenson.

September 28: Dominic Searles (USC)
Root-theoretic Young diagrams and Schubert calculus

We discuss some results in Schubert calculus obtained using the combinatorial model of root-theoretic Young diagrams (RYDs). In joint work with A. Yong, we give nonnegative rules for the Schubert calculus of the (co)adjoint varieties of classical type, and use these rules to suggest a connection between planarity of the root poset and polytopality of the nonzero Schubert structure constants. We also present an RYD rule for the Belkale-Kumar product for flag varieties of type A (after the puzzle rule of A. Knutson-K. Purbhoo).

September 29: Julia Pevtsova (University of Washington)
Varieties of elementary subalgebras for modular Lie algebras

Motivated by questions in representation theory, Carlson, Friedlander and the speaker instigated the study of projective varieties of abelian $p$-nilpotent subalgebras of a fixed dimension $r$ for a $p$-Lie algebra $\mathfrak{g}$. These varieties are close relatives of the much studied class of varieties of $r$-tuples of commuting $p$-nilpotent matrices which remain highly mysterious when $r > 2$. In this talk, I'll present some of the representation-theoretic motivation behind the study of these varieties and describe their geometry in a very special case when it is well understood: namely, when $r$ is the maximal dimension of an abelian $p$-nilpotent subalgebra of $\mathfrak{g}$ where $\mathfrak{g}$ is a Lie algebra of a reductive algebraic group. This is joint work with J. Stark.

October 5: Alexander Soibelman (USC)
Quiver representations, parabolic bundles, and the Deligne-Simpson problem

The additive and multiplicative formulations of the Deligne-Simpson problem ask, respectively, if a collection of complex matrices with prescribed conjugacy classes has sum zero or product the identity. Both versions may be restated as a single question about the existence of a regular singular connection on the projective line. We approach this question by generalizing Crawley-Boevey's moment map construction for quivers to representations of squid algebras and by using a technical property coming from Beilinson and Drinfeld's work on the geometric Langlands correspondence.

October 12: Daniele Rosso (UC Riverside)
Heisenberg categorification and wreath products of Frobenius algebras

Heisenberg algebras are important in quantum mechanics and in the theory of affine Lie algebras. Starting with the work of Khovanov, there has been interest in constructing categorifications of such algebras, using categories of planar diagrams. Here we give a general construction of a diagrammatic category that depends on the choice of a Frobenius (graded super-)algebra $B$. This gives a categorification of a certain quantum lattice Heisenberg algebra (also depending on $B$). For specific choices of $B$, we can recover existing results of Khovanov and Cautis-Licata. This is joint work with Alistair Savage.

October 26: Carl Mautner (UC Riverside)
Hypertoric and Matroidal Schur algebras

(joint with Tom Braden) Schur algebras are remarkable finite-dimensional algebras that interpolate the representation theory of the symmetric groups and the general linear groups and are most interesting in small characteristics. Motivated by Springer theory and a duality for symplectic singularities, we introduce a new family of algebras associated to hypertoric varieties (and more generally matroids). We show that these new algebras share a number of the nice properties of Schur algebras—in particular, they are quasi-hereditary and behave well under Ringel duality.

November 2: Arnav Tripathy (Stanford)
Symmetric powers and the étale Dold-Thom theorem

After reminding everyone why the symmetric powers $\operatorname{Sym}^nX$ of a scheme arise and are interesting from the point of view of the Weil conjectures, I'll recall the Dold-Thom theorem of algebraic topology, which governs the behavior of symmetric powers of a topological space. I'll then explain how the notion of étale homotopy allows us to compare these two realms of arithmetic geometry and algebraic topology, providing a homotopical refinement of a small part of the Weil conjectures.

November 9: Jason Fulman (USC)
Cohen-Lenstra heuristics and random matrix theory over finite fields

Two permutations are conjugate if and only if they have the same cycle structure, and two complex unitary matrices are conjugate if and only if they have the same set of eigenvalues. Motivated by the large literature on cycles of random permutations and eigenvalues of random unitary matrices, we study conjugacy classes of random elements of finite classical groups. For the case of $\operatorname{GL}(n,q)$, this amounts to studying rational canonical forms. This leads naturally to a probability measure on the set of all partitions of all natural numbers. We connect this measure to symmetric function theory, and give algorithms for generating partitions distributed according to this measure. We describe analogous results for the other finite classical groups (unitary, symplectic, orthogonal). We were excited to learn that (at least for $\operatorname{GL}(n,q)$), exactly the same random partitions arise in the “Cohen-Lenstra heuristics” of number theory.

November 16: Alfonso Zamora Saiz (CSU Channel Islands)
GIT characterizations of Harder-Narasimhan filtrations

We will discuss constructions of moduli spaces in algebraic geometry by using Geometric Invariant Theory (GIT). When performing such constructions we usually impose a notion of stability for the objects we want to classify and another notion of GIT stability appears, then it is shown that both notions coincide. For an object which is unstable there exists a unique canonical filtration, called the Harder-Narasimhan filtration. On the other hand, GIT stability is checked by 1-parameter subgroups by the classical Hilbert-Mumford criterion, and it turns out that there exists a unique 1-parameter subgroup giving a notion of maximal unstability in the GIT sense. We show how to prove that this special 1-parameter subgroup can be converted into a filtration of the object and coincides with the Harder-Narasimhan filtration, hence both notions of maximal unstability are the same. We will present the correspondence for the moduli problem of classifying coherent sheaves on a smooth complex projective variety. A similar treatment can be used to prove the analogous result for many other moduli problems: pairs, Higgs sheaves, rank 2 tensors, quiver representations and constellations.

November 30: Birge Huisgen-Zimmerman (UC Santa Barbara)
The finitistic dimension conjectures: an update

I'll start by recalling the big and little finitistic dimensions of an algebra and by sketching the history that gave rise to the conjectures. Several examples will provide illustration. Then I'll outline a “representative” selection of results to date, with emphasis on the most recent ones.

December 4: Alexander Shapiro (UC Berkeley)
Quantized Grothendieck-Springer resolution and cluster
algebraic approach to quantum groups

A Poisson-Lie group $G$ with its standard Poisson structure admits a family of cluster coordinates with the defining property of having log-canonical Poisson brackets: ${X_i, X_j} = a_{ij} X_i X_j$. On the level of quantum groups, these coordinates become a family of $q$-commuting generators $X_i X_j = q^{a_{ij}} X_j X_i$ for (a localization of) the quantized algebra $O_q[G]$ of functions on $G$. Showing that a localization of the quantum group $U_q(\mathfrak{g})$ is isomorphic to certain quantum torus algebra (i.e. an algebra with $q$-commuting generators) is a much desired property known as the quantum Gelfand-Kirillov conjecture. In a joint work with Gus Schrader, we construct an embedding of the quantum group $U_q(\mathfrak{g})$ into a quantum torus algebra naturally defined from the quantum double Bruhat cell $O_q[G^{w_0,w_0}]$. Our construction is motivated by Poisson geometry of the Grothendieck-Springer resolution and is closely related to the global sections functor of the quantum Beilinson-Bernstein theorem. I will explain our work and discuss its applications to representation theory of quantum groups.

December 7: Doosung Park (UC Berkeley)
Triangulated categories of motives over fine saturated log schemes

Ayoub's thesis and an unpublished work of Voevodsky constructed triangulated categories of motives over schemes with rational coefficients and their six operations formalism. In this talk, I will discuss the generalization of this to fine saturated log schemes.

February 2: Jesse Burke (Australian National University)
The “Quillen stack” of a finite group

Let $G$ be a finite group and $k$ an algebraically closed field of characteristic dividing the order of $G$. Quillen's famous stratification theorem describes $V_G$, the spectrum of the cohomology ring of $G$ over $k$, in terms of $V_E$, for $E$ ranging over all elementary abelian $p$-subgroups of $G$. Implicit in Quillen's description is a quotient by a (usually non-free) $G$-action. We make this explicit and, using Quillen's theorem, construct a scheme $X_G$ with a $G$-action whose quotient is $V_G$. This is motivated by the study of $\pi$-points, which are certain $k$-linear endomorphisms of the group ring $kG$. Friedlander and Pevtsova introduced an equivalence relation on $\pi$-points and showed there is a bijection between equivalence classes of $\pi$-points and closed points of $V_G$. However, given a point of $V_G$, there does not seem to be a canonical $\pi$-point representing it, i.e. there isn't a universal $\pi$-point living over $V_G$. We show that there is instead a $G$-equivariant universal $\pi$-point living over $X_G$, so specializing at any point of a $G$-orbit of $X_G$ gives a representative for the corresponding $\pi$-point. Using this universal $\pi$-point, we can mimic a construction of Friedlander and Pevtsova in the case of infinitesimal group schemes, to construct equivariant vector bundles on $X_G$ from a broad class of $kG$-modules, in particular modules of constant Jordan type. This gives geometric invariants of these modules and, we hope, constructions of interesting vector bundles on $X_G$.

This is joint work with Eric Friedlander and Julia Pevtsova.

February 8: Julia Plavnik (Texas A&M)
On the classification of weakly integral modular categories

Modular categories are enriched monoidal categories that appear in many mathematical subjects, like topological quantum field theory, conformal field theory, representation theory of quantum groups, von Neumann algebras, and vertex operator algebras. One of the main motivations for studying modular categories comes from their application in condensed matter physics and quantum computing.

Modular categories are quantum analogues of finite groups: one replaces the symmetry of the tensor product of representations with a non-degenerate braiding on the tensor product of objects. Classification of low rank modular categories is a first step in a structure theory for modular categories parallel to group theory.

In this talk, we will present the basic definitions, examples and some general results of modular categories. We will also give the classification of modular categories of dimension 4m, where m is an odd square-free integer. If the time allows us, we will also discuss some consequences related to the classification of low rank weakly integral modular categories. These results are part of a joint work with P. Bruillard, C. Galindo, S.-H. Ng, E. Rowell and Z. Wang (J. Pure Appl. Algebra, preprint arXiv:1411.2313). The focus of the talk will be on the fundamental ideas, trying to avoid technicalities.

February 29: Nina Yu (UC Riverside)
Rationality, regularity and $C_2$-cofiniteness

Rationality, regularity and $C_2$-cofiniteness are three most important concepts in representation theory of vertex operator algebras. In this talk, I will talk about connections among these three notions and recent progress in proving the conjecture that rationality implies $C_2$-cofinitenss.

March 7: Alexei Entin (Stanford)
Monodromy of Hurwitz Spaces

We compute the monodromy action for the Hurwitz space parametrizing simply ramified genus $0$ degree $n$ covers of the Riemann sphere, viewed as a cover of the configuration space of $n$ points on the sphere. We show that it is the full alternating group whenever $n \geq 5$. We make use of the classification of $2$-transitive finite group actions which in turn relies on the classification of finite simple groups. Joint work with Robert Guralnick and Chris Hall.

March 21: Richard Ng (Louisiana State University)
On the invariance of the order of the antipode of a Hopf algebra

Let $S$ be the antipode of a finite-dimensional Hopf algebra $H$ over the field of complex numbers. If follows from the invariance of the Frobenius-Schur indicators and the semisimplicity of $H$ that the traces of $S$ and $S^2$ are invariants of the monoidal category of representations of $H$. However, the question of whether the trace of any power of $S$ is a categorical invariant remains open. An affirmative answer to this question would imply the invariance of the order of $S$, which is another open problem. In this talk, we will discuss the invariance of the order of $S$ when the Jacobson radical of $H$ is a Hopf ideal. This is a joint work with C. Negron.

April 4: Pablo Solis (Caltech)
Degenerations of Loop Groups and Moduli Spaces

Many important varieties in algebraic geometry come in some way from algebgraic groups. Examples include Abelian varieties, toric varieties, flag varieties or more generally spherical varieties. An important example of the latter is the so called wonderful compactification of a semisimple adjoint group. In this talk I will discuss a compactification of a Kac-Moody group associated to a loop group that in many ways generalizes the wonderful compactification of a semisimple group. I will also explain a modular interpretation of the partial compactification in terms of bundles on nodal curves.

April 18: Martin Gallauer Alves de Souza (UCLA)
Motivic Hopf algebras

We will first try to introduce the audience to the philosophy of motives and the motivic Hopf algebra. The main goal is then to explain why the proposed candidates for the latter are in fact isomorphic. This is joint work with Utsav Choudhury.

April 25: Yilong Yang (UCLA)
Diameter bounds for finite simple groups of large rank

Given any non-abelian finite simple group $G$ and any generating set $S$, it is conjectured by Babai that the Cayley graphs should always have diameters $O(\log|G|)$. This conjecture has been verified for all finite simple groups of Lie type with bounded ranks, but little progress is made in the cases with large ranks. Motivated by the methods of Helfgott and Seress for symmetric groups, we obtained improved diameter bounds for finite simple groups of Lie type with large ranks of $O(1)^{O(n(\log n)^3)}$, when the size of the base field is bounded. This is joint work with Arindam Biswas from University of Paris-Sud.

April 29: Matt Hogancamp (Indiana University)
Categorical diagonalization

In this talk I will discuss joint work with Ben Elias in which we categorify the notion of a diagonalizable linear operator. The main application so far is to the diagonalize the action of the full twist on the category of Soergel bimodules, which gives rise to categorified Young symmetrizers.