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# USC Algebra Seminar

## Fall 2016/Spring 2017

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

## Upcoming talks

 Summer 2017 West Coast Lie Theory Workshop Hopf algebras and Tensor Categories Location: UC San Diego Dates: June 3-4, 2017 Further information: website

## Past talks

 September 12: Alexander Neshitov (USC) Framed Correspondences and the Milnor-Witt K-theory The theory of framed motives developed by Garkusha and Panin gives a tool to construct fibrant replacements and hence to do explicit computations in stable $\mathbf{A}^1$-homotopy theory. In the talk we will discuss how this construction identifies certain motivic stable homotopy groups of the base field with its Milnor-Witt K-theory. This identification can be considered as a ‘framed’ analog of the Suslin-Voevodsky theorem which identifies $(n,n)$-motivic cohomology of the base field with its Milnor K-theory. September 26: Jesse Levitt (USC) Discriminants of Polynomial Identity Quantized Weyl Algebras We present explicit formulas for the the discriminants of Polynomial Identity (PI) quantized Weyl algebras over a general class of central subalgebras along with several applications. The work follows a program proposed by Chan, Young and Zhang for investigating filtered PI algebras whose associated graded algebras are skew polynomial algebras. We first classify the centers of PI quantized Weyl algebras, and examine the case where these algebras are then free over their centers. Two distinct approaches arise for calculating their discriminants, with one coming from deformation theory and Poisson geometry, while the other is based in the methods of quantum cluster algebras. Both formulations allow all such algebras to be classified and the discriminant is found to be both locally dominating and effective with applications to the automorphism and isomorphism problems for tensor products of these algebras. This is a joint work with Milen Yakimov. October 3: Fei Xie (UCLA) Algebraic K-theory of toric surfaces over any field We study toric varieties over any field in the Merkurjev-Panin motivic category. In 1997, Merkurjev and Panin showed that a toric variety $X$ is always a direct summand of a separable algebra in the motivic category and it is isomorphic to one if and only if $K_0(X_{\text{sep}})$ is a permutation Galois module. We will verify the condition for toric surfaces and construct the separable algebra in terms of a permutation basis of $K_0(X_{\text{sep}})$. This computes the algebraic K-theory of toric varieites whenever the condition is true. This also generalizes Blunk's study of del Pezzo surfaces of degree $6$ over any field. October 7: Yan Soibelman (Kansas State University) (Joint Categorification/Algebra/Topology Seminar) Riemann-Hilbert correspondence, schobers on surfaces and Fukaya categories Classical Riemann-Hilbert correspondence establishes an equivalence of the category of holonomic $\mathcal{D}$-modules with regular singularities, and the category of perverse sheaves. The notion of perverse schober (here “schober” is the German word for “stack”) was proposed recently by Kapranov and Schechtman as a categorification of the notion of perverse sheaf. Differently from the latter, there is no general definition of perverse schober. The approach of Kapranov and Schechtman is based on an explicit description of perverse sheaves on a disc in terms of quivers. In my talk I am going to discuss a joint project with Dyckerhoff, Kapranov and Schechtman, in which we define perverse schobers on Riemann surfaces, possibly with punctures and boundary. The main idea of our approach is a combination of the original approach of Kapranov and Schechtman with some ideas from symplectic topology (Fukaya-Seidel categories). As an application, one can associate with a punctured Riemann surface a category (called topological Fukaya category), which is the category of global sections of a certain constructible sheaf of categories on a graph with endpoints at the punctures. October 10: Hans-Jürgen Schneider (LMU Munich) Nichols algebras with finite Weyl groupoid This is joint work with I. Heckenberger. It was shown by Lusztig that the plus part $U^+_q(\mathfrak{g})$ of the quantum group $U_q(\mathfrak{g})$, $\mathfrak{g}$ a semisimple Lie algebra, $q$ not a root of one, is a Nichols algebra with a PBW-basis generated by the positive root vectors. The Hopf algebra $U_q(\mathfrak{g})$ is pointed, that is, its simple comodules are one-dimensional. It is well known that Nichols algebras play a crucial role in the classification theory of pointed Hopf algebras, an active area of research since about 20 years (for a survey, see the invited talk by N. Andruskiewitsch at the ICM in Seoul 2014). I will explain the Nichols algebra $B(V)$ of a Yetter-Drinfeld module $V$. Based on a general result on braided monomial categories we extend Lusztig's results to Nichols algebras with finite Weyl groupoid. The main technique is a stepwise construction of right coideal subalgebras. In particular, we obtain a new interpretation of (part of) the Lusztig automorphisms as braided Hopf algebra isomorphisms and new proofs in the quantum group case (no case by case arguments). An application is the description of he right coideal subalgebras which implies a conjecture of Kharchenko on $U^+_q(\mathfrak{g})$. Our categorical construction allows to introduce the useful notion of a Nichols system. As applications we obtain new and more conceptual proofs of several important know results (by Lusztig, Rosso, Angiono) which say that a given braided Hopf algebra is Nichols. October 17: Ozlem Ejder (USC) Fermat Surfaces and Modular Symbols for Fermat Curves We compute the modular symbols and the set of cuspidal modular symbols for the Fermat curve $x^n+y^n=z^n$. We use these computations to compute the monodromy of a certain fibration where the fibers are Fermat curves using the modular realization of Fermat curves. October 31: Julia Pevtsova (University of Washington) Local duality for the stable module category of a finite group scheme For $G$ a finite group scheme over a field $k$ of positive characteristic, we consider the stable module category of $G$, $\operatorname{StMod} G$. The building blocks of this category are minimal localizing subcategories $\Gamma_p(\operatorname{StMod} G)$ which are precisely subcategories of $G$-modules supported at a single point. We show that the category $\operatorname{StMod} G$ has a Gorenstein property which amounts to a version of Serre duality for the minimal localizing subcategories $\Gamma_p(\operatorname{StMod} G)$. This generalizes results of Benson and Benson-Greenlees for finite groups, although as standard with finite group schemes we have to use completely different methods, producing conceptually new proofs for the case of finite groups on the way. Joint work with D. Benson, H. Krause and S. Iyengar. November 7: Alexander Soibelman (USC) Motivic Classes for Moduli of Connections In their paper, “On the motivic class of the stack of bundles”, Behrend and Dhillon were able to derive a formula for the class of a stack of vector bundles on a curve in a completion of the K-ring of varieties. Later, Mozgovoy and Schiffmann performed a similar computation in order to obtain the number of points over a finite field in the moduli space of twisted Higgs bundles. We will briefly introduce motivic classes. Then, following Mozgovoy and Schiffmann's argument, we will outline an approach for computing motivic classes for the moduli stack of vector bundles with connections on a curve. This is a work in progress with Roman Fedorov and Yan Soibelman. November 14: Harold Williams (UT Austin) A Symplectic Viewpoint on the Affine Grassmannian The affine Grassmannian plays a central role in modern representation theory, for example through its appearance in the celebrated geometric Satake correspondence. However, current developments in mathematical physics suggest that its geometry is controlled at least in part by a priori unrelated objects in symplectic geometry. In this talk we explain in detail an example of this emerging picture: we identify the equivariant K-theory of the affine Grassmannian (of $\operatorname{PGL}_n$) with the coordinate ring of a moduli space associated to a particular Legendrian link. Moreover, this space has a special kind of combinatorial structure—a so-called cluster structure—which relates the classification of exact Lagrangian fillings of the link to the category of perverse coherent sheaves on the Grassmannian. Our discussion is based on joint work with S. Cautis and separately with V. Shende, D. Treumann, and E. Zaslow. December 5: Matthew O'Dell (UC Riverside) Integrable representations of equivariant map algebras associated with Borel-de Siebenthal pairs Borel and de-Siebenthal classified the maximal connected subgroups of maximal rank of a connected compact Lie group. This result can be rephrased in terms of automorphisms of the semisimple Lie algebra and the subalgebra of fixed points. They also give rise in a natural way to a maximal parabolic subalgebra of an affine Lie algebra. In the case when the automorphism is non-trivial, we shall see that the parabolic subalgebra is isomorphic to an equivariant map algebra. We develop the theory of integrable representations of such Lie algebras. In particular, we define and study global Weyl modules. These are closely related to the module category of a commutative associative algebra. In this talk, we give a presentation of this algebra, and give partial results toward a dimension formula for local Weyl modules. This is joint work with Vyjayanthi Chari and Deniz Kus. January 9: Brian Hwang (Cornell) Modelling the local geometry of moduli spaces with degenerations Degeneration techniques have been used to simplify calculations since the classical days of algebraic geometry. For example, one can determine that two lines intersect four given lines in 3-space by specializing to a “degenerate” case where the four given lines are assumed to be two pairs of intersecting lines; the two lines are then (1) the line joining the two points of intersection and (2) the line formed by the intersection of the two planes spanned by the respective intersecting pairs. The same answer holds true generically via “the principle of conservation of number,” despite the fact that we calculated the number in a drastically simplified setting. Indeed, one reason to seek compact moduli spaces is that they allow us to interpret such degenerations as points “close to the boundary”. We want to highlight a recent application of this philosophy where a degeneration technique can be used not only to study compact moduli spaces ”near the boundary” but to study the local geometry of a general neighborhood of a moduli space. As a concrete illustration, we will explain how to use degenerations of Grassmannians to study some moduli spaces of abelian varieties (or Hodge structures) which are closely linked to representation theory (e.g. certain Shimura varieties, like modular curves). These specific degenerations turn out to be (a) defined not only over fields, but over rings; (b) closely related to certain constructions in rigid analytic and tropical geometry; and (c) predict interesting interactions between the geometry and the concomitant algebraic and arithmetic structures. February 1: Anthony Blanc (Max Planck Institute for Mathematics) Topological K-theory of noncommutative spaces I'll review the construction of a topological invariant associated to a dg-category over the complex numbers, called topological K-theory. The main motivation is from noncommutative Hodge theory, where it conjecturally provides a lattice in periodic cyclic homology. In fact we can define a semi-topological K-theory, a noncommutative analog of Friedlander--Walker semi-topological K-theory of complex varieties, which interpolates between algebraic and topological K-theory. In the case of perfect complexes on a variety, we recover Friedlander--Walker theory and after inverting the Bott element the KU theory of the complex points. I will talk about other geometrical examples, the relation with the moduli space of objects in the dg-category and the case with finite coefficients. February 6: Michiel Kosters (UC Irvine) Slopes of $L$-functions of $\mathbf{Z}_p$-covers of the projective line Let $P: \cdots \to C_2 \to C_1 \to \mathbf{P}^1$ be a $\mathbf{Z}_p$-cover of the projective line over a finite field of characteristic $p$ which ramifies at exactly one rational point. In this talk, we study the $p$-adic Newton slopes of $L$-functions associated to characters of the Galois group of $P$. It turns out that for covers $P$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ for $n$ large, the Newton slopes are uniformly distributed in the interval $[0,1]$. Furthermore, for a large class of such covers $P$, these slopes behave in an even more regular way. This is joint work with Hui June Zhu. February 13: Mark Shusterman (Tel Aviv University) Rank gradient of direct products of finitely generated groups The number of generators of a finite index subgroup $U$ of a finitely generated group $G$ is at most $(d(G)-1)[G : U] + 1$. We define the rank gradient of a sequence $G_n$ of finite index subgroups of $G$ to be $\operatorname{inf}_i (d(G_i) - 1)/[G : G_i]$ thus measuring the strictness of the aforementioned inequality. The study of this combinatorial invariant is motivated by combinatorial group theory, ergodic theory, 3-manifold theory, and number theory. A vivid example is the group $G = SL_2(\mathbf{Z}[i])$ where the rank gradient (of congruence sequences) controls both Galois representations, and covers of a 3-manifold. Group theoretically $G$ is (almost) a semidirect product of $\mathbf{Z}$ acting on a free group, and calculating the rank gradient for semidirect products is a major challenge. We will see that this task can be accomplished in the easier case of direct products, using the classification of finite simple groups, building on works of Mann, and Guralnick-Kantor-Kassabov-Lubotzky on presentation of finite groups with few relations. This is a joint work with Nikolay Nikolov and Zvi Shemtov. February 17: Abdenacer Makhlouf (Université Haute-Alsace) On graded Hopf algebras classification and twisted Hopf algebras February 27: Yuri Bahturin (Memorial University of Newfoundland and Moscow State University) Classification of real graded division algebras Let $G$ be a finite abelian group. In this joint work with Mikhail Zaicev of Moscow State University, we classify all real $G$-graded division algebras of finite dimension, up to equivalence. March 3: Deniz Kus (University of Bonn) Graded tensor products for Lie (super)algebras In this talk I will discuss the construction of graded tensor products for the current algebra associated to a Lie (super)algebra. For the ortho-symplectic Lie superalgebra we will show that these representations can be filtered by the corresponding graded tensor products for the underlying reductive Lie algebra. In the second part of my talk, I will discuss the appearance of graded tensor products in PBW theory and categorification. One of the future goals is to understand which $2$-representation of the categorified quantum group corresponds to graded tensor products. April 10: Cris Negron (MIT) Small quantum groups associated to Belavin-Drinfeld triples I will introduce the small quantum group associated to a simple Lie algebra $L$. This is a finite dimensional Hopf algebra introduced by Lusztig in the 90's. Small quantum groups have a number of fantastic properties from the perspective of conformal field theory and low dimensional topology. (For example, they are factorizable and ribbon.) I will explain how decorations of the Dynkin diagram for $L$ lead to new (non-isomorphic) Hopf algebras with the same fantastic properties. I will also explain how these new Hopf algebras fit into recent studies of non-semisimple tensor categories, and provide us with new examples of tensor equivalences which are not reducible to well-understood semisimple phenomena. No deep knowledge of Hopf algebras or tensor categories will be assumed.