# Algebra Seminar 2015

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

### Spring Quarter, 2016

- April 5,
**Ben Elias** (UO)

Diagonalization in categorical representation theory
- April 12,
**Nick Addington** (UO)

Some new rational cubic 4-folds
**Abstract**: I will discuss some work in progress, joint with Hassett, Tschinkel, and Varilly-Alvarado, in which we produce new rational cubic 4-folds for the first time in 20 years. After a brief tour of cubic 4-folds in general, I will spend some time discussing cubics containing a plane, a beautiful story due to Voisin, Hassett, and Kuznetsov; this will then serve as a model for our new examples, which are cubics containing sextic del Pezzo surfaces and elliptic ruled surfaces.

- May 10,
**Lewis Topley** (Padua)

Modular finite W-algebras
**Abstract**: Finite W-algebras have found their most famous applications in the representation theory of finite dimensional complex semisimple Lie algebras, however they were introduced to mathematics in the modular realm. In this talk I shall describe a joint work with Simon Goodwin in which we develop the theory of finite W-algebras in a purely modular setting, and use our construction to classify the minimal simple modules for the general linear Lie algebra with a fixed p-central character.

- May 18,
**Vinoth Nandakumar** (Utah)

**Special Day/Location:** Wednesday, 11am, in 103 Deady.

Modular representations with two-block nilpotent central character
**Abstract**: We will study the category of modular representations of the special linear Lie algebra with a fixed two-block nilpotent p-character. Building on work of Cautis and Kamnitzer, we construct a categorification of the affine tangle calculus using these categories; the main technical tool is a geometric localization-type result of Bezrukavnikov, Mirkovic and Rumynin. Using this, we give dimension formulae for the irreducible modules, and a description of the Ext algebra. This Ext algebra is an “annular” analogue of Khovanov’s arc algebra, and can be used to give an extension of Khovanov homology to links in the annulus.

### Winter Quarter, 2016

- January 5,
**Paul Johnson** (University of Sheffield)

Topology of Hilbert schemes and the Combinatorics of Partitions
**Abstract**: The Hilbert scheme of n points on a complex surface is a smooth manifold of dimension 2n. Their topology has beautiful structure related to physics, representation theory, and combinatorics. For instance, Göttsche’s formula gives a product formula for generating functions for their Betti numbers.

Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and when G is abelian their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology is well understood and in terms of cores and quotients of partitions, and related to representation theory of quantum groups. Following Gusein-Zade, Luengo and Melle-Hernández we study general abelian G, stating a conjectural product formula, and proving a homological stability result using a generalization of cores and quotients.

- January 12,
**Michael Thaddeus** (Columbia University)

Compactifications of reductive groups as moduli stacks of bundles
**Abstract**: I will explain how bi-equivariant compactifications of reductive groups may be realized as moduli spaces of principal bundles on chains of lines, reporting on joint work with Johan Martens. I will speculate on possible applications to global problems in the moduli theory of bundles on curves.

- January 26,
**Victor Ostrik** (UO)

Green rings and cyclotomy
**Abstract**: I will talk about following result: if the green ring (= split Grothendieck ring) of a symmetric tensor category has finite rank over the integers, then this ring is cyclotomic,

i.e. any homomorphism from it to the complex numbers lands in a cyclotomic extension of rational numbers. The proof

uses Adams operations in a version due to Dave Benson, and Chebotarev density theorem.

- February 9,
**Douglas Lind** (University of Washington)

The algebra of algebraic actions
**Abstract**: Many, perhaps most, dynamical questions about actions of one or several commuting group automorphisms can be directly translated, via duality, into questions in commutative algebra, number theory, and algebraic geometry. I’ll start by sketching how this works in several concrete examples and the kinds of questions that arise. I’ll then focus on the property of expansiveness, and finer questions about expansive subdynamics, whose complete solution involves amoebas, both complex and p-adic. Finally, I’ll describe analogous problems for actions of the discrete Heisenberg group, where there are many open problems.

- February 16,
**Emanuele Macri** (Northeastern University)

Stability conditions on the three dimensional projective space
**Abstract**: I will present joint work in progress with Benjamin Schmidt on stability for complexes on the projective three-dimensional space and on their moduli spaces.

- March 8,
**Henry Kvinge** (UC Davis)

The influence of the Kirillov-Reshetkhin crystal $B^{1,1}$ on the structure of simple cyclotomic KLR modules
**Abstract**: Khovanov-Lauda-Rouquier (KLR) algebras were invented to categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. It was later shown by Lauda-Vazirani that the simple modules of the cyclotomic KLR algebra, $R^{\Lambda}$, carry the structure of the highest weight crystal $B(\Lambda)$. It follows from this that any properties of $B(\Lambda)$ should be the shadow of some module-theoretic property of simple $R^{\Lambda}$-modules.

In classical affine type, highest weight crystals (which are infinite) have the remarkable property that they can be constructed from the tensor product of the much more tractable perfect crystals (which are finite). In this talk I will describe the algebraic analogue of this phenomenon in terms of simple $R^{\Lambda}$-modules in the case where the perfect crystal is the Kirillov-Reshetkhin crystal $B^{1,1}$ and $\Lambda$ is the fundamental weight $\Lambda_i$.

This is joint work with Monica Vazirani.

### Fall Quarter, 2015

- September 29,
**Justin Hilburn ** (UO)

Boundary Conditions in QFT and Modules in Catergory O
**Abstract**: It has been observed that all known dual pairs of symplectic singularities arise as the Higgs and Coulomb branches of the moduli space of vacua for 3d N=4 SUSY gauge theories, but so far very little is understood about the physical interpretation of the associated categories O. I will describe a new construction of projective and tilting modules in hypertoric category O inspired by the study of boundary conditions in abelian gauge theory.

- October 6,
**Nick Addington ** (UO)

Exoflops
**Abstract**: Consider a contraction pi: X -> Y from a smooth Calabi-Yau 3-fold to a singular one. (This is half of an “extremal transition;” the other half would be a smoothing of Y.) In many examples there is an intermediate object called an “exoflop” — a category of matrix factorizations, derived-equivalent to X, where the critical locus of the superpotential looks like Y with a P^1 sticking out of it, and objects of D(X) that will be killed by pi_* correspond to objects supported at the far end of the P^1. I will discuss one or two interesting examples. This is joint work with Paul Aspinwall.

- October 9,
** Ivan Loseu ** (Northeastern)

**Special Day/Location**: 4pm in 208 Deady Hall

Cacti and cells
**Abstract**: Cells (left, right and two-sided) are subsets in a Weyl group W that play an important role in several branches of Lie Representation theory. Cactus group of W is a group that should be thought as a crystal analog of the braid group. I will produce commuting actions of two copies of the cactus group on W that behave nicely on cells. The actions comes

from the categorical actions of the braid group on the BGG categories O.

This talk is based on

http://arxiv.org/abs/1506.04400
- October 13,
**Alexander Shapiro ** (Berkeley)

Towards cluster structure on quantum groups
**Abstract**: 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(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 have constructed an embedding of the quantum group U_q(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.

- October 20 and 27,
**Se-Jin Oh ** (UO)

Reduced expressions of finite Weyl group and categorification of quantum group of finite types
**Abstract**: In this talk, we will consider the relationship between reduced expressions and categorification of quantum group for finite types.

This talk includes the basic property of root systems and recent progress on categorification theory for quantum group simultaneously.

If time permits, I will talk about my “expectations” on this research area.

- November 3,
**Ben Elias** (UO)

Webs, ladders, and clasps
**Abstract**:

Webs – a way to describe morphisms between tensor products of fundamental representations of a lie algebra.

Ladders – rigid versions of webs; light ladders are a new basis for webs.

Clasps – idempotents projecting from a tensor product to its top irreducible summand. There’s now a conjectural recursive formula for them in type A.

- November 9,
**Pavel Etingof ** (MIT)

**Special Day/Location**: 3pm in 102 Deady Hall

The Weyl algebra has no finite quantum symmetries
**Abstract**: Let A_n be the n-th Weyl algebra over a field of characteristic zero (i.e., the algebra of polynomial differential operators in n variables).

Then any finite dimensional Hopf algebra action on A_n is trivial, i.e., factors through a group action. I will sketch a proof of this result, which is based on reduction modulo prime powers and the theory of actions of finite dimensional Hopf algebras on division rings. This argument easily generalizes to filtered quantizations of commutative domains. I will also explain how to use algebraic number theory to generalize this result to the case of q-Weyl algebras.

This is joint work with Juan Cuadra and Chelsea Walton.

- November 17,
**Dev Sinha** (UO)

Cohomology of symmetric and alternating groups
**Abstract**: I’ll review the connection between group cohomology and invariant theory, discuss Hopf ring and related structures, and define Fox-Neuwirth resolutions for symmetric groups and their subgroups. I will then use these to present the cohomology of symmetric groups (respectively alternating groups) all together using two products, one of which is the usual cup (Yoneda) product.

While this will build at times on topics from the topology seminar, the development should be understandable independently. For example, while bialgebra axioms will be verified in the topology seminar using covering spaces, the results can be stated in algebra where analogous formulae for induction-restriction are well known.

- November 24,
**Jon Brundan** (UO)
- November 30,
**Sabin Cautis** (UBC)

**Special Day/Location**: 11am in 203 Chapman Hall

K-theoretic geometric Satake
**Abstract**: The geometric Satake equivalence relates the category of perverse sheaves on the affine Grassmannian and the representation category of a semisimple group G. We will discuss a quantum K-theoretic version of this equivalence. In this setup the representation category of G is replaced with (a quantum version) of coherent sheaves on G/G.

This is joint work Joel Kamnitzer.

- December 1,
**Inna Entova-Aizenbud **

Schur-Weyl duality in complex rank
**Abstract**: Let V be a finite dimensional vector space. The classical Schur-Weyl duality describes the relation between the actions of the Lie algebra gl(V) and the symmetric group S_n on the tensor power V^{\otimes n}.

We will discuss Deligne categories Rep(S_t), which are extrapolations to complex t of the categories of finite dimensional representations of the symmetric groups. I will then present a generalization of the classical Schur-Weyl duality in the setting of Deligne categories, which involves a construction of “a complex tensor power of V”, and gives us a duality between the Deligne category and a Serre quotient of a parabolic category O for gl(V).

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