## Joint UOttawa/Carleton

## Algebra Seminar

### Winter 2023

#### Usual time: Fridays, 2:30-4:00pm (coffee 2:30-2:45; talk 2:45-3:45)

**January 27** **(Carleton University, HP 4325)**

Kathlyn Dykes (Carleton University)

Title: MV polytopes of highest vertex w

Abstract: Mirkovic and Vilonen provide a geometric interpretation of the

representation theory of an algebraic group, G, using certain varieties

known as Mirkovic-Vilonen (MV) cycles. As geometric objects, these

varieties have been difficult to understand but through the work of

Kamnitzer, the set of MV polytopes gives a purely combinatorial

description of these MV cycles. Goncharov and Shen take this one step

further by explicitly showing that the set of MV polytopes corresponds to

the tropical points of the unipotent group of G.

In this talk, we consider a certain subclass of these polytopes, called

MV polytopes of highest vertex w, and show that they are in

correspondence to the tropical points of reduced double Bruhat cells of

the unipotent group. We will explore the combinatorics of this subclass

of polytopes and show that the vertices can be labelled by elements in

the Weyl group which are less than w in the Bruhat order.

**February 3 (Carleton University, HP 4325)**

Henrique Rocha (Carleton University)

Title: Annihilators of $Amathcal{V}$-modules and differential operators

Abstract: In this talk, we will present a recent result on the category

of finitely generated modules over the ring of functions of a smooth

algebraic variety that has a compatible action of the Lie algebra of

polynomials vector fields on such variety. We show that any object in

this category sheafifies to an infinitesimally equivariant bundle, and

the associated representation of the Lie algebra is given by a

differential operator of order depending on the rank of the module.

For this purpose, we will present certain annihilators of these modules which

play a central role in the proof of these two results.

**February 10 (Carleton University, HP 4325)**

Yuli Billig (Carleton University)

Title: Sheaf of algebras governing representations of vector fields on

quasi-projective varieties.

Abstract: We construct a quasicoherent sheaf of associative algebras

which controls the category of finitely generated $AV$-modules over a

smooth quasiprojective variety. We establish a local structure theorem,

proving that in ‘etale charts these associative algebras decompose into

a tensor product of the algebra of differential operators and the

universal enveloping algebra of the Lie algebra of power series vector

fields vanishing at the origin. This is joint work with Colin Ingalls.

**February 17 (Carleton University, HP 4325)**

Emile Bouaziz (Carleton University)

Title: Calculus On Loop Spaces

Abstract: We explain how some simple meditations on abstract mirror

symmetry for representations of the N=2 vertex algebra lead to a

construction of a loop space version of the usual “calculus” package of

forms and polyvectors. We discuss the resulting algebraic structures and

how we can use these to construct invariants of Poisson manifolds.

**March 10** **(University of Ottawa, STEM464)**

Houari Benammar Ammar (UQAM)

Title: Fibred algebraic surface with low slope.

Abstract: Let “f : S to C” be a morphism with connected fibers from a smooth complex projective surface to a smooth complex projective curve, we describe the notion of slope inequality as defined in the reference below. We explain the work in progress and some results obtained very recently.

**March 17 ** **(University of Ottawa, STEM464 at 4:00pm)**

Nikolay Bogachev (University of Toronto)

Title: On geometry and arithmetic of discrete groups

Abstract: Modern methods of the theory of discrete subgroups of Lie groups usually combine ideas and approaches from general group theory, geometry and topology, number theory, dynamics, classical theory of algebraic groups and Lie groups, and representation theory. And lattices in Lie groups, meaning discrete subgroups of finite co-volume with respect to the Haar measure, play an important role here. There was a famous conjecture by Selberg and Piatetski-Shapiro that irreducible lattices are actually arithmetic groups, i.e. integer points of algebraic groups. This conjecture was proved in 1974 by Margulis (due to his famous Superrigidity and Arithmeticity Theorems) for lattices in Lie groups of real rank > 1.

However, in real rank 1 this conjecture is not true, and thus a rich theory of arithmetic and non-arithmetic lattices arises in Lie groups PO(n,1) = Isom H^n and PU(n,1). The problem of detecting arithmeticity of a hyperbolic n-manifold H^n/Г only through its geometry has been remained open in general for several decades. This question attracted many famous experts, but only very recently several revolutionary solutions were suggested to this problem. Different superrigidity approaches were suggested by Margulis and Mohammadi in dimension 3, and by Bader, Fisher, Miller and Stover for all n>2. In our joint paper with Misha Belolipetsky, Sasha Kolpakov and Leone Slavich we also developed a large industry connecting arithmeticity of a lattice Г in PO(n,1) = Isom H^n with geometric properties of a hyperbolic manifold H^n/Г.

In this talk, I will give an overview of this area with some focus on real hyperbolic geometry, and also will present the above mentioned recent results. If time permits, I will also briefly discuss the highlights of Vinberg’s theory of hyperbolic reflection groups which provide many interesting and nice examples in this field.

**March 24 ** **(Carleton University)**

Matthew Rupert (University of Saskatchewan)

Title: Recent progress on the Kazhdan-Lusztig correspondence for vertex

operator algebras and quantum groups.

Abstract: The Kazhdan-Lusztig correspondence is a series of

equivalences between representation categories of vertex operator

algebras and quantum groups. In this talk I will present recent

categorical results which aid in proving such equivalences with

applications to the singlet vertex algebra in particular. Based on joint

work with Thomas Creutzig and Simon Lentner.