## Algebra Seminar

### Winter 2023

#### 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 $A\mathcal{V}$-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.

March 10 (University of Ottawa)

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)

Nikolay Bogachev (University of Toronto)

Title: TBA

Abstract: TBA