Fall 2021

Thursday, September 16, 4:00-5:00pm (UOttawa, STEM201)

Kirill Zaynullin (UOttawa)

Introduction to Schubert Calculus

I am aiming to explain the basics of Schubert calculus, the branch of modern algebraic geometry that studies intersections of linear subspaces. My goal is to relate it to a couple of recent results on cohomology/K-theory of flag varieties. All graduate students (or last year undergrads) who are interested in algebra and geometry are welcome.

Friday, October 8, 2:30-3:30pm (UOttawa, STEM201)

Colin Ingalls (Carleton)

Classification of Algebraic Surfaces

This will be an expository talk.

Friday, October 22, 3:00-4:00pm (UOttawa, STEM201)

Nathan Grieve

About linear series on projective varieties

In this introductory lecture, I will explain the concept of linear series on projective varieties. It will be a follow-up to the earlier lectures of Kirill and Colin. I will place some emphasis on the case of curves and intend to explain the statement of the main theorems from Brill-Noether theory. Time permitting, I will hint at some of the combinatorics and algebra that arise in their proof (e.g., Schubert calculus and Determinantal Varieties).

Friday, October 29, 2:30-3:30pm (UOttawa, STEM201)

Allan Francis Merino (UOttawa)

Schur-Weyl duality, Spin representation and Lie superalgebras

In my talk, I would like to explain few results that we can get directly by using the Schur-Weyl duality. More precisely, one can obtain the Howe duality for the pair (GL(n, C), GL(m,C)) in Sp(2nm, C) and a similar duality for the spinorial representation of O(nm, nm, C). It turns out that the previous results can be extended to other pair of subgroups of the real or complex symplectic and orthogonal groups.
In my talk, I am going to explain what is known concerning the previous dualities (by avoiding the technical details). Time permitting, I will give a possible extension of the previous results to Lie superalgebras.

Friday, November 5, 2:30-3:30pm (UOttawa, STEM201)

Erhard Neher (UOttawa)

What is a linear algebraic group? 

Linear algebraic groups are a central topic in algebra and a very active research area. For example, they appeared in Kirill’s talk in the algebra seminar on Grassmannains and Schubert calculus and they will likely re-appear in other talks of the algebra seminar. 

The goal of my talk is to introduce students to the structure theory of linear algebraic groups (experts in algebraic groups are discouraged to attend). I will not assume anything beyond a senior undergraduate algebra course, like the University of Ottawa’s MAT3143.

Friday, November 12, 2:30-3:30pm (UOttawa, STEM201)

Monica Nevins (UOttawa)

Welcome to p-adic representation theory

Representation theory gives us the capacity to use tools from linear algebra over complex vector spaces to understand abstract groups. Of special importance are algebraic groups over a field F; when F=R these include many Lie groups, but if F is p-adic, we get groups crucial to number-theoretic problems such as the Langlands’ programme. In this introductory talk, we’ll aim to convey some of the spirit of, and big open questions in, p-adic representation theory today.

Friday, November 19, 2:30-3:45pm (UOttawa, STEM201)

(a small coffee will be offered 2:30-2:45; the talk will start at 2:45)

Yuli Billig (Carleton)

Topological Quantum Computing

Topological quantum computing is based on the process of braiding of anyons. Anyons are quasiparticles that acquire a non-trivial phase shift when moved around other quasiparticles. The advantage of topological quantum computing to “traditional” quantum computing is that the result of a computation depends only on the homological class of the braiding trajectory and is thus not susceptible to small errors. Accumulation of errors in the process of computation is a major challenge for “traditional” quantum computing. We will discuss the connection of topological quantum computing to quantum groups and braided tensor categories.

Friday, November 26, 2:30-3:45pm (UOttawa, STEM201)

(a small coffee will be offered 2:30-2:45; the talk will start at 2:45)

Chris Dionne (Queen’s)

Numerical restrictions on Nagata curves and geometric consequences of small Seshadri constants

Seshadri constants are numbers associated with ample line bundles on smooth projective varieties (but we’ll stick to surfaces). These are very interesting numbers that show up in many different contexts. Examples include the study of plane curves, symplectic packing, and many others. We’ll give a quick introduction to the topic and discuss some of these connections.

The Seshadri constant is bounded above by the “Nagata bound”, and in cases where it is not equal to this maximal value there is a set of “Nagata curves” which “break the Nagata bound”. We’ll explain what this means and give some necessary conditions for a curve to be a Nagata curve. We’ll also give a strategy for generating lower bounds on the Seshadri constant based on these conditions. Finally, we’ll go over some cases where we can learn about the geometry of the surface if we have a small enough upper bound on the Seshadri constant.

Friday, December 10, 2:30-3:45pm (UOttawa, STEM201)

(a small coffee will be offered 2:30-2:45; the talk will start at 2:45)

Hadi Salmasian (UOttawa)

The faithful dimension of finite groups and a “logical” argument

Given a finite group G, what is the smallest integer n such that G can be embedded in the group of invertible n by n complex matrices? The number n is called the faithful dimension of G. Perhaps surprisingly, this elementary notion appears in “advanced” research directions in combinatorics, number theory, and algebraic geometry.

In this talk, I will first introduce some applications of the faithful dimension. Then I will explain a not-so-well-known approach to understanding representations of finite groups that is inspired by symplectic geometry. Finally, I will talk about a recent result on “polynomiality” of the faithful dimension, which is proved using model theory.

This talk will be fully accessible to graduate and advanced undergraduate students. Part of this lecture is based on a forthcoming joint work with Bardestani and Mallahi-Karai.