Graduate Seminar

Speaker: Nicholas Meadows (Western)

"the Hilbert scheme of Cohen-Macauley curves"

Time: 11:20

Room: MC 106

After reviewing very quickly some algebraic geometry, I will define the Hilbert scheme which parameterizes closed subschemes of projective space $P_{k}^{n}$ and state its basic properties, for k an algebraically closed field of characteristic 0. I will then define various notions of deformations (deformations sheaves, deformations over the dual numbers etc). Finally, I will use obstruction theory for a local ring to prove a lower bound on the dimension of irreducible components of the Hilbert scheme of Cohen-Macauley curves of genus g and degree d in $P_{k}^{3}$

Geometry and Topology

Speaker: Chris Kapulkin (Western)

"Internal languages for higher categories"

Time: 15:30

Room: MC 107

Every category $C$ looks locally like a category of sets, and further structure on $C$ determines what logic one can use to reason about these "sets". For example, if $C$ is a topos, one can use full (higher order) intuitionistic logic. Similarly, one expects that every higher category looks locally like a higher category of spaces. A natural question then is: what sort of logic can we use to reason about these "spaces"? It has been conjectured that such logics are provided by variants of Homotopy Type Theory, a formal logical system, recently proposed as a foundation of mathematics by Vladimir Voevodsky. After explaining the necessary background, I will report on the progress towards proving this conjecture.

Analysis Seminar

Speaker: Thomas Ransford (U. Laval)

"Capacity and coverings"

Time: 14:30

Room: MC 107

I shall discuss two elementary inequalities relating capacity to coverings. They provide an approach to determining whether a set has positive capacity and, if so, to estimating the value of the capacity. (Joint work with Quentin Rajon, Jeremie Rostand and Alexis Selezneff).

Homotopy Theory

Speaker: Cihan Okay (Western)

"Homotopy groups of the circle"

Time: 12:00

Room: MC 106

(Note the unusual day, time, and room.) I will talk about homotopy type theoretic proofs of a well known topological fact that the fundamental group of the circle is the set of integers. There are two closely related proofs. The homotopy-theoretic proof follows a similar reasoning used in the classical proof in topology, whereas the encode-decode proof is more type theoretic.

Colloquium

Speaker: David Riley (Western)

"Hopf algebra actions, gradings, and identical relations"

Time: 15:30

Room: MC 107

I will begin by discussing how and when the action of a Hopf algebra $H$ on an algebra $A$ can be viewed as a grading of $A$. For example, if $G$ is a finite group and $H$ is the dual of the group algebra $K[G]$, then $A$ is an $H$-algebra precisely when $A$ is group-graded by $G$. I will then discuss the identical relations of an algebra with a Hopf algebra action. In particular, I will address the following question: when does the existence of an $H$-identity on $A$ imply the existence of an ordinary polynomial identity on $A$?

Algebra Seminar

Speaker: Martin Frankland (Western)

"Locally presentable categories and applications"

Time: 14:30

Room: MC 107

We will survey some characterization theorems for locally presentable categories and variants thereof. Then we will discuss some applications of locally presentable categories to homological and homotopical algebra.