Transformation Groups Seminar

Speaker: Matthias Franz (Western)

"The integral cohomology of smooth toric varieties"

Time: 10:30

Room: MC 204

We present a proof that the integral cohomology of a smooth toric variety is additively isomorphic to a torsion product involving the Stanley-Reisner ring of the fan defining the toric variety. Ingredients are a result of Gugenheim-May about the cohomology of pull-backs of principal torus bundles and a formality result for Davis-Januszkiewicz spaces.

Final Presentation

Speaker: Meagan James (Western)

"An Introduction to Mapping Class Groups"

Time: 13:30

Room: MC 108

Given a surface S, the set of all homeomorphisms from S to itself which fix the boundary and preserve orientation form a group under composition; this is known as the group of homeomorphisms and is denoted Homeo+(S, ∂S). The mapping class group of S, denoted Mod(S), can be understood as Homeo+(S, ∂S) modulo homotopy. Mapping class groups are often studied using simple closed curves in the surface, that is, embeddings of the form f : S1 → S. More specifically, given a collection of simple closed curves in S, we can understand the behaviour of an element of the mapping class group f ∈ Mod(S) by observing what happens to the simple closed curves after applying a representative homeomorphism ϕ of the class f to the surface S. The curve graph, denoted C(S), is a graph whose vertices correspond to isotopy classes of essential simple closed curves in S and edges join vertices whose isotopy classes have disjoint representatives. In this talk, we will become familiarized with curves in surfaces in order to better understand mapping class groups of different surfaces. We will then discuss the nature of the curve graph and how it can be used as a combinatorial model of Mod(S).

Geometry and Topology

Speaker: Luuk Stehouwer (Dalhousie University)

"Cutting and pasting manifolds"

Time: 15:30

Room: MC 107

In this talk, we will explore an equivalence relation on manifolds through cutting and pasting along boundaries. This relation leads to the definition of an abelian group known as the SKK-group, which was introduced by Karras, Kreck, Neumann, and Ossa in the 1970s. We will discuss the connection between the SKK-group and the bordism category, as well as the bordism group. This connection is made clearer through the examination of the fundamental group of the geometric realization of categories, described using zig-zags. Additionally, I will present joint work with Simona Veselá and Renee Hoekzema, where we compute the SKK groups for particular cases. If time allows, we will also briefly explore the relationship between the SKK-group and invertible topological quantum field theory.

Colloquium

Speaker: Ajneet Dhillon (Western)

"Basic Notions: The Borel-Bott-Weil theorem"

Time: 15:30

Room: MC 107

The Borel-Bott-Weil theorem describes the cohomology of line bundles on homogeneous varieties in terms of the representation theory linear algebraic groups. This talk will be a gentle introduction to this theorem in the case of the general linear group.

The first part of the talk will be an extensive motivation for why one is interested in line bundles on projective varieties and their cohomologies. In an nutshell, line bundles turn the extrinsic problem of studying maps to projective space into an intrinsic one.

The second part of the talk will discuss the representation theory of the general linear group and will end with a statement of the theorem in this special case.

Graduate Seminar

Speaker: Curtis Wilson (Western)

"The Golod-Shafarevich Theorem"

Time: 16:30

Room: MC 108

We introduce the class field tower infinite issue and state its solution, the Golod-Shafarevich theorem. We prove the theorem, and provide a refinement for the graded case. We discuss some examples and finish with an application involving finitely generated infinite torsion groups.