Pure Mathematics Honours Talks (I)

Speaker: Joel Springer (supervised by Mircea Voineagu)

Title: Cech cohomology as a cohomology theory

Time: 1:00-1:30pm  (Tue 15/Nov/2022)

Abstract

We will look to the construction of Cech cohomology often used in the study of topological manifolds, after a brief explanation of the direct limit and singular cohomology.

We will then address the question of to what extent Cech cohomology satisfies the Eilenberg-Steenrod axioms for cohomology theory to see how much it diverges from singular cohomology and other equivalent cohomology theories. We will finish with a discussion of fully relative Poincaré duality and how Cech cohomology can be nicer computationally than singular cohomology.

Speaker: Adam He (supervised by Jie Du)

Title: Representation theory of Dynkin quivers and the Ringel-Hall algebra

Time: 1:30pm-2:00pm

Abstract: We discuss the representations that arise from quivers of finite type, the so called Dynkin quivers. We begin by establishing the elegant ADE classification of Dynkin graphs and then introduce Gabriel's theorem which develops a profound connection with Lie theory. Finally we outline the construction of a Hall algebra and Ringel's application to quiver algebras resulting in the quantum group.

Speaker: Tal Zwikael (supervised by Daniel Chan)

Title: Geometric Invariant Theory and Quiver Representations

Time: 2:00-2:30 p.m

Abstract: Alastair D. King extends the notion of stability from geometric invariant theory into the notion of quiver representations in his 1994 paper “Moduli of Representations of Finite Dimensional Algebras”. We summarise his contributions in the representation of quivers and explore a simple quiver which offers extensive analysis through the lens of affine invariant theory.

Speaker: Chandler Corrigan (supervised by Lee Zhao)

Title: On the distribution of zeros for families of Dirichlet L-functions 

Time: 2:30-3:00pm

Abstract: With the generalised Riemann hypothesis currently unresolved, we instead turn our interests to bounding the number of zeros that could lie off the critical line. Results of this type are generally presented as averages over a family of primitive Dirichlet characters. We give results for averages over characters of certain fixed orders, and characters with conductor belonging to certain sparse sets.