Pure Mathematics Honours Talks (II)

Speaker: Alexander Bednarek (supervised by Behrouz Taji)

Title: Canonical Metrics in Complex Geometry

Time: 1:00-1:30pm (Wed 16 Nov 2022)

Abstract: Complex geometry is the study of geometric objects constructed out of the complex numbers such as complex manifolds and holomorphic vector bundles. One can provide additional geometric structure in the form of a metric and an interesting question is for which manifolds particular canonical metrics exist. For certain Kahler manifolds, the existence of Kahler-Einstein metrics is given by the Calabi-Yau theorem for which Shing-Tung Yau received the Fields medal in 1982.

The Riemann uniformization theorem states that any simply-connected Riemann surface is biholomorphic to either P^1(C), C or the unit disc D. This is an extension of the Gauss-Bonnet theorem which provides a connection between topology and geometry in terms of genus and curvature. As every compact Riemann surface is a 1-dimensional Kahler manifold, one generalizes the Riemann uniformization theorem to higher dimensions via the use of Kahler-Einstein metrics to show that certain topological constraints control the curvature of a Kahler manifold. This thesis provides explicit local calculations for this partial generalization which are inaccessible for those who are not well-versed in the field.

Speaker: Jodie Lee (supervised by Catherine Greenhill)

Title: Colouring Random Graphs

Time: 1:30pm-2:00pm

Abstract: The chromatic number of a graph is the smallest number of colours needed for a vertex colouring where no two adjacent vertices are coloured the same. If I choose a graph randomly, what can we expect its chromatic number to be? Now, if I choose another graph according to the same random model, how different is the chromatic number from this graph compared to the first?

In this talk, we answer these questions with the binomial or Erdos-Renyi random graph G_{n,p}. We will see that the chromatic number behaves quite differently depending the choice of probability p = p(n). I will give an overview of the major breakthroughs and ideas which help uncover the interesting behaviour of this graph parameter.