Pure Honours Talks -- Honours Talks

All academic staff, current and prospective Honours students are invited to attend. Monday 16 October, Carslaw Lecture Room 451

14:00-14:20 William Magarey

Semigroups and Differential Equations on the Space of Borel Measures

Abstract: In this talk, we firstly introduce a locally convex topology which we call the mixed topology \(\tau_{\mathcal{M}}\) on the space of bounded continuous functions on some Polish space E. From this we can explore the dual space, which is the space of signed Borel measures on E \(M(E)\). We then look at semigroups on \(M(E)\) and use duality to prove certain results about these semigroups. From here we then look at differential equations and stochastic differential equations proving the uniqueness and equivalence of weak and mild solutions of these equations to certain different types of equations on this space.

14:25-14:45 Matthew Hanna

The Calabi-Futaki Invariant: An Obstruction to Kahler-Einstein Metrics

Abstract: The Calabi-Futaki invariant is a significant integral within Kahler geometry, as an obstruction to the existence of Kahler-Einstein metrics. The so called ’invariance’ arises from the property that the integral remains unaltered when using any metric from its Kahler class. We aim to discuss the necessary techniques to demonstrate a proof of the invariance and examine how it obstructs Kahler-Einstein metrics.

14:50-15:10 Tiernan Cartwright

Kahler Geometry and the Kahler-Ricci Flow

Abstract: The Ricci flow is a PDE which evolves the metric on a Riemannian manifold and aims to improve it. This talk introduces the Ricci flow on Kahler manifolds, which are manifolds which admit compatible Riemannian, complex and symplectic structures. The focus is on the geometric aspects of the theory, namely about relevant theorems from Kahler geometry, and how they can simplify results for the Kahler—Ricci flow.

15:15-15:35 Yuxuan Cheng

The graded representation theory of symmetric groups and the graded decomposition number

Abstract: Computing the decomposition number is one of the main outstanding questions in the representation theory of the symmetric group \(\frak{S}_n\) of order \(n\geqslant 1\). In 1996, Lascoux, Leclerc and Thibon conjectured and produced an algorithm that compute the decomposition number in the Iwahori-Hecke algebra. The conjecture is proved as a theorem and the (LLT) algorithm provide the decomposition number as a polynomial which brings more insight of grading structure to the Iwahori-Hecke algebra. In this talk, we will introduce the graded decomposition number of KLR algebra (which is closely related to the Iwahori-Hecke algebra) which is naturally graded. We will then exhibit the LLT algorithm with an example that computes a canonical basis element labeled by a small partition; this will give the graded decomposition number of this partition. The talk will also introduce a lot of combinatorial objects such as partitions, diagrams, and Tableaux etc. with examples.