Pure Mathematics Honours Talks (III)

Speaker: Jee Hyo Lee (supervised by Michael Cowling)

Title: L^p Convergence of Fourier Series in N-dimensional Spheres

Time: 9:00-9:30am Thurs 17/Nov/2022

Abstract: Trigonometric functions form an orthogonal basis in L^2 ([0, 1]). This can be further extended to the case where we replace [0, 1] with the torus. We show that Fourier series converges for L^2 -norms. We then prove that Fourier series converges for L^p norms with 1 < p < ∞ and convergence fails for p = 1 and p = ∞. Fourier series approximation on a torus is extended to Fourier-Laplace approximation on n-spheres. We prove that Fourier-Laplace series converges for L 2 -norms. We then show Fourier-Laplace series does not converge for p ̸= 2. Cesaro mean is then introduced and conditions for the convergence of Cesaro means will be found.

Speaker: Chris Zeng (supervised by Alina Ostafe)

Title: The Uniform Boundedness Conjecture in Arithmetic Dynamics

Time: 9:30-10:00 a.m

Abstract: Arithmetic dynamics is the study of dynamics over fields that are of interest in number theory. We focus on a particularly interesting conjecture in this field called the Uniform Boundedness Conjecture, which posits uniform bounds for the number of preperiodic points (a special type of point analogous to torsion points on algebraic objects) of rational functions over number fields. We will discuss partial results obtained by allowing an additional parameter corresponding to primes of bad reduction.  This will involve either a reduction to Diophantine equations or to the tools of complex and non-archimedean dynamics.

Speaker: Fahim Rahman (supervised by Daniel Chan)

Title: An Investigation in Invariant Theory - The Shephard Todd Chevalley Theorem

Time: 10:00-10:30am

Abstract: The study of Invariants is a relatively old branch of mathematics that has been revived throughout the course of its existence in numerous ways. I hope to share some insight into the invariants of polynomial rings, some basic properties, and how we can study them by exploiting their graded structure and their respective Hilbert series. The main aim is to introduce notions and machinery that allows us to appreciate a jewel of invariant theory: the Shephard Todd Chevalley theorem, which states that the invariant ring of a polynomial ring is also a polynomial ring if and only if the associated group is generated by pseudoreflections. This all takes place in a commutative setting and as such, I also hope to shed some insight on some more modern work about how the Shephard Todd Chevalley Theorem behaves in a non-commutative setting with a focus on revising our initial hypotheses. We also refine the notion of a pseudoreflection to that of quasi-reflections and mystic reflections.

Speaker: Dallas Yan (supervised by Catherine Greenhill)

Title: Approximately counting colourings of sparse linear uniform hypergraphs

Time: 10:30-11:00 am

Abstract: A graph colouring is an assignment of colours to vertices so that adjacent vertices receive distinct colours. Many problems in industry and academia can be modelled as graph colourings. A simple example is ensuring there is no overlap between nearby radio stations transmitting at the same frequency. Counting colourings is also of interest in statistical physics, where the anti-ferromagnetic Potts model assigns colours to vertices and gives colourings a weight. 

The problem of counting colourings of graphs is #P-complete, which is generally considered strong evidence that there is no way of efficiently solving the problem exactly. For this reason we instead use efficient approximation algorithms. Various methods of efficiently approximately counting colourings have been studied, and we will focus on one particular method which involves Markov chains.

A hypergraph is a generalisation of graphs where an edge may contain any number of vertices. A k-uniform hypergraph is one where every edge contains k vertices. We introduce two Markov chains which can be used to approximately count colourings on graphs and discuss extending them to k-uniform hypergraphs.

The area involves a mixture of graph theory, probability and theoretical computer science.