Welcome to the webpage for the Mathematical Postgraduate Seminar Series (MaPSS). We’re committed to fostering a friendly atmosphere in the school of Mathematics and Statistics. All maths postgraduate students are encouraged to present. It’s an excellent opportunity to hone presentation skills and talk about fun new topics. Most of all, it’s a great way of getting to know your fellow students. So come along and meet some friends over free pizza!

If you or anyone you know is interested in presenting, or for any other enquiries, please contact the MaPSS organisers: Grace Garden, Yunwei Zhang and Wenqi Yue.

For a record of MaPS talks in previous years please visit here

See also the postgraduate reading groups.

# Seminars in 2022, Semester 1

Seminars are held usually at 5:00 pm on Thursdays in the Access Grid Room(AGR) on Level 8 of the Carslaw Building. Free pizza & soft drinks are provided at the end of the talks.

Below is a list of presenters with the titles and abstracts of their presentations

## Thursday, August 18th

Grace Garden (The University of Sydney) — Why should I care about character varieties?

Character varieties are a rich area of study and have been shown to reflect geometric and topological information of manifolds. This talk will serve as an introduction to character varieties from this perspective. I will go through the definition, some useful examples, and some motivation and tools for studying these objects.

## Thursday September 1st

Marco Lotz (Otto von Guericke University Magdeburg) — Reflection length in non-affine Coxeter groups

Coxeter groups are finitely generated reflection groups, classified roughly by the type of space they act on. This talk illustrates the geometric nature of Coxeter groups by means of the reflection length. Besides introductory definitions and results, we look at some hyperbolic pictures.

## Thursday September 15th

James Morgan (The University of Sydney) — A Gentle Introduction to Hyperbolic Knot Theory

When studying knots and links we often look to their complements. These are 3-manifolds for which we have a myriad of tools at our disposal in order to study them. More often than (k)not, we study these manifolds from a purely topological perspective. However, it has been established that the complements of many knots and links admit a complete hyperpolic structure, and many that don’t can be decomposed into pieces which do.

The aim of this talk will be to introduce topologically ideal triangulations of these hyperbolic knots through two main examples - the figure 8 knot (the simplest hyperbolic knot), and another knot that I’m not specifying here. We won’t prove hyperbolicity, but hopefully some intuition as to why we should think they’re hyperbolic will be given. Time permitting, we will also look at a large family of hyperbolic links - two bridge links.

Overall this talk will consist of many pictures, no proofs, and hopefully minimal written words.

## Thursday September 29th (Cancelled)

Vladimir Jakovljevic (The University of Sydney) — Confocal surfaces and Pseudo-Euclidean Billiards on Spheres in Four-Dimensional Pseudo-Euclidean Spaces

The n-dimensional Euclidean space is equipped with bilinear form diag(1,1,…,1), generating well-known geometrical visualisation of orthogonality, called perpendicularity. Changing this form to G=diag(-1,…,-1,1,..1), we lose the intuition of orthogonality and stay only with the analytical tool to explore it. Set R^n equipped with G is called Pseudo-Euclidean space with signature (k,n-k), where k represents how many -1 we have in G. Since the orthogonality plays an essential role in the law of reflection, pseudo-Euclidean billiards differ to Euclidean in terms of trajectory. Nevertheless, previous research has shown that, in many cases, they have Poncelet-like characteristics and are integrable.

In this talk, I observe pseudo-Euclidean billiards on two spheres: one is in a four-dimensional Pseudo-Euclidean space with signature (1,3), and the other is in a four-dimensional P-E space with signature (2,2). Imposing the condition for their boundaries to be compact and generated by cones with vertex at the origin, my goals are to:

- describe and classify them;
- explore the existence and (potentially) describe their caustics;
- determine their integrability in the sense of Liouville;
- explore Poncelet-like characteristics;
- provide analytical conditions for periodical orbits.

## Thursday October 13th

Tomas Lasic Latimer (The University of Sydney) — An Intro to Riemann-Hilbert Problems

The talk is aimed to be a brief introduction to Riemann-Hilbert Problems(RHPs) and their applications. First, I?ll talk about how differential equations can be reframed as RHPs and what?s gained from this. Then, I?ll talk about how RHPs can be used to describe orthogonal polynomials. Finally, I?ll discuss RHPs in the context of difference equations. If time permits, I?ll also talk about some of the complex analysis techniques used in RHP theory