# MaPS – MaPSS Seminar Series

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: Eric Hester, Alexander Kerschl, Nathan Duignan, and Giulian Wiggins.

MaPSS also ran in 2015, 2016, 2017, and 2018.

# Seminars in 2019, Semester 1

All seminars will be held at 5:00 pm on Mondays in Carslaw Room 535A, with free pizza and soft drink after the talk.

## Monday, March 4th

Eric Hester (The University of Sydney) — Understanding multi-scale Partial Differential Equations on arbitrary domains using the straightforward differential geometry of the signed distance function

Partial Differential Equations (PDEs) are a core part of mathematical modelling in science, industry and engineering; the applications are endless! And often the most interesting problems involve complex geometries. We normally model them using PDEs which live on separate domains, with boundary conditions applied at the infinitesimal interfaces.

But that’s math, not reality. Reality is smooth! Things get fuzzy at the micro and nanoscale. And it can also be useful to do simulations this way – don’t try to simulate your PDEs on complicated domains. Instead, perturb your PDEs to implicitly model your boundary conditions.

Having smooth (but small) transitions between domains means we are considering inherently multi-scale singular perturbations of PDEs. So these approximations are only true asymptotically. What we need to know is how these smooth approximations behave in the limit. Do they tend to the correct answer? How fast? And do they work in arbitrary geometries?

This talk will examine a really useful coordinate system for analysing such multi-scale PDEs. It all comes from the straightforward differential geometry of the signed distance function. I’ll be focussing on examples from my research on modelling moving objects in fluid dynamics. No background in differential geometry or fluid dynamics is required.

## Monday, March 11th

Giuliam Wiggins (The University of Sydney) — Abstract Voting Theory

We look at some cute applications of representation theory to the study of election procedures and voting paradoxes.

## Monday, March 18th

Nathan Duignan (The University of Sydney) — Simultaneous Binary Collisions and the Mysterious 8/3

Of central importance in the N-body problem is the fact that isolated binary collisions can be regularized: that a singular change of space and time variables (first written down by Levi-Civita) allows trajectories to pass analytically through binary collisions unscathed. The resulting flow is smooth with respect to initial conditions. Curiously, we are not so lucky with simultaneous binary collisions. In a landmark paper, Martinez and Simo gave strong evidence that the best one can hope is 8/3 differentiability of the flow in a neighborhood of simultaneous binary collisions in the 4 body problem. In this talk we follow Easton by linking regularizability to the behaviour of the flow in Conley isolating blocks around the collisions. We show the 8/3 is produced from the first resonant monomial with nonzero coefficient near a degenerate singularity formed when the two binaries are separately Levi-Civita regularized. To show this, we blow-up the singularity and study the flow near the resulting two, 3:1 resonant, normally hyperbolic manifolds connected by heteroclinics. A lengthy normal form computation confirms the conjecture.

## Monday, March 25th

Alice Bellière (The University of Namur) — Desynchronization of the Kuramoto model using the collective coordinates.

We witness in our surrounding that populations of humans or atoms or else animals are likely to synchronize their behaviour. We think for example about the neuronal synchronization in the brain. The aim of this synchronization is performing some actions like moving an arm. The action wouldn’t be possible if the neurons hadn’t synchronized such that the information can flow through the brain and reach their purpose. However, the synchronization of these systems is sometimes counterproductive. For example, the epileptic seizures are an over synchronization of the neurons and the brain can’t turn off by itself the message sent to the muscles. We will present the setup of a control in the Kuramoto model, model widely used for neuronal systems. That control, that we try to be as minimal as possible, should desynchronize the oscillators. Our approach is using the collective coordinates framework.

## Monday, April 1st

Claire Jiao (The University of Sydney) — Duality-based approach to stochastic control

We apply the Lasserre hierarchy of relaxations to stochastic optimal control problems to obtain tight pointwise bounds and global bounding functions for the value function. The primal minimization corresponds to the well studied moment problem based upon a set of necessary equality constraints on the occupation and boundary measures, whereas the dual maximization is built on a set of sufficient inequality constraints on the test polynomial function with a flexible choice of optimality criteria. The dual maximization is particularly effective in two senses: it only requires a single implementation to yield a tight global bound in the form of a polynomial function over the whole problem domain; an optimal solution to a dual problem can sometimes be reused directly as a feasible solution to different dual problems. We provide some numerical results to illustrate the effectiveness of this framework in a variety of stochastic control problems built on different Markov processes.

## Monday, April 8th

Carol Badre (The University of Sydney) — An Introduction to Bass-Serre Theory

One of the most fundamental parts of geometric group theory is Bass-Serre theory. Bass-Serre theory was introduced by Serre (and developed further by Bass) in the 1970’s in order to understand how groups act on simplicial trees. This theory allows for the extraction of information of the structure of a group from its action on a tree, characterising free groups, amalgamated products and their generalisations as groups acting on simplicial trees. Bass-Serre theory has inspired developments in the accessibility of groups, JSJ decomposition of finitely presented groups, lattices in automorphism groups of trees, as well as results in the world of CAT(0)-cube complexes and R-trees. Today I will give a gentle introduction to Bass-Serre theory.

## Monday, April 15th

Joel Gibson (The University of Sydney) — An introduction to error-correcting codes

Error-correcting codes are used all over the digital world to detect and recover from various errors, whether they be a noisy connection, a hard-drive failure, a scratched-up CD, or even a partially-revealed QR code. Almost all of these algorithms are algebraic at heart, relying on the properties of linear algebra and polynomials over finite fields to operate. In this talk, I cover some of the basics of error detection and correction, before showing how an optimal encoding algorithm can be built out of basic algebraic structures, such as vector spaces over Z/pZ, the integers modulo a prime number.

## Monday, April 29th

Connor Smith (The University of Sydney) — Introduction to Feature Stability and Selection in Regression Models

This talk about regression modelling is designed to cover a few basic concepts of both variable selection and stability. We will first cover the idea of feature selection and a few ways this is conducted; exhaustive searches, stepwise searches and shrinkage methods. Briefly covering resampling methodology, we will discuss the idea of how small changes in the data can lead to the selection of different models otherwise known as stability. We will then quickly touch on a new way of identifying which variables are important in a regression setting and how this is useful in more complex situations. By then end of the talk , you should understand a range of different methods used to identify and select important variables in a regression setting.

## Monday, May 6th

Yeeka Yau (The University of Sydney) — Coxeter systems for which the Brink-Howlett automaton is minimal

In their celebrated 1993 paper, Brink and Howlett proved that all finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognising the language of reduced words in a Coxeter group. This automaton is not minimal in general, and recently Christophe Hohlweg, Philippe Nadeau and Nathan Williams stated a conjectural criteria for the minimality. In this talk we will explain these concepts, and prove the conjecture of Hohlweg, Nadeau, and Williams. The talk should be accessible to anyone with an elementary knowledge of groups. This work is joint with James Parkinson.

## Monday, May 13th

Jacqui Ramagge (The University of Sydney) — My Life in Maths

The reading of a play in three acts outlining my life as a mathematician. It will be the story of my life since I started studying mathematics at university punctuated by the statement of some theorems related to the mathematics I was doing at various times and with a summary of life learnings from the experiences to date.

## Monday, May 20th

Kane Townsend (The University of Sydney) — Minimal parabolic subsystems containing Sylow subgroups in Weyl groups and finite groups of Lie type

Parabolic subgroups of Weyl groups and Levi subgroups of finite groups of Lie type are both associated to a subset of the generating reflections of the Weyl group. We will consider the minimal such parabolic subgroups and Levi subgroups containing the \ell-Sylow subgroups in the special case that the prime power q associated to the finite group of Lie type has order 1 modulo . A link between these two subgroups will be found by considering the order formulas of both groups which both depend on the degrees of invariants of the Weyl group and a well known fact about the prime divisors of cyclotomic polynomials.

## Monday, May 27th

Eric Hester (The University of Sydney) — Life Hacks for Your Research

Ever get sick of changing from your shiny laptop to a school computer for printing, or tuteroll? Hoping for an easier way to download papers and sort out your bibliography? Baffled by BibTex vs Biblatex for your arXiv submission? Well, this is the talk for you!

I’m going to show you some of the time saving tools I’ve amassed over my PhD, (especially Zotero) so you don’t have to spend years figuring it out like I did. This is going to be a very interactive demonstration, so please bring your laptops and we’ll get these tools installed!

## Monday, June 3rd

Michael Denes (University of New South Wales) – Lagrangian Coherent Structures in the Ocean

Ocean flows are dominated by Lagrangian Coherent Structures (LCSs), like gyres and eddies, with lifetimes longer than typical dynamical timescales. Due to their capacity to transport water and material over large distances, LCSs play an important role in climate, biogeochemistry, and small-scale mixing. In this talk, I will cover motivations for identifying LCSs, and a brief comparison of methods used to extract LCSs from high resolution ocean model output using simulated drifter trajectories. I will also provide some preliminary results on both a global scale and in the region where the Agulhas Current meets the Antarctic Circumpolar Current.

# Seminars in 2019, Semester 2

## Monday, August 12th

Sidney Holden (The University of Sydney) — Fundamental eigenmodes of the wave equation on a membrane and on a disc-shaped graph

We aim to determine which collections of 1D structures most closely reproduce the behaviour of `equivalent’ 2D structures when subjected to the wave equation. We solve the wave equation on a star graph and find that though the fundamental eigenmode exhibits similar behaviour to that on the 2D membrane, the speeds at which this behaviour occurs differ significantly. We attempt to approximate the fundamental eigenmode of the wave equation on the membrane by increasing the density of vertices on a random disc-shaped graph. We expect that as the density of vertices increases, the resulting collection of edges approximates 2D continuous space to the extent that the fundamental eigenvalue of the random disc-shaped graph approaches that of the wave equation on the membrane.

## Monday, August 19th

Zacharia Issa (The University of Sydney) — On Piles of Sand and Exotic Derivatives: An Introduction to Robust Finance

The field of robust finance was born in the wake of the global financial crisis, partly due to mathematical models employed by practitioners at the time being unable to adequately account for downside risks present in some financial assets. One of the key insights in this field was highlighting the link between no-arbitrage bounds for exotic payoffs and the optimal transportation problem of Gaspard Monge, which was first posed in 1781. This link allowed for many of the well-established techniques in optimal transport to be adapted for use in a financial context.

My motivation for this talk is to help give an intuition between why the problems of financial risk management and optimal transportation are so closely related. I will first explain some financial concepts, followed by a discussion of the Monge-Kantorovich optimal transport problem. Finally, I will then show how the robust finance problem marries these two seemingly disparate fields of research.

Note I am not assuming any understanding of financial mathematics or probability/measure theory; although, a background in either of these two fields will be helpful.

## Monday, August 26th

Haruki Osaka (The University of Sydney) — Overview of Likelihood Inference in Non-regular Settings

Likelihood based methods are popular in parametric statistical inference due to its well established theory and its intuitive interpretation. Analysis of the likelihood function determines the asymptotic behaviour of the maximum likelihood estimator and its associated test statistics. When the likelihood can be assumed to satisfy certain regularity conditions, these test statistics have a normal or chi-squared distributed limiting distribution and are efficient. However, when these conditions do not hold, the asymptotic theory becomes less widely known, as there is no result that encompasses all problems that avoid the narrow specification of regularity. We review some common situations where one or some of the regularity conditions which underlie likelihood based parametric inference fail. We’ll also talk about likelihood ratio tests in normal mixtures as a well studied problem of this kind.

## Monday, September 2nd

Vishnu Mangalath (The University of Sydney) — A Brief Introduction to Gauges

In this talk I will explain the concept of a “gauge”. Gauges, while being mostly used in theoretical physics, are no more than a choice of local “coordinate system” which varies according to some base space. After presenting some basic definitions, I will work through several examples of gauge freedom, and time permitting give an application of gauge transformations in a differential geometric setting. Knowledge of basic undergraduate algebra and differential geometry/topology will be helpful but not required.

## Monday, September 9th

Jonathan Mui (The University of Sydney) — Spectral Decomposition of Linear Operators

I will present a proof for the Jordan decomposition of matrices using only the most elementary results from complex analysis. The proof rests entirely on the Laurent expansion of the resolvent operator about an eigenvalue. This result is fairly old, and can be found already in a classic text by Kato (1966). However, it does not seem to be well-known outside of the field of operator theory. I will also discuss some applications to infinite dimensional Banach spaces.

## Monday, September 16th

Mitchell Curran (The University of Sydney) — Eigenvalues of the linearised Nonlinear Schrodinger Equation on a compact interval

In 1988 Jones proved a theorem regarding the existence of a positive eigenvalue for the linearised operator associated with the nonlinear Schrodinger equation with spatial domain the real line. Specifically, one linearises this complex-valued second order partial differential equation about a standing wave and splits the system into real and complex parts. The resulting operator N is not self-adjoint, and much of its spectrum lies on the imaginary axis; however, it can be written in terms of 2 self-adjoint operators (L_+, L_-) whose spectra are real. With P being the number of positive eigenvalues of L_+ and Q the number of positive eigenvalues of L_- (both well-defined quantities), we arrive at the neat relationship: P - Q = the number of positive real eigenvalues of N (well, almost). I am looking at this statement for the case when the spatial domain is a compact interval - we will see some pretty plots which shows the relationship holds true in this case. What remains is to rigorously prove the statement!

## Monday, September 23th

Hazel Browne (The University of Sydney) — A Generalisation of the McKay Correspondence

“If we needed to make contact with an alien civilization and show them how sophisticated our civilization is, perhaps showing them Dynkin diagrams would be the best choice!” ~ Etingof et al, Introduction to Representation Theory.

The Dynkin diagrams are famous because of their tendency to appear in classification problems across disparate areas of mathematics. One example is the McKay Correspondence: a natural bijection between conjugacy classes of finite subgroups of SL_2(C) and the affine simply-laced Dynkin diagrams.

We will begin by explaining this bijection, using some basic Representation Theory. Then we’ll introduce a generalisation of the map, and the motivating question of this honours project: is the generalised map a bijection? (Unfortunately it isn’t, but there are still plenty of interesting things we can say about it!)

I will make sure there are lots of pictures and fun for all the applied mathematicians who came to watch Edda’s talk and are stuck watching mine as well :)

Edda Koo (The University of Sydney) — Coherent Structures and Solitary Waves in Geophysical Fluid Flows

Coherent structures such as the Great Red Spot in the Jovian atmosphere or blocking highs in the atmosphere are prominent features in rapidly rotating fluids. Such fluids are described by the so-called quasi-geostrophic equations. We seek to find a reduced model description of coherent structures. We first attempt to model them as solitary waves and aim to derive the two-dimensional Zakharov-Kuznetsov equation - an extension of the one-dimensional Korteweg-de-Vries equation - which supports coherent stable lump solitary waves. We do so in a weakly nonlinear analysis of the quasi-geostrophic equations in several geophysically relevant scenarios. We will see that the QG equations do not allow for reduced solitary wave equations. In a second approach we will use collective coordinates to model coherent structures based on exact modon solutions of the quasi-geostrophic equations for constant mean flows.

## Tuesday, October 8th

Joshua Ciappara (The University of Sydney) — An introduction to p-adic analysis

Beginning with a discussion of ideas due to Kurt Hensel, we motivate and then construct the field of p-adic numbers _p. En route, we will witness the unusual geometry of ultrametric spaces and a classification theorem due to Ostrowski. To conclude, we mention some applications, including a result of Hasse–Minkowski, Hensel’s Lemma, and directions of further study (rigid geometry, p-adic differential equations, …)

## Monday, October 14th

Timothy Collier (The University of Sydney) — Existence of mild solutions for accretive operators

Partial differential equations have a number of different notions of solutions in part due to the variety of methods constructed to obtain results of existence, uniqueness and regularity. In this talk I will introduce the analysis of PDEs including one such notion well suited to evolution equations, namely mild solutions. In understanding these we will see the importance of accretive operators such as the negative Laplacian for which we can obtain mild solutions. We will then see some relationships between this and other notions of solutions.

## Monday, October 21st

Xiangnan Xu (The University of Sydney) — Introduction to Deep Learning

Deep learning has achieved tremendous success in recent years including classification, data generation and reinforcement learning. In this talk, we will introduce some basic ideas of deep learning and some network structures that are commonly used in deep learning. Also, we will show a network which converts photos from real world to animation style image using CycleGAN. Finally, if time permits, we will discuss some combination of deep learning with bioinformatics.

## Monday, October 28th

Eric Hester (The University of Sydney) — Examples of Duality in Mathematics

Vector and dual vector, matrix and transpose, covariant and contravariant. Time and frequency, position and momentum, temperature and entropy. Maths (and physics) are full of “dual” concepts. This talk will show some examples of this duality, and draw some connections between them.

## Monday, November 4th

Joel Gibson (The University of Sydney) — Quaternions, rotations, and Spin groups

The quaternions are a four-dimensional number system, famously discovered by William Hamilton in 1843 whilst vandalising a bridge in Dublin. Despite their dubious origins, they have become ubiquitous in computer graphics because of their close relationship to 3D rotations. In this talk I will cover different ways to encode 3D rotations, including unit quaternions and their higher-dimensional analogues built from Clifford algebras (the Spin groups), and conclude by comparing the underlying geometry of all of these parameter spaces.

The talk will include lots of pictures, discussion of vague questions like “what is an 11-dimensional rotation, really?”, and most probably a mislabelled angle in a planar geometry proof.