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, and 2017.

See also the postgraduate reading groups.

# Seminars in 2018, 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 5th

Zeaiter Zeaiter (Sydney University) — The Effect of Thermoregulation on Honey Bee Colony Health and Survival

In recent years honey bee colonies have been experiencing increased loss of hives. One cause of hive loss is colony collapse disorder (CCD). Colony collapse disorder is characterised by a previously healthy hive having few or no adult bees but with food and brood still present. This occurs over several weeks. It is not known if there is an exact cause of CCD but rather it is thought to be the accumulation of multiple stressors placed on a hive. One of theses stressors is the breakdown of thermoregulation inside the hive. The bee life cycle begins with eggs that hatch into larvae that become brood. The hive contains combs which are made up of multiple cells; these cells house the brood. Pupal cells are capped off by adult bees (and so are known as capped brood) and they undergo changes to develop into an adult bee. In order for these capped rood to develop correctly, physically and mentally, the temperature within the hive must be regulated by the hive bees to ensure optimal development of the capped brood. Variations in the temperature, caused by the breakdown of thermoregulation, lead to deformations in the adults that emerge from capped brood. This later leads to these bees becoming inefficient foragers which also have shorter life spans. We model the effect of thermoregulation on hive health using a system of DDEs which gives insights into how varying hive temperatures have an effect on the survival of the colony.

## Monday, March 12th

Joel Gibson (Sydney University) — A Brief Introduction to Differential Forms

The language of differential forms was developed in order to do calculus on (oriented) manifolds, particularly in more than three dimensions, where a plane is no longer determined by a normal vector. In this talk, I will give an introduction to integration using differential forms, with many examples in three dimensions relating back to the usual curve, surface, and volume integrals. Using this language, the gradient, curl, and divergence operators are replaced by a single operator, and Stokes’, the divergence, and fundamental theorems of calculus are replaced by a single equation. Time permitting, I will mention de Rham’s theorem, relating the cohomology of a manifold to solutions of differential equations.

## Monday, March 19th

Pantea Pooladvand (Sydney University) — Do T cells compete for antigen?

When a pathogen invasion begins, our bodies immune response is two-fold. First, the T cells will go through a rapid expansion phase, in order to fight off the intruders, followed by a contraction phase which subsequently contributes to immunological memory. It is difficult to assess the contribution of initial T cell numbers to the total T cell numbers at the peak of the response due to the widely differing views in recent publications. Does the initial number of T cells determine the peak or is the T cell response limited by the amount of antigen present? Inspired by new experimental results from our collaborators, we introduce a system of ODEs to investigate this problem by considering that T cells compete for limited amount of pathogen. We propose that this competition between T cells limits the peak of response and we compare the dynamics from this system to our collaborators’ data. To further explore this problem, we consider a published model based on the opposing view, that T cell replication is an inbuilt developmental program. Can this model also explain the experimental results or is competition a better explanation of this phenomenon?

## Monday, March 26th

Lucy Klinger (Sydney University) — Using mean field games to study cost-per-click advertising behaviour

In online sponsored search mechanism, advertisements are sold based on a cost-per-click model and where each ad unit sold is measured per click. The pricing of each click is determined by the multi-keyword sponsored search auction mechanism which involves multiple generalised second price auctions running simultaneously for each query. In a dynamic game setting, a continuum of advertisers participate in a sequence of multi-keyword sponsored search auctions, and their bidding behaviour can be analysed as a non-cooperative game of incomplete and imperfect information. Each advertiser has a private valuation that is modelled by a stationary stochastic process, and the motion of cost state is driven by the optimal drift, which can be derived from the ex-post Bayesian Nash equilibrium bids generated by the static version of the game. Though the induced dynamic game is complex, we can simplify the analysis of the market using an approximation methodology known as mean field games, to study a specific example. The methodology assumes that advertisers optimise only with respect to long run average estimates of the distribution of other advertisers’ bids. Closed-form analytic solutions do not exist; however, I developed a numerical method for computing both stationary and time-varying equilibria. The problem can be broken down into a system of coupled PDEs, where an individual advertiser’s bidding choices can be analysed by solving Hamilton-Jacobi-Bellman equations, and the evolution of joint distribution of costs and valuations can be characterised by Fokker-Planck equations. I also show that a mean-field equilibrium exists, and that it is a good approximation to the rational advertisers’ behaviour when the number of advertisers is large. This was then followed by computing the hypothetical best response via solving a mixed-integer nonlinear problem to produce optimal bids.

## Monday, April 9th

Andrew Swan — Timid frogs and hyperbolic magnets are isomorphic

The aim of this talk will be to make sense of the title. To be less cryptic: I wish to outline a few recent results concerning a precise but strange connection between random walks (frogs) and supersymmetric sigma models (magnets), two classes of object which on the face of things are completely unrelated. Using this connection, we can transfer problems concerning walks into problems concerning magnets, allowing us to prove that the vertex reinforced jump process is recurrent in two dimensions.

This talk is based on joint work with Roland Bauerschmidt and Tyler Helmuth (see this preprint ).

## Monday, April 16th

Sara Loo (Sydney University) — Mathematical modelling of the evolution of human behavioural strategies

Human behaviour has evolved from the behaviour of our ancestors, adapting to new and changing environments, and changes to population behaviours. Hypotheses of the origins and mechanisms of these uniquely human behaviours have been presented throughout anthropological literature. We attempt to quantify these hypotheses using techniques such as differential equations and systems analysis. One such question is that of human pair bonding. Where paternal care has been widely assumed to be the cause of pair bonding, we develop a model that favours the alternate explanation of mate guarding. We show that pairing is favoured when payoffs to mate guarding rise with the mating sex ratio. A further question considers the evolution of large-game hunting in human life history. Another counter-intuitive behaviour, investment into this costly behaviour pays-off by way of increased likelihood of paternity, despite the cost to offspring care.

## Monday, April 23rd

Ali Mohammadi (Sydney University) — On collinear triples formed by grids over finite fields

We consider the problem of finding nontrivial upper bounds on the number of collinear triples formed by cartesian products of sets contained in finite fields of prime power cardinality. Aksoy Yazici, Murphy, Rudnev and Shkredov (2016) have obtained a strong estimate in prime fields, however their method does not extend to arbitrary finite fields. We conclude the talk with some number theoretic applications of our results.

## Monday, April 30th

Jonathan Mui (Sydney University) — Symmetry in Musical Structures

It is often claimed that mathematics and music have a lot in common — for example, both disciplines value the interplay between rigour and intuition, and both have their respective ideas on what is considered “beautiful”. These are however quite philosophical comparisons, and it is perhaps less well-known how mathematical and musical structures interact in a concrete way. One such way is through symmetry. In this presentation, I give a mathematical introduction to the music of the Austrian composer Arnold Schoenberg (1874-1951), who developed a method of composing with “twelve-tone rows” (Zwölftonreihen) that incorporates permutations and symmetry as a fundamental part of the musical language, i.e. not merely as an aesthetic consideration thrown onto a pre-existing framework. I will also talk more generally about how some simple mathematical objects — mainly algebraic and geometric — can provide interesting and perhaps profound insights into musical language, in particular to the theory of harmony.

## Monday, May 7th

Steven Luu (Sydney University) — Sums for divergent series

“Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.” - N. H. Abel, 1828.

Divergent series have appeared in various branches of mathematics where they primarily appear as asymptotic series. In asymptotic theory, a function asymptotic to a divergent series can, under certain conditions, approximate the function to exponential accuracy. The infamous result, \displaystyle 1 + 2 + 3 + \cdots = -\frac{1}{12}, has been used in the context of quantum field theories, for which experimental evidence has been found to agree with the theoretical predictions up to an astonishing degree of accuracy . In this talk, we discuss how divergent series can be `summed’ by extending the notion of summation, which is consistent with the usual summation rules. Although many summation methods for divergent series exist, we will consider Borel summation and use this to reproduce the result \displaystyle 1 - 1! + 2! - 3! + 4! - 5! + \cdots = 0.59637255\ldots, originally derived by Euler (1760).

## Monday, May 14th

Sarah Romanes (Sydney University) — An Introduction to Machine Learning

This talk about Machine Learning (ML) is designed to be a fun and lighthearted talk covering the basic concepts of various aspects of ML. In this talk, we will cover the paradigms of Machine Learning (statistics vs computer science), the types of classifiers used in ML (supervised vs unsupervised, generative vs discriminative), as well as touching briefly on a number of ML algorithms used in industry and academia alike, such as logistic regression, support vector machines, and neural networks. By the end of the talk, you should be able to recognise algorithms when used in context as well as know when they should (and shouldn’t be) applied to solve problems in the real world.

## Monday, May 21st

Adrian Toshar Miranda (Macquarie University) — Internal categories and factorisation

The notion of a category is ubiquitous across mathematics. Among other things, categories model the composition of homomorphisms between sets with extra structure. It turns out that if a category E has finite limits, then we can make sense of the notion of a category inside E itself. When E is the category of sets and functions, this gives the ordinary notion of a category. In this talk we will assume enough structure on E so that it resembles the category of sets, and look at the totality of categories inside E and structure preserving maps between them, called internal functors. Of particular interest to us is the phenomenon of factorisation, where we express an arbitrary internal functor as the composite of two internal functors from particular nice classes, known as the ’final functors’ and the ’discrete fibrations’. The route we take to this factorisation was originally due to Ross Street and Dominic Verity.

## Monday, May 28th

Zeina Libnan Haidar (Sydney University) — Periodic Trajectories of Pseudo-Integrable Billiard within Circle with a Barrier

Mathematical billiard in a domain describes the motion of a particle with elastic reflections from the boundary. In this talk, we will focus on billiards within a circular domain with a barrier. Trajectories of such biulliards always have a circular caustic, however the motion is not integrable since the reflection is not defined at the end-point of the barrier. We will discuss periodic trajectories, the effect of the rotation number and the length of the barrier on them, and give a topological description of their phase space leafs.

Adarsh Kumbhari (Sydney University) — What’s the point in exercising? A cell-biology perspective

Mitochondria are specialised organelles that produce adenosine triphosphate (ATP), a molecule used by cells as an energy source. Mitochondria form dynamic networks that constantly undergo fission and fusion in response to increased ATP demand. There is a lack of high-resolution data tracking the reorganisation mitochondrial networks in a beating heart cell. We use an agent-based model to simulate a mitochondrial network reorganising in beating heart cell. Our visualisation highlights how mitochondrial networks respond to stressors such as bouts of sustained vigorous physical activity.

## Monday, June 4th

Samuel Jelbart (Sydney University) — TBA

TBA

# Seminars in 2018, Semester 2

## Monday, July 30th

Eric Hester (Sydney University) — How do you simulate a PDE? (with Dedalus)

Partial Differential Equations are used to understand myriad phenomena: the motion of birds in flight, wave generation by ships, bacterial growth and motility, global weather systems, volcanic eruptions, animal pattern formation, and even traffic jams. Despite this common mathematical framework, most codes only focus on how to solve a very restricted subset of these problems.

This talk will be about Dedalus, an open-source, high-performance spectral code for solving almost arbitrary PDEs in multiple dimensions. I’ll begin the talk with a mini demo showing how Dedalus can use Chebyshev spectral methods to simulate the KdV equation. Then I’ll go through and explain the key mathematics underlying the code, and finish with a couple fun movies of how I’ve used Dedalus in my research.

## Monday, August 6th

Dickson Annor (Sydney University) — Representation Varieties for SL(3,C): The Trefoil Knot

Following the foundational work of Thurston, Culler and Shalen, the varieties of representations and characters of 3-manifold groups into SL(2,C) have been extensively studied, as both reflect geometric and topological properties of 3-manifolds. However, much less is known about the representations of 3-manifold groups into other Lie groups, notably SL(n,C) with n >= 3.

In this talk, we will study representation varieties of the trefoil knot group into SL(3,C). Let G be the fundamental group of the complement of the trefoil knot in S^3. This has presentation G = <x,y | x^2 = y^3> = <s,t | sts = tst>. In particular, we will determine all representations up to conjugacy, using results of Fricke for SL(2,C) and Lawton for SL(3,C).

## Monday, August 13th

Haruki Osaka (Sydney University) — Introduction to Singular Learning Theory

Many statistical models used in data science have hierarchical structures and hidden variables, for example, mixture models, hidden Markov models, neural networks and so on. Although such models are widely used in practice, no sound theoretical foundation for the large sample behaviour of these models has been established. The main reasons for this difficulty is that of non-identifiability and a degenerate Fisher Information metric of these models, which are basic regularity conditions required for Fisher’s asymptotic normal theory. Such statistical models are called singular. Statistical inference for singular models are different to that of regular models. In addition, popular information criteria used in model selection such as AIC and BIC are not valid for singular models.

Singular Learning Theory is a new area of mathematics which attempts to use algebraic techniques to study singular models. Using these results, Drton and Plummer (2017) recently proposed the singular Bayesian information criterion (sBIC) that is valid for singular models. In this talk, I will give an overview of some typical problems that may occur in singular models and how singular learning theory attempts to resolve them.

## Monday, August 20th

Madeleine Cartwright (Sydney University) — Power Grids and the Second Order Kuramoto Model

Power grids transmit power using alternating current electricity and can be modelled as a network of coupled oscillators. These oscillators must be synchronised to prevent cascading failures and widespread power loss. I will talk about how an extension of the paradigmatic model for coupled oscillators, the Kuramoto model, can be applied to the power grid, and how collective coordinates can be used to further reduce this model.