Yueyi Sun is currently a third year Science student in the Mathematics and Statistics, at The University Of Sydney. She will graduate in Dec. 2017 with a Bachelor of Science (advanced mathematics) and minor in IT. She is interested in financial mathematics and risk management. More specifically, her work examines games theory.

The rst part of this project will be devoted to learning basic concepts of game theory. This will require reading selected sections from [1]. An emphasis will be put on Nash equilibria and their properties. Nash equilibrium is a very nice mathematical idea but it leads to many mathematical diculties and it is far from obvious that it describes behaviour of real economic agents [4]. It is usually not unique and very unstable and these properties undermine its applicability. On the other hand it is very conservative towards risk taking while real agents are typically much more optimistic. We will learn about some modications of the concept of Nash equilibrium, such as maxmin strategies [5] or evolutionary games, [4]. An associated concept of (bounded) rationality will also be studied. We will follow here the classical works by Aumann, Simon and Myerson.

In the second part of the project we will study some modern extensions of the game theory developed for situations when the number of players (economic agents) is very large. The starting point for this new development is the assumption that the economic agents are “exchangeable”, that is, the joint probability distribution of the positions of all players does not change after arbitrary permutation of the players. Then an intractable (high-dimensional) problem of nding the Nash equilibrium can be reduced to a game against a single opponent representing the mean behaviour of the population of players. This approach has been borrowed from the mean eld theory in statistical physics but the optimisation problem brings new mathematical questions and a wide range applications in economics, mathematical nance, biology and social sciences for example in modelling crowd behaviour). It is known as the mean eld games (MFG) approach (or theory).

In this project we will focus on some recent attempts [2, 3] to use the MFG theory in order to describe how the market equilibrium arises from actions of a large (innite) number of economic agents. It turns out that this approach provides important insights into the high frequency trading and predatory economic behaviour. We will concentrate on the single-queue model for the price formation in high frequency trading, as presented in Section 3 of [3]. The main simplifying assumption made in this section is that the process of arrivals of traders is described by an exogenous Poisson process with a xed intensity . We will investigate a situation, when the intensity (x) depends on the size x of the queue. To this end, we will try to modify arguments presented in [3]. Since in [3] the innitesimal arguments at equilibrium are used to derive the basic mean eld game equation, we expect that such an extension is achievable. In the project we still assume that the intensity function x ! (x) is exogenously given but our assumption that depends on x is a modest attempt to take into account the fact that behaviour of the traders depends on the state of the market. An interesting question how this function arises as part of the equilibrium is beyond this project.

References

[1] Aliprantis, Charalambos D.; Chakrabarti, Subir K. Games and decision making. OUP, 2011

[2] Carmona R.; Webster K. High Frequency Market Making Making,

https://arxiv.org/abs/1210.5781

[3] Lachapelle, Aime; Lasry, Jean-Michel; Lehalle, Charles-Albert; Lions, Pierre-Louis. Eciency of the price formation process in presence of high frequency participants: a mean eld game analysis. Math. Financ. Econ. 10 (2016), no. 3, 223-262

[4] Mailath, George J. Do People Play Nash Equilibrium? Lessons from Evolutionary Game Theory. Journal of Economic Literature 36, No. 3 (Sep., 1998), pp. 1347-1374

[5] Pruzhansky, Vitaly. Some interesting properties of maximin strategies. Internat. J. Game Theory 40 (2011), no. 2, 351-365

Quinn has just completed his third undergraduate year in a bachelor of advanced mathematics at University of Wollongong. Quinn will be completing a project on differential operators on manifolds and positive scalar curvature alongside fellow student Angus Alexander under supervisors Adam Rennie and Alan Carey.

The project will begin by developing the theory of differential operators on manifolds.

The research component will be applying the theory to the effect of curvature on the index of the Hodge De Rham operator, namely the Euler characteristic.

Lachlann O’Donnell is a student at the University of Wollongong undertaking a Bachelor of Mathematics. He graduated in 2017 and is doing an honours project in geonetric analysis in 2018. His interests lie in the areas of Differential Geometry, Topology and Analysis with particular emphasis on Differential Geometry.

The area of curvature flow was developed to deal with problems that arise in geometry, for example the Poincare conjecture was resolved through the use of Ricci flow. The project aims at discussing specific types of curvature flows i.e. contraction and expansion flows, culminating in considering the cases of fully nonlinear curvature flows.

Angus is currently in his third year of a double degree studying mathematics and physics. He intends to pursue honours in maths in 2019. His main interests are subjects related to mathematical physics, in particular differential geometry and operator theory.

The project will begin by developing the theory of differentiable operators on manifolds. It will then apply the theory to problems in relativity, especially the index of the relativistic Dirac operators on cylinders.

Vishnu Mangalath is a Bachelor of Philosophy student from the University of Western Australia majoring in Mathematics and Physics. He has a wide range of interests in pure mathematics and physics, particularly, algebraic topology and geometry, statistical mechanics and quantum mechanics. He will be completing his honours in mathematics in 2018 in pure mathematics.

This project will start by investigating the basics of simplicial homology theory, a well-established feild of algebraic topology, up to and including the Mayer-Vietoris exact sequence. During this investigation, the student will gain the required back- ground for further investigation, such as constructing simplicial complexes from the nerve of an open cover of a topological space.

The research component will predominantly involve calculating the homology groups of the nerve of an open cover found from experimental data. For example, this data may be points from the state space of some unknown dynamical system. From this we can take an open cover of this data and therefore construct a simplicial complex via the nerve. Calculating the homology groups of these simplicial complexes will give insight into the shape of the underlying state space.

Yilun He is a BSc (Adv. math) student in University of Sydney. He is studying statistics and computer science. He specializes in the study of hypothesis testing and computer algorithms.

At the moment Yilun receives a scholarship from Victor Chang Cardiac Research Institute and is working on a individual project.

He also received a summer scholarship from Data61 and commenced research on interactive theorem prover with functional programming implementation. In addition, he received a summer research scholarship from department of mathematics and statistics of University of Sydney.

He is interested in bioinformatics, financial mathematics and statistics.

The aim of this project is to explore the property of false discovery rate (FDR): the expected percentage of true null hypotheses among all the rejected hypotheses. FDR is widely used in modern large hypothesis testing. A deeper understanding in the concept is very important to the correctness of these inference.

Syamand Hasam is a 2nd-3rd year undergraduate student at the University of Sydney studying a BSc (Advanced) double majoring in Mathematics and Statistics. He also holds a completed Bachelor of Computer Science and Technology from the University of Sydney. Academic interests are still wide ranging across pure and applied mathematics, theoretical computer science, statistics and the application of such in the fields of medicine and biology.

In a problem to do with mass spectrometry analysis, we wish to investigate whether recent methods in controlling the FDR (False Discovery Rate) for predicting the best-scoring-peptide match, are justified in their assumptions by looking at these methods in relation to real data, and questioning the assumptions used.

Ruebena Dawes is from Sydney, Australia, and is due to complete a Bachelor of Science (Advanced Mathematics) at The University of Sydney in 2017, with majors in applied mathematics and biochemistry. She began her degree with a broad interest in mathematics, and although she retains a deep appreciation and wonder for pure mathematics, she has decided her passion is in applying mathematics and mathematical principles to issues in the medical sciences. In addition, in the past year she has been learning a lot about computer science and big data and is excited to begin a career at the intersection of these three disciplines.

Effective radiotherapy is dependent on being able to (i) visualise the tumour clearly, and (ii) deliver the correct dose to the cancerous tissue, whilst sparing the healthy tissue as much as possible. In the presence of motion, both of these tasks become increasingly difficult to perform accurately – increasing the likelihood of incorrect dose delivered to cancerous tissue and exposure of healthy tissue to unnecessary radiation, causing adverse effects. This project will develop mathematical optimisation tools to improve the quality of diagnostic images and treatment accuracy, and requires some programming experience.

Leo is currently an undergraduate student at the University of Sydney with majors in mathematics and chemistry. His mathematical interests include category theory, knot theory, and ergodic theory, and he would like to know more about representation theory. In 2018 he is planning to pursue Honours in pure mathematics.

The goals of this project are:

– to define and classify Niemeier lattices as they arise from gluing root lattices

– to categorify root lattices, that is, present them as Groethendieck groups of module categories

– to understand the obstacles in lifting gluing constructions to the categorical level (original research)

Riley Cooper is an undergraduate student in the School of Mathematical and Physical Sciences at the University of Newcastle, studying a Bachelor of Mathematics degree with specialisation in Applied Mathematics and Statistics. He is interested in Applied Mathematics with focuses on Optimisation, Programming and Computation and Statistics with focuses on Bayesian Analysis, Markov Chains and Data Science. Flexibility of the Bachelor of Mathematics degree has allowed Riley to study a range of topics in Pure Mathematics on top of the Applied Mathematics and Statistics Majors. Riley has also established programming skills in different languages in relation to his university study. By being placed on the Faculty of Science and Information Technology Commendation List for every year of tertiary study to date, Riley has proven strong academic results. Riley was granted the Faculty of Science and Information Technology Summer Vacation Scholarship in 2016 and was able to pursue his interest in Mathematical Research. Under the supervision of Dr Thomas Kalinowski, Riley completed a research project titled “Allocation of Indivisible Goods”. Studying abroad at the University of Leeds in 2017 gave Riley the opportunity to experience university life in a different country, travel, and build many personal skills. Riley intends to complete an Honours degree in Applied Mathematics or Statistics and to then to pursue work and/or graduate study.

Production planning and scheduling problems are a key component in many supply chains. A recurring model in these problems is the multi-item lot sizing model in which a schedule of production is determined for each item being produced so that the demand for each item is satisfied subject to constraints on production, inventory and the operation of the machinery. In this project we will investigate a variant of the classical deterministic single-item lot sizing problem in which the underlying network is a cycle. Such a variant is motivated by strategic production planning and scheduling problems in which it is often convenient to assume that the planning horizon wraps around on itself, thereby eliminating the need to specify boundary conditions and avoiding possible end effects. Such an approach can be viewed as a form of steady state model.

In this project we will introduce the single-item lot sizing problem on a cycle. We will propose a mixed-integer programming formulation for the problem, explore the structural properties of the optimal solutions, and use them to establish the computational complexity of the problem and develop efficient algorithms for its solution. Finally we will investigate the polyhedral structure of the convex hull of the set of feasible solutions to the problem and propose extended formulations and strong reformulations for the problem.