# Vacation Scholarships Projects

## Algebra

Listed on this page are current research projects being offered for the Vacation Scholarship Program.

For more information on this research group see: Pure Mathematics

### Working with Macaulay2

The goal of this project is, as a first step, to learn how to use Macaulay2, a software for computations in algebraic geometry and commutative algebra.

The project can then be extended in two directions:

- go on to more advanced programming with Macaulay2 and learn how to contribute to an open source project.
- explore some simple computational algebraic geometry problems with Macaulay2, e.g., related to Groebner degenerations, toric varieties, etc.

**Contact:** Paul Zinn-Justin pzinn@unimelb.edu.au

## Analysis

Listed on this page are current research projects being offered for the Vacation Scholarship Program.

For more information on this research group see: Pure Mathematics

### Wave propagation in flat space

In this project we learn about the classical wave equation, the representation formula by spherical means, and its interpretation in different dimensions. The student learns the basics about the formulation of the initial value problem, and derives the asymptotics of general solutions. There are various applications of this material in physics, and also non-linear wave equations can be explored.

**Contact:** Volker Schlue volker.schlue@unimelb.edu.au

### Topics in general relativity

**Contact:** Volker Schlue volker.schlue@unimelb.edu.au

## Applied Mathematics

Listed on this page are current research projects being offered for the Vacation Scholarship Program.

For more information on this research group see: Applied Mathematics

### Fluid mechanics of active granular systems

Dense arrays of microorganisms exhibit striking collective motions, jamming and turbulent-like properties, and their behaviour underpins a range of processes in biology. However, the physical principles giving rise to their collective properties are not well understood.

Co-supervised by Dr. Douglas Brumley and Professor Antoinette Tordesillas, this project will investigate the bulk properties of active suspensions, and develop simplified models to predict the flow of nearly-jammed arrays of microorganisms. Some background in fluid dynamics and computational work would be beneficial. No biological background necessary.

**Contact:** Douglas Brumley d.brumley@unimelb.edu.au, Antoinette Tordesillas atordesi@unimelb.edu.au

### What kind of random walk are these biological cells doing?

Biological cell motility is a key aspect of embryonic growth, homeostasis and disease processes. It is possible to observe isolated cells migrating, apparently in some random way, on planar substrates. How well can simple models of this random motion describe what is seen? If the model is “correct”, how accurately can the underlying parameters be deduced from limited data? Can limited data distinguish clearly between several competing plausible models?

This is an opportunity for a student with appropriate skills with Matlab or or other computational tools and an interest in probability to learn about various random walk models and run computer experiments to investigate these interesting questions.

**Contact:** Barry Hughes barrydh@unimelb.edu.au

## Discrete Mathematics

For more information on this research group see: Discrete Mathematics

### Hamiltonicity of vertex-transitive graphs

A question of Lovasz in 1969 asked whether every connected Cayley graph with more than two vertices have a Hamilton cycle. This question is still wide open, even for Cayley graphs on some fairly simple families of groups. More surprisingly, there are only a few examples of vertex-transitive graphs known to have no Hamilton cycle, which leads to the conjecture that there are only finitely many vertex-transitive graphs with no Hamilton cycles. The aim of this project is to investigate the Hamiltonicity for some special families of vertex-transitive graphs, especially Cayley graphs.

**Contact:** Binzhou Xia binzhoux@unimelb.edu.au

## Geometry and Topology

For more information on this research group see: Pure Mathematics

### Flatification for monoid schemes

A monoid scheme is a geometric object whose functions can only be multiplied, not added - for example, the monoid affine space of dimension d admits only the monomials in d variables as functions. Because these objects are much more rigid than the usual objects studied in algebraic geometry, they can be described combinatorially, and monoid analogues of theorems in algebraic geometry are typically easier to prove. Flatification - usually referred to by its French name "platification par éclatement" - is a crucial theorem in algebraic geometry that should admit a good monoid analogue. In this project, you will learn the language of monoid schemes and attempt to formulate and prove an appropriate flatification result.

**Contact:** Christian Haesemeyer christian.haesemeyer@unimelb.edu.au

### Configuration spaces and shuffle algebras

This project involves studying the topology of configuration spaces and related objects. Concretely, we want to write down complexes that compute their homology. The configuration space of a manifold is the space of different arrangements of points in a manifold, where no two points can overlap. We can generalise these spaces by adding algebraic labels to the points and allowing points to collide in different ways. These spaces appear geometry, algebra, physics - anytime you want to study points in a system really! - but our particular interest is motivated by counting problems from arithmetic geometry.

**Contact:** TriThang Tran trtran@unimelb.edu.au

### Enumerative geometry and physics

Mirror symmetry is one of the most important and influential problems in mathematics and mathematical physics. At the simplest level mirror symmetry realises solutions of enumerative problems from mathematical physics in two quite different ways. This project involves concrete calculations related to geometry that give an accessible approach to mirror symmetry for students. It involves techniques from geometry, complex analysis, combinatorics and simple programming.

**Contact:** Paul Norbury norbury@unimelb.edu.au

### Super hyperbolic surfaces

An ideal triangulation of a cusped hyperbolic surface is a triangulation with no vertices, and all faces triangles. The number of faces and edges is 4g-4+2n, respectively 6g-6 +3n for a surface of genus g with n cusps. Coordinates of the space of hyperbolic structures are given by positive numbers assigned to edges of the triangulation. These generalise to super coordinates assigned to edges and faces. This project will aim to generalise classical results to analogous results with super coordinates

**Contact:** Paul Norbury norbury@unimelb.edu.au

### Super Hurwitz numbers

Hurwitz numbers count the number of genus g surfaces that cover the two-sphere with prescribed branching. This project will look at a generalisation to super genus g surfaces.

**Contact:** Paul Norbury norbury@unimelb.edu.au

### Geodesics on cones

Geodesics are curves between two points on a manifold that extremize the length functional. In this project the student will learn about the basics of variational calculus which is concerned with the extremization of functions of functions (such as curves) and apply the techniques to the surprisingly rich question of how to find and describe geodesics on cones. If time permits the project will also involve having a closer look at the properties of the singularity at the tip of the cone.

**Contact:** Thomas Quella thomas.quella@unimelb.edu.au

### Wave propagation in flat space

In this project we learn about the classical wave equation, the representation formula by spherical means, and its interpretation in different dimensions. The student learns the basics about the formulation of the initial value problem, and derives the asymptotics of general solutions. There are various applications of this material in physics, and also non-linear wave equations can be explored.

**Contact:** Volker Schlue volker.schlue@unimelb.edu.au

### Topics in general relativity

**Contact:** Volker Schlue volker.schlue@unimelb.edu.au

## Learning and Teaching Innovation

For more information on this research group see: Learning and Teaching Innovation

### Visual blocks to help students learn syntax

Learning mathematical syntax is essential for understanding and communicating mathematics, but it is something students often find hard to master. Maths Blocks is a system of visual blocks intended to help students with mathematical syntax. This project will investigate ways to extend Maths Blocks to support additional areas of mathematics, such as functions and their inverses, sets, or other topics. It will involve some mathematics (such as formal language theory and type theory), some learning theory, some software design, and (potentially) some programming.

**Contact:** Anthony Morphett a.morphett@unimelb.edu.au

### Mathematical modelling of a classroom disease epidemic simulation

The 'Handshake game' is a classroom simulation of an infectious disease outbreak, which can be used in teaching infectious disease modelling. In this project, we will model the progression of the handshake game using ODE (ordinary differential equation) models. In particular, we will investigate how well the classic SIR model describes the progression of the handshake game, and explore variants of the SIR model which may give a better fit. This will involve some mathematical modelling, numerical solving of ODEs, and model fitting using MATLAB, Python or similar software.

**Contact:** Anthony Morphett a.morphett@unimelb.edu.au

### Analysis of student sampling strategies in an activity with chocolate

Chocs and Blocks is a statistical sampling activity where students try to choose representative samples from a population of chocolate pieces to estimate a population mean. There are several popular strategies for selecting a sample, such as selecting a mix of small and large pieces. This project will investigate the sampling distributions of several common strategies, and attempt to model data sets of samples from large lecture classes.

**Contact:** Anthony Morphett a.morphett@unimelb.edu.au

## Mathematical Biology

For more information on this research group see: Mathematical Biology

### Fluid mechanics of active granular systems

Dense arrays of microorganisms exhibit striking collective motions, jamming and turbulent-like properties, and their behaviour underpins a range of processes in biology. However, the physical principles giving rise to their collective properties are not well understood.

Co-supervised by Dr Douglas Brumley and Professor Antoinette Tordesillas, this project will investigate the bulk properties of active suspensions, and develop simplified models to predict the flow of nearly-jammed arrays of microorganisms. Some background in fluid dynamics and computational work would be beneficial. No biological background necessary.

**Contact:** Douglas Brumley d.brumley@unimelb.edu.au, Antoinette Tordesillas atordesi@unimelb.edu.au

### Statistical analysis of emerging biological data

Technological improvements have allowed for the collection of data from different types of molecules (e.g. genes, proteins, metabolites, microorganisms) resulting in multiple ‘omics data (e.g. transcriptomics, proteomics, metabolomics, microbiome) measured from the same set N of biospecimens, individuals, or cells. Our group is interested in developing computational and statistical methods for the analysis of such data, and tackle problems such as data integration, feature selection, and mathematical modelling. Integrating data include numerous challenges – data are complex and large, each with few samples (N < 50) and many molecules (P > 10,000), and generated using different technologies.

More details on our research can be found on our webiste and examples of methods we have developed for data mining and integration. Projects range from methods development, computational implementation in R and applications to case studies we have available through our collaborators.

**Contact:** Kim-Anh le Cao kimanh.lecao@unimelb.edu.au

### Mathematical models of nanoparticle-cell interactions

Nanoparticles are a promising tool for the targeted delivery of medicine. However, the complex biological and physical processes that influence nanoparticle-cell interactions are not well understood. This project will develop mathematical models of nanoparticle transport (differential equations) and cell behaviour (differential equations or agent-based models). These models will help us understand which biological and physical processes dictate whether the targeted delivery of medicine via nanoparticles will be successful.

**Contact:** Stuart Johnston stuart.johnston@unimelb.edu.au

## Mathematical Physics

For more information on this research group see: Mathematical Physics

### Algebra and calculus of random matrix integrations

The study of topics of contemporary interest in mathematical physics, such as the scrambling of information in black holes, and quantum many body thermalisation requires methods for integration over combinations of elements of random matrices. The aim of this project is investigate mathematical methods for this purpose.

**Contact:** Peter Forrester pjforr@unimelb.edu.au

### Density matrices, quantum entanglement and random matrices

Quantum entanglement is an essential resource in the design of quantum computers. Its mathematical description relies on a density matrix formulation of quantum mechanics, which for finite dimensional systems relates to positive definite matrices. This project aims to introduce these concepts, and show the applicability of random matrix theory when dealing with a random quantum state.

**Contact:** Peter Forrester pjforr@unimelb.edu.au

### Enumerative geometry and physics

Mirror symmetry is one of the most important and influential problems in mathematics and mathematical physics. At the simplest level mirror symmetry realises solutions of enumerative problems from mathematical physics in two quite different ways. This project involves concrete calculations related to geometry that give an accessible approach to mirror symmetry for students. It involves techniques from geometry, complex analysis, combinatorics and simple programming.

**Contact:** Paul Norbury norbury@unimelb.edu.au

### Topological invariants in quantum systems

The physical properties of a quantum system generally depend on parameters which determine the strength of various interactions, e.g. the coupling to a magnetic field. Upon variation of these parameters the system exhibits different physical phases with qualitatively different features. Some of these phases can be distinguished by a discrete invariant that takes one value in one phase and another one in a second. This observation provides a link to the mathematical field of topology which studies the properties of geometric objects, such as knots, up to continuous deformations. In view of this connection, one frequently speaks about topological phases of matter. There are various prominent examples which have only been discovered in the last couple of years - first theoretically, then also experimentally.

Building on the example of Kitaev's so-called Majorana chain, a simple free fermion model of a 1D superconductor, the Vacation Scholar will develop some intuition about the associated topological invariant which, essentially, counts the number of Majorana edge modes. She or he will then apply these insights to a closely related system of so-called parafermions and try to derive a topological invariant for these. While the project has a strong analytical/mathematical component, there will also be the possibility to analyse different parafermion systems using computer algebra in case of interest.

**Contact:** Thomas Quella Thomas.Quella@unimelb.edu.au

### Classifying phases of many-body systems using machine learning

Matter can exist in various different phases. Water for instance can exist in a frozen, a liquid or a gaseous state depending on external parameters such as temperature and pressure. Other materials may exhibit a very complicated phase diagram involving lots of parameters and many distinct phases, potentially even phases of topological origin. When looking at a specific Hamiltonian describing the dynamics of a classical or quantum system with a large number of particles it is usually highly non-trivial to determine the phase the system resides in for a given set of parameters.

In this project the vacation scholar will explore how to describe phases of matter mathematically and use machine learning techniques to map out the phase diagrams of some model systems. Affinity to physics and basic programming experience will be assumed but besides numerical work (with Python) there will also be ample opportunity to gain new analytical insights.

**Contact:** Thomas Quella thomas.quella@unimelb.edu.au

### Wave propagation in flat space

In this project we learn about the classical wave equation, the representation formula by spherical means, and its interpretation in different dimensions. The student learns the basics about the formulation of the initial value problem, and derives the asymptotics of general solutions. There are various applications of this material in physics, and also non-linear wave equations can be explored.

**Contact:** Volker Schlue volker.schlue@unimelb.edu.au

### Topics in general relativity

**Contact:** Volker Schlue volker.schlue@unimelb.edu.au

## Operations Research

For more information on this research group see: Operations Research

### The price of anarchy, the price of stability, and the price of communication in interacting intensive care units

The price of anarchy (PoA), the price of stability (PoS), and the price of communication (PoC) are measures of how inefficient the system is if interacting intensive care units (ICUs) do not communicate, communicate but do not cooperate, and communicate and cooperate, respectively. For this project we model the interaction between two ICUs as a continuous-time Markov chain and calculate the PoA, PoS, and PoC, and interpret the results. The model can be extended to incorporate more ICUs, or units and wards in the same hospital.

**Contact:** Mark Fackrell fackrell@unimelb.edu.au

### Floods, fires and explosions: how to design survivable networks in the modern age

Much of society’s critical infrastructure takes the form of large-scale networks. Think of examples such as the power grid, the NBN, gas and water pipelines, and transportation networks. All such networks are potentially vulnerable to natural disasters, or even terrorist attacks. Significant interruption to these networks can wreak havoc. So the question is: how do we design these networks to be robust against local, regional destruction, without blowing the national budget?

In this project we will use planar geometric graph models for this problem and analyse survivability when the destruction region is modelled as a circular disk. In particular, we would like to find algorithms for optimally designing networks that are survivable against failures of a given maximum radius. The project will use mathematical tools from graph theory, optimisation, computer science and just a little bit of Euclidean geometry.

**Contact:** Charl Ras cjras@unimelb.edu.au

### Charging coordination for plug-in electric vehicle fleets

With an increasing uptake of Plug-in Electric Vehicles (PEVs), it is becoming increasingly important for aggregate charging behaviour to be coordinated in way that does not place undue stress on electricity distribution infrastructure. On the other hand, PEV owners typically make charging decisions based on individual factors (such as minimising electricity costs) rather than factors affecting electricity distribution infrastructure as a whole. To reconcile these competing interests, this project will examine decentralised algorithms based on game theory for making coordinated optimal charging decisions in fleets of non-cooperation PEVs.

**Contact:** Matthew Tam matthew.tam@unimelb.edu.au

### Study of the relation between built environment and traffic congestion

This project proposes to identify related factors in the Built Environment as independent variables through an integrative model, and to investigate their effects on the dependent variables reflecting traffic capacity and congestion level using an inferential statistical model. The established relationship between the variables will be utilised to support the development of congestion mitigation and optimal BE design strategies. Using spatial-temporal statistical analysis, we aim to investigate how congestion of one area can spatially interact with other area, and how dynamics of congestion temporally evolve.

**Contact:** Joyce Zhang lele.zhang@unimelb.edu.au, Tingjin Chu

### Design of a loading zone reservation system with travel and service times uncertainty

Couriers undertaking deliveries in Central Business Districts (CBDs) often have difficulty conducting efficient routes due to the uncertainty associated with the availability of loading zones. A booking system can better utilise the limited resource and improve the efficiency of the delivery system. This project will develop a reservation system for loading zones scattered in CBDs and takes into account the uncertainty in vehicle travel times and courier stay durations. The objective is to maximise the delivery efficiency and the service reliability.

**Contact:** Joyce Zhang lele.zhang@unimelb.edu.au

## Statistics

For more information on this research group see: Statistics

**Visualisation and modelling pedestrian flows in Melbourne**

There are around 40 sensors installed in Melbourne’s Central Business District (CBD), and the pedestrian number is closely monitored and recorded hourly. This dataset is available to the public.

This project will involve visualising and analysing this data: (1) Visualising the typical daily and weekly pattern of the data through interactive tools (e.g., RShiny, Plotly/Dash); (2) Modelling the datasets through time series methods (e.g., ARIMA and VAR).

**Contact:** Tingjin Chu tingjin.chu@unimelb.edu.au, Joyce Zhang lele.zhang@unimelb.edu.au

### Classifying phases of many-body systems using machine learning

Matter can exist in various different phases. Water for instance can exist in a frozen, a liquid or a gaseous state depending on external parameters such as temperature and pressure. Other materials may exhibit a very complicated phase diagram involving lots of parameters and many distinct phases, potentially even phases of topological origin. When looking at a specific Hamiltonian describing the dynamics of a classical or quantum system with a large number of particles it is usually highly non-trivial to determine the phase the system resides in for a given set of parameters.

In this project the vacation scholar will explore how to describe phases of matter mathematically and use machine learning techniques to map out the phase diagrams of some model systems. Affinity to physics and basic programming experience will be assumed but besides numerical work (with Python) there will also be ample opportunity to gain new analytical insights.

**Contact:** Thomas Quella thomas.quella@unimelb.edu.au

## Stochastic Processes

For more information on this research group see: Stochastic Processes

### The price of anarchy, the price of stability, and the price of communication in interacting intensive care units

The price of anarchy (PoA), the price of stability (PoS), and the price of communication (PoC) are measures of how inefficient the system is if interacting intensive care units (ICUs) do not communicate, communicate but do not cooperate, and communicate and cooperate, respectively. For this project we model the interaction between two ICUs as a continuous-time Markov chain and calculate the PoA, PoS, and PoC, and interpret the results. The model can be extended to incorporate more ICUs, or units and wards in the same hospital.

**Contact:** Mark Fackrell fackrell@unimelb.edu.au