# Vacation Scholarships Projects

## Vacation Scholarships Projects

## Current projects

This section showcases projects, sorted by research area, that are available for the current year. The final list of projects will be confirmed annually **on the day** of our Information Session.

## 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

### Viscoplastic biofluids: active mixing of yield-stress fluids

Active fluid flows occur in a range of systems from coral reefs and the human respiratory tract to engineered systems and micro-robots. This project will investigate the structure of flows in unbounded and confined geometries, and assess how flow fields are modified by non-Newtonian rheology (e.g., yield stress fluid). The project will draw on skills across multiple domains, including analytical modelling, dynamical systems, asymptotic analysis and numerical simulations.

**Contact:** Edward Hinton edward.hinton@unimelb.edu.au

## Discrete Mathematics

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

For more information on this research group see: Discrete Mathematics

### Lattice models of polymer systems

Long chain polymers like DNA can be modelled by walks, polygons, trees, and various other combinatorial structures embedded in lattices. This project aims to investigate new polymer models. This can be approached using exact solution techniques or computational methods like series enumeration and random sampling.

**Contact:** Nick Beaton nrbeaton@unimelb.edu.au

### Graph Colourings and Oriented Graphs

By defining graph colouring using graph homomorphism, we can build a definition for graph colouring for directed graphs that, in some sense, takes into account the direction of the arcs. In this analogue, our intuition for how graph colourings should be behave often is mistaken. Well understood results and bounds, like Brooks' Theorem or the Four-Colour Theorem no longer hold. In this project we will look at some subgraphs of complete graphs that, surprisingly, can only be coloured by assigning. every vertex its own colour. We examine what change is possible when we reverse the direction of a subset of arcs.

**Contact:** Christopher Duffy christopher.duffy@unimelb.edu.au

## Learning and Teaching Innovation

There is no current research project in this area being offered for the Vacation Scholarship Program.

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

## Mathematical Biology

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

For more information on this research group see: Mathematical Biology

### Viscoplastic biofluids: active mixing of yield-stress fluids

Active fluid flows occur in a range of systems from coral reefs and the human respiratory tract to engineered systems and micro-robots. This project will investigate the structure of flows in unbounded and confined geometries, and assess how flow fields are modified by non-Newtonian rheology (e.g., yield stress fluid). The project will draw on skills across multiple domains, including analytical modelling, dynamical systems, asymptotic analysis and numerical simulations.

**Contact:** Edward Hinton edward.hinton@unimelb.edu.au

### Applying deep learning to problems in genetic epidemiology

In phylogenetics, we use genomic data from pathogens to study infectious disease. In this project the student will investigate using neural networks to tackle computational problems in phylogenetics.

**Contact:** Alex Zarebski azarebski@unimelb.edu.au

### A portrait of intercellular communication in Waddington’s landscape

Waddington’s epigenetic landscape is an illustrative metaphor proposed by the biologist C.H. Waddington in the mid-20th century to describe cell development. The metaphor suggests that cell development is analogous to a marble rolling down a hill. As a marble will descend down a hill until eventually coming to rest in a (local) valley, so too will a cell develop along trajectories of an epigenetic landscape until it has become a fully differentiated (or ‘developed’) cell. The features (peaks and troughs) of the epigenetic landscape are determined by the gene expression of the cell. Traditional mathematical models of the Waddington landscape used a deterministic approach that can only feasibly be applied to low-dimensional gene regulatory networks that are known in advance. These models did not account for stochasticity, nor the influence of intercellular communication on gene expression — both of which are crucial for determining a cell’s future state, or fate. To address these limitations (and more!), this project will use a statistical mechanics approach to describe Waddington’s epigenetic landscape.

Contact: Michael Stumpf mstumpf@unimelb.edu.au

## Mathematical Physics

For more information on this research group see: Mathematical Physics

### Lattice models of polymer systems

Long chain polymers like DNA can be modelled by walks, polygons, trees, and various other combinatorial structures embedded in lattices. This project aims to investigate new polymer models. This can be approached using exact solution techniques or computational methods like series enumeration and random sampling.

**Contact:** Nick Beaton nrbeaton@unimelb.edu.au

### Learning to unkot

Studing a recent paper by Gukov et al (https://arxiv.org/abs/2010.16263) you will learn about the use of natural language processing AI into the study of knot theory. Through the braid word representation of knots the UNKNOT problem will be studied, i.e. determining whether or not a given knot is knotted or not. Interested students require some proficiency in coding and Mathematica.

**Contact:** Jan de Gier jdgier@unimelb.edu.au

### Projects in the area of quantum information, quantum computing and related areas

My research uses tools from quantum information theory applied to a range of different areas, including quantum computing, condensed matter physics and fundamental theory. Projects could explore the black hole information paradox, typical entanglement in many body quantum systems, the energy cost of entanglement extraction or circuit complexity in quantum fields. There is also the possibility to work with IBM Quantum, where one performs quantum operations on a quantum computer in the cloud. If this sounds interesting, please contact me attaching a brief CV and a current transcript.

**Contact:** Lucas Hackl lucas.hackl@unimelb.edu.au

### Matrix Product States and exactly solvable quantum systems

In quantum physics it is an essential problem to find the ground state of a given quantum system and to be able to analyze its properties. This is an extremely challenging problem since the underlying Hilbert space grows exponentially with system size. Matrix Product States (MPS) provide a novel tool to solve that problem for large classes of toy models.

In this project the vacation scholar will explore the mathematics of Matrix Product States as well as their physical relevance and try to construct interesting families of exactly solvable quantum systems. A strong affinity to physics will be assumed.

**Contact:** Thomas Quella Thomas.Quella@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 examples of the SSH model and 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 invariants and learn how they are connected to fundamental concepts such as the Berry phase in quantum mechanics.

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

### Assorted projects in "Conformal Field Theory, String Theory and Perturbation Theory"

Details: https://researchers.ms.unimelb.edu.au/~dridout@unimelb/vacschol.html

**Contact: **David Ridout david.ridout@unimelb.edu.au

### Inference and learning with spin glass models

Spin glasses arise in statistical physics as models of systems of interacting variables, the most famous of which is the Ising model, a classical model of ferromagnetism in which magnetic sites in a lattice are coupled via spin-spin interactions. Spin glass models have been shown to be a powerful tool for understanding complex systems, and have been well studied in the machine learning literature under the name of "energy-based models". This project will focus on the problem of inferring spin glass models of gene regulation from single-cell gene expression data. Genes can be modelled as sites in an

unstructured graph which have a spin of +1 (gene is on) or -1 (gene is off). The state of a system evolves according to a spin glass Hamiltonian. We want to solve the inverse problem of inferring the coupling matrix, which in the genetic setting contains information about which pairs of genes interact, and is unknown. There are a wealth of related problems which are of interest -- for example, one can relax the state of the system to be continuous rather than discrete, giving rise to Hopfield networks. Having a continuous model allows for more tractable inference and opens the way to investigate a wider range of loss functions; and we want to use these models to understand the gene regulatory programs underpinning cellular behaviour.

**Contact:** Michael Stumpf mstumpf@unimelb.edu.au

## Operations Research

For more information on this research group see: Operations Research

### 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

## Pure Mathematics

For more information on this research group see: Pure Mathematics

### Higher symmetries and their realisation in physics

We are used to symmetries being described by groups. However, it was recently found that groups are not sufficient to realise the full symmetry of certain physical systems. In this project the vacation scholar will study novel so-called “higher symmetries” including symmetries associated with higher-degree differential forms and non-invertible symmetries that can be described in terms of certain types of categories. Depending on the interest of the vacation scholar this project may either focus entirely on the mathematical foundations or also aim at unveiling the physical context in which these symmetries arise.

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

### Assorted projects in Lie Theory, Group Theory and Representation Theory

Details: https://researchers.ms.unimelb.edu.au/~dridout@unimelb/vacschol.html

**Contact:** David Ridout david.ridout@unimelb.edu.au

## Statistics and Data Science

For more information on this research group see: Statistics

### Using generalised additive models to model timeseries data

Statistical models are often used to characterise and predict temporal trends in quantities of interest. This includes, for example, modelling the changing patterns of human mobility and contacts during the COVID-19 pandemic to understand the risks of transmissions and the impact of interventions, such as lockdowns. Regression models with splines and smoothing terms, such as generalised additive models (GAMs), are useful for this application, because of their flexibility in capturing unpredictable temporal trends, and the ease of implementing step-change parameters. However, real-life timeseries data are almost always limited in terms of frequency of sampling, and in terms of providing unbiased and representative samples of the population of interest. It is important to thoroughly test how models behave when fitted with challenging data, so that we can use the models most appropriately.

In this project, we will fit GAMs designed for modelling timeseries data to a variety of simulated timeseries datasets of human contact patterns. We will simulate data with specific issues, such as biased and missing data, and we will explore model behaviour with these data in terms of predictive accuracy and ability to capture key characteristics of the timeseries. The learnings of this project will inform research software design with real-life applications.

**Contact: **Jennifer Flegg jennifer.flegg@unimelb.edu.au

### Applying deep learning to problems in genetic epidemiology

In phylogenetics, we use genomic data from pathogens to study infectious disease. In this project the student will investigate using neural networks to tackle computational problems in phylogenetics.

**Contact:** Alex Zarebski azarebski@unimelb.edu.au

### A portrait of intercellular communication in Waddington’s landscape

Waddington’s epigenetic landscape is an illustrative metaphor proposed by the biologist C.H. Waddington in the mid-20th century to describe cell development. The metaphor suggests that cell development is analogous to a marble rolling down a hill. As a marble will descend down a hill until eventually coming to rest in a (local) valley, so too will a cell develop along trajectories of an epigenetic landscape until it has become a fully differentiated (or ‘developed’) cell. The features (peaks and troughs) of the epigenetic landscape are determined by the gene expression of the cell. Traditional mathematical models of the Waddington landscape used a deterministic approach that can only feasibly be applied to low-dimensional gene regulatory networks that are known in advance. These models did not account for stochasticity, nor the influence of intercellular communication on gene expression — both of which are crucial for determining a cell’s future state, or fate. To address these limitations (and more!), this project will use a statistical mechanics approach to describe Waddington’s epigenetic landscape.

**Contact:** Michael Stumpf mstumpf@unimelb.edu.au

### Hypergraph Animal Decomposition of Complex Networks

Hypergraph animals are small sub-networks which capture the local neighbourhoods of vertices in complex hypergraphs.The combine aspects of classical lattice animals and network motifs. We understand their combinatorial properties and their frequency spectra in random hypergraphs, and a next step in their analysis is to study their frequency spectra in real-world hypergraphs. Determining empirical distributions of hypergraph animals (so-called hypergraph zoos) and comparing them between different types of networks/hypergraphs will allow us to distill their functional relevance. This project will compare hypergraph zoos corresponding to metabolic and biochemical reaction networks in different species, in order to explore the functional role of hypergraph animals in real systems.

**Contact:** Michael Stumpf mstumpf@unimelb.edu.au

## Stochastic Processes

For more information on this research group see: Stochastic Processes

### Strongly Correlated Percolation Models

Percolation is the study of the connectivity properties of disordered media, for instance how water `percolates' through soil. Percolation models with strong correlations can behave very different to the classical models with short-range correlations. This project will explore strongly correlated percolation models, how to efficiently simulate them, and how to estimate their critical exponents.

**Contact:** Stephen Muirhead smui@unimelb.edu.au

### Hypergraph Animal Decomposition of Complex Networks

Hypergraph animals are small sub-networks which capture the local neighbourhoods of vertices in complex hypergraphs.The combine aspects of classical lattice animals and network motifs. We understand their combinatorial properties and their frequency spectra in random hypergraphs, and a next step in their analysis is to study their frequency spectra in real-world hypergraphs. Determining empirical distributions of hypergraph animals (so-called hypergraph zoos) and comparing them between different types of networks/hypergraphs will allow us to distill their functional relevance. This project will compare hypergraph zoos corresponding to metabolic and biochemical reaction networks in different species, in order to explore the functional role of hypergraph animals in real systems.

**Contact:** Michael Stumpf mstumpf@unimelb.edu.au

### Inference and learning with spin glass models

Spin glasses arise in statistical physics as models of systems of interacting variables, the most famous of which is the Ising model, a classical model of ferromagnetism in which magnetic sites in a lattice are coupled via spin-spin interactions. Spin glass models have been shown to be a powerful tool for understanding complex systems, and have been well studied in the machine learning literature under the name of "energy-based models". This project will focus on the problem of inferring spin glass models of gene regulation from single-cell gene expression data. Genes can be modelled as sites in an

unstructured graph which have a spin of +1 (gene is on) or -1 (gene is off). The state of a system evolves according to a spin glass Hamiltonian. We want to solve the inverse problem of inferring the coupling matrix, which in the genetic setting contains information about which pairs of genes interact, and is unknown. There are a wealth of related problems which are of interest -- for example, one can relax the state of the system to be continuous rather than discrete, giving rise to Hopfield networks. Having a continuous model allows for more tractable inference and opens the way to investigate a wider range of loss functions; and we want to use these models to understand the gene regulatory programs underpinning cellular behaviour.

**Contact:** Michael Stumpf mstumpf@unimelb.edu.au