# 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: Algebra

### Lattice reduction and continued fractions

Taking the integer span of a set of vectors defines a lattice. The lattice reduction problem is to determine the vector with smallest length in the lattice. Already in the 2-dimensional case this relates to hyperbolic geometry and continued fractions. Less well studied are when the vectors have entries over the complex or quaternion number field. This project will explore aspects of these cases.

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

### 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 eg related to Groebner degenerations, toric varieties, etc.

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

### Some problems in Automorphic Forms

**Contact:** Chenyan Wu chenyan.wu@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

### Topics in wave equations

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

### Spatially homogeneous models in cosmology

This project explores the predictions of general relativity for the geometry of a spatially homogeneous universe. We learn about the simplest models in cosmology introduced by Friedmann, Lemaitre, Robertson & Walker. The dynamics are governed by ordinary differential equations, and encompass scenarios of collapse, and accelerated expansion. This is classical topic in mathematical general relativity, with many links to current research.

**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

### Lava flow through a forest

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. From a linear algebra perspective this amounts to finding the vector that minimizes the eigenvalue of a specific linear map called the Hamiltonian. This problem sounds simple but is extremely challenging since in practical applications one is interested in vector spaces of dimension 2^N where N is typically of the order of 100 or larger. In recent years neural networks have been proposed as an efficient way to approximate the ground state. The corresponding variational ansatz is known as “Neural Network Quantum States” (NQS).

In this project the vacation scholar will explore neural network quantum states and relations to important concept from quantum theory such as entanglement. 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:** Edward Hinton ehinton@unimelb.edu.au

### Spin glasses and random matrices

Spin glass theory is a very active area of research in contemporary mathematical physics and applied mathematics with a wide range of interdisciplinary applications, e.g. in machine learning, algorithm optimisation, and neuroscience. Modern mathematical developments have replaced heuristic methods introduced by physicists in the 1970s and 1980s with a rigorous mathematical framework. The aim of this project is to use techniques from random matrix theory to analyse complex energy landscapes arising from "simple" mean field spin glass models.

**Contact: **Jesper Ipsen jesper.ipsen@unimelb.edu.au

### Neural Network Quantum States

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. From a linear algebra perspective this amounts to finding the vector that minimizes the eigenvalue of a specific linear map called the Hamiltonian. This problem sounds simple but is extremely challenging since in practical applications one is interested in vector spaces of dimension 2^N where N is typically of the order of 100 or larger. In recent years neural networks have been proposed as an efficient way to approximate the ground state. The corresponding variational ansatz is known as “Neural Network Quantum States” (NQS).

In this project the vacation scholar will explore neural network quantum states and relations to important concept from quantum theory such as entanglement. 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

## Discrete Mathematics

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

## Geometry and Topology

For more information on this research group see: Geometry and Topology

### Lattice reduction and continued fractions

Taking the integer span of a set of vectors defines a lattice. The lattice reduction problem is to determine the vector with smallest length in the lattice. Already in the 2-dimensional case this relates to hyperbolic geometry and continued fractions. Less well studied are when the vectors have entries over the complex or quaternion number field. This project will explore aspects of these cases.

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

### Spatially homogeneous models in cosmology

Description: This project explores the predictions of general relativity for the geometry of a spatially homogeneous universe. We learn about the simplest models in cosmology introduced by Friedmann, Lemaitre, Robertson & Walker. The dynamics are governed by ordinary differential equations, and encompass scenarios of collapse, and accelerated expansion. This is classical topic in mathematical general relativity, with many links to current research.

**Contact: **Volker Schlue volker.schlue@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

### Topics in wave equations

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

## 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

### 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

### 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

### 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

### 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

### Topics in wave equations

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

### Spatially homogeneous models in cosmology

This project explores the predictions of general relativity for the geometry of a spatially homogeneous universe. We learn about the simplest models in cosmology introduced by Friedmann, Lemaitre, Robertson & Walker. The dynamics are governed by ordinary differential equations, and encompass scenarios of collapse, and accelerated expansion. This is classical topic in mathematical general relativity, with many links to current research.

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

## Operations Research

For more information on this research group see: Operations Research

### Self-organised Criticality in Real World Systems

The phenomenon of self-organised criticality (SOC) in real world systems is where interacting components combine to cause the system to reach a “critical” state, where any small perturbation can cause a major disturbance. For example, tectonic plates in the earth’s crust may move slowly, but when pressure builds up, an earthquake occurs. The Bak-Tang-Wiesenfeld (BTW) model (or "sand-pile model") has been used to model SOC. In the two-dimensional BTW model, grains of sand are added randomly, one at a time, to cells in a rectangular grid, until any “sand-piles” that are too high topple over, potentially causing a significant cascading toppling effect. The BTW model is not only rich in mathematical properties, but can be applied to many real world phenomena, such as earthquakes, forest fires, and epidemics. In this project we apply the BTW model to describe SOC in real world systems.

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

### Continuous dynamical systems associated with iterative algorithms in optimisation

The study of continuous time dynamical systems associated with iterative algorithms for solving optimization problems has a long history which can be traced back at least to 1950s. The relationship between the continuous and discrete versions of an algorithm provides a unifying perspective which gives insights into their behavior and properties, as well as providing a tool to derive and analyse new methods. The aim of this project is to investigate continuous version of recently discovered gradient-based methods with adaptive stepsize rules.

**Contact:** Matthew Tam matthew.tam@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

## Statistics and Data Science

For more information on this research group see: Statistics

### Stochastic models for populations with competition for resources

Many biological populations experience logistic growth: the population per capita growth rate decreases as the population size approaches a maximum imposed by limited resources in the environment, known as the carrying capacity.

The main objective is to study different stochastic models of population-size dependent branching processes, and develop parameter estimation methods to fit these models to real data. We are mainly working with data on bird populations living on small islands.

This research area can be decomposed into several sub-projects which involve (among others):

- Studying different model outputs such as the distribution of the time until extinction and the total progeny size
- Investigating the sensitivity of the model outputs with respect to the choice of the offspring distribution
- Designing optimal strategies for the reintroduction of species in a new area (this would require some Operations Research tools)
- Developing multi-type models with a different carrying capacity for each type, and estimate the model parameters based on real data (this would require some Statistics tools)
- Comparing different parameter estimation methods for population-size-dependent branching processes (this would require some Statistics tools).

These questions will be tackled using a combination of simulation studies and theoretical developments.

**Contact:** Sophie Hautphenne sophiemh@unimelb.edu.au

## Stochastic Processes

For more information on this research group see: Stochastic Processes

### Stochastic models for populations with competition for resources

Many biological populations experience logistic growth: the population per capita growth rate decreases as the population size approaches a maximum imposed by limited resources in the environment, known as the carrying capacity.

The main objective is to study different stochastic models of population-size dependent branching processes, and develop parameter estimation methods to fit these models to real data. We are mainly working with data on bird populations living on small islands.

This research area can be decomposed into several sub-projects which involve (among others):

- Studying different model outputs such as the distribution of the time until extinction and the total progeny size
- Investigating the sensitivity of the model outputs with respect to the choice of the offspring distribution
- Designing optimal strategies for the reintroduction of species in a new area (this would require some Operations Research tools)
- Developing multi-type models with a different carrying capacity for each type, and estimate the model parameters based on real data (this would require some Statistics tools)
- Comparing different parameter estimation methods for population-size-dependent branching processes (this would require some Statistics tools).

These questions will be tackled using a combination of simulation studies and theoretical developments.

**Contact:** Sophie Hautphenne sophiemh@unimelb.edu.au

### Self-organised Criticality in Real World Systems

The phenomenon of self-organised criticality (SOC) in real world systems is where interacting components combine to cause the system to reach a “critical” state, where any small perturbation can cause a major disturbance. For example, tectonic plates in the earth’s crust may move slowly, but when pressure builds up, an earthquake occurs. The Bak-Tang-Wiesenfeld (BTW) model (or "sand-pile model") has been used to model SOC. In the two-dimensional BTW model, grains of sand are added randomly, one at a time, to cells in a rectangular grid, until any “sand-piles” that are too high topple over, potentially causing a significant cascading toppling effect. The BTW model is not only rich in mathematical properties, but can be applied to many real world phenomena, such as earthquakes, forest fires, and epidemics. In this project we apply the BTW model to describe SOC in real world systems.

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