This section records projects from past years.
Algebra and Number Theory
Understanding the Riemann zeta function
Can we make sense of the divergent series 1^s+2^s+3^s+… when s>=1? This project studies the Riemann zeta function using modular forms.
Contact: Chenyan Wu chenyan.wu@unimelb.edu.au
Structure in space: aperiodic tiling
We can tile a plane using squares for example. The pattern repeats itself. This project considers ways of tiling the plane without repetition using number theory and representation theory.
Contact: Chenyan Wu chenyan.wu@unimelb.edu.au
Multipartitions and level-rank duality
Contact: Ting Xue ting.xue@unimelb.edu.au
Cyclic quivers and distinguished orbits
Contact: Ting Xue ting.xue@unimelb.edu.au
Applied Mathematics
Autonomous propulsion of nearly spherical nanoparticles trapped in acoustic fields
When asymmetric nanoparticles are trapped at the pressure node/velocity antinode of an acoustic field, they can swim autonomously due to a phenomenon called ‘acoustic streaming’. While this phenomenon has been explained for arbitrary shapes, analytical formulae only exist in the highly idealised cases of a perfect sphere and a two-sphere system. This project aims to develop an analytical formulation for how a nearly spherical particle behaves in an acoustic trap. The project will use asymptotic methods on the Navier-Stokes equations alongside domain perturbations on nearly spherical geometries.
Contact: Jesse Collis jesse.collis@unimelb.edu.au
Asymptotic properties of random knots
Knotted structures are ubiquitous in nature and of interest in many areas of pure and applied mathematics. There are several different ways of representing a knot, each with its own advantages and drawbacks. In this project, we will investigate large scale properties of random knots and how these change according to the model used. Time permitting, we will discuss applications to biology and biophysics.
Contact: Daniele Celoria daniele.celoria@unimelb.edu.au (joint with Agnese Barbensi)
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
Lava flow through a forest
Recent field studies have identified that vegetation can have a strong effect on the flow of lava as it migrates downslope from a volcanic vent. Forests act as a network of obstacles that can slow and divert the advancing lava. This project will investigate the interaction of shallow viscous flows with arrays of obstructions to determine how their formation influences the flow field and the force exerted. Existing numerical codes are available for the student and the main aim will be to develop simple analytical solutions to describe the flow physics.
Contact: Edward Hinton ehinton@unimelb.edu.au
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
Counting pattern-avoiding permutations
A pattern-avoiding permutation is a permutation whose entries are restricted to avoid one or more substructure. They have simple descriptions but the problem of counting them can range from trivially easy to devilishly difficult. They are connected to a range of other combinatorial objects like lattice paths and binary trees. This project will look at some open problems in pattern-avoiding permutations and related objects. Some experience with Mathematica and/or Python would be helpful.
Contact: Nick Beaton nrbeaton@unimelb.edu.au
Recursive computations for random matrix theory using Mathematica
Higher level mathematical structures, such as recursions and Pfaffians, show themselves in computing statistical quantities relevant to random matrix theory.
This project is to undertake implementation of these structures using the computer algebra system Mathematica. As a side topic, there will be opportunity to explore the use of Mathematica in the context of large language models.
Contact: Peter Forrest pjforr@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
Case studies for “RealStat”
The teaching resource “RealStat” is a collection of case studies involving statistics and data science. In all of the projects, the Statistical Consulting Centre at the University of Melbourne was involved, by providing assistance, advice, analysis and support to the researchers.
The case studies have rich material that provides context and background, which typifies the usual presentation of a project to a statistician. This makes them ideal for teaching students about authentic statistical practice, and they have been used in a variety of ways in subjects at undergraduate and postgraduate level.
The vacation scholar project involves working on some new case studies, to add to the current set.
We are seeking a student with a strong background in statistics and good writing skills. Experience with R and web programming would be beneficial.
Contact: Ian Gordon irg@unimelb.edu.au (jointly with Sue Finch)
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 (www.mathsblocks.com) 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
Mathematical Biology
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
Go with the flow: mathematically modelling hormonal fluctuations throughout the menstrual cycle
No one menstrual cycle looks exactly like another. How a person experiences their period is influenced by how their hormones fluctuate throughout the menstrual cycle. Understanding a person’s unique menstrual cycle is crucial to monitoring health outcomes and assessing the efficacy of hormonal contraception at an individual level.
Previous mathematical studies of the menstrual cycle have often neglected the individual experience. This project aims to explore how we can model the hormonal fluctuations throughout a menstrual cycle, and how these fluctuations are impacted by individual menstrual cycle characteristics.
Contact: Adriana Zanca adriana.zanca@unimelb.edu.au and Jennifer Flegg jennifer.flegg@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
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
Multicellular Systems Biology
My research is on the interface between applied mathematics numerical methods scientific computing and biology. We use theoretical tools to try to get a better understanding of organ and tissue development and disease.
Due to recent increases in the amount and quality of cell level imaging data, and matching advances in computational power, multicellular modelling has become ever more popular. Multicellular modelling considers cells as discrete entities and represents their interactions using mathematical formalisms, both stochastic and mechanics based. This allows tissues to be simulated, with tissue level behaviour and properties being emergent rather than imposed.
Various projects are available focusing on modelling and on numerical methods. See my website for examples of my work.
Contact: James Osborne jmosborne@unimelb.edu.au
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
Recursive computations for random matrix theory using Mathematica
Higher level mathematical structures, such as recursions and Pfaffians, show themselves in computing statistical quantities relevant to random matrix theory.
This project is to undertake implementation of these structures using the computer algebra system Mathematica. As a side topic, there will be opportunity to explore the use of Mathematica in the context of large language models.
Contact: Peter Forrest pjforr@unimelb.edu.au
Assorted projects
A variety of projects on solvable vertex models, quantum spin chains, algebraic combinatorics and diagram algebras with applications to mathematical physics. Projects on applications of natural language processing AI to pattern recognition in the symmetric group for students with very strong coding skills.
Contact: Jan de Gier jdgier@unimelb.edu.au
Entanglement of completely symmetrized quantum states
Dicke states are collective quantum states of identical two-level atoms (or qubits) interacting coherently with a common electromagnetic field, resulting in enhanced collective phenomena like superradiance. They are represented in the symmetric subspace of the total Hilbert space, characterized by fixed total angular momentum. In this project the vacation scholar will try to quantify the entanglement properties of Dicke states using a variety of entanglement measures. On the way, there is ample opportunity to learn about the representation theoretic aspects underlying their construction as well as their applications in quantum computing and information processing.
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
Symmetric polynomials and vertex operators
Symmetric polynomials with several variables are important objects in physics, they enter into the expression of quantum wave functions, but also into the non-perturbative analysis of certain quantum field theories. Around 1990, Naihuan Jing developed a method to study these symmetric polynomials based on vertex operators, an important tool of 2d conformal field theories (statistical physics, string theory). The goal of this project is to learn Jing's technique and apply it to generalize the family of Hall-Littlewood polynomials. The main references for this project are Jing's 1991 paper "Vertex Operators and Hall-Littlewood symmetric functions" and part of Macdonald's lectures on "Symmetric Functions and Orthogonal Polynomials". A good understanding of basic linear algebra will be essential for this project.
Contact: Jean-Emile Bourgine jean-emile.bourgine@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
Quantum entanglement
In this project, we will study the concept of quantum entanglement, famously coined by Albert Einstein as "spooky action at a distance". The goal is to understand and classify how different parts of a quantum system can be quantum mechanically correlated, which cannot be described by classical physics. The project requires a solid background in linear algebra and will give the opportunity to learn some advanced topics (e.g., in Lie theory and random matrix theory), while no prior physics knowledge is required.
Contact: Lucas Hackl lucas.hackl@unimelb.edu.au
Black hole information paradox
In this project, we will explore the famous black hole information paradox, which asks the question what happens to the information of matter falling into a black hole once the black hole is completely dissolved into thermal Hawking radiation. The goal is to understand the paradox, analyze several potential solutions and attempt to relate them to the concept of quantum entanglement. The project requires a solid background in linear algebra and will give the opportunity to learn some advanced topics (e.g., in differential geometry and Lie theory), while no prior physics knowledge is required.
Contact: Lucas Hackl lucas.hackl@unimelb.edu.au
Individual projects in the areas of field theory, supersymmetry, string theory and the mathematics behind
Contact: Johanna Knapp johanna.knapp@unimelb.edu.au
Quantum groups and integrable lattice models
In this project the student will learn about some algebraic constructions behind solvable physical models.
Contact: Sasha Garbali alexandr.garbali@unimelb.edu.au
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
Algorithms for distributed, decentralised optimization
The field of distributed optimisation studies algorithms suited to solving optimisation problems which can exploit computing architectures such as those used in high performance and cluster computing. Within this paradigm, the basic computing model considers a fleet of "simple" devices, connected via some network topology, which work collaboratively to solve a complex problem. Each device has some partial knowledge of the problem, which is private to that particular device, and the ability to do work. This project will study algorithms for solving optimisation problems in distributed, decentralised systems using recent advances in the field of monotone operator splitting.
Contact: Matthew Tam matthew.tam@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
Projects in K-theory, Algebra, and Topology
K-theory is a way to answer the following question: is an idempotent matrix of functions necessarily diagonalisable? Depending on what we mean by "functions", this question relates to algebraic topology; algebraic geometry; group theory; number theory; and other topics.
Contact: Christian Haesemeyer christian.haesemeyer@unimelb.edu.au
Riemann surfaces and complex algebraic curves
The theory of Riemann surfaces, or equivalently, complex algebraic curves, sits at the confluence of differential geometry, complex analysis, and algebraic geometry. For this reason, the study of these objects has led to the development of many broader theories with wide-ranging implications across mathematics and broader sciences. This project will focus on gaining an understanding of these objects from one or both of these perspectives with potential projects (depending on interests and background) in rational billiards, enumeration and existence of branched covers, holomorphic differentials and flat geometry, line bundles, divisors, Riemann-Roch theorem, Riemann-Hurwitz theorem, Teichmüller dynamics, projective complex geometry, and the moduli space of Riemann surfaces or algebraic curves.
Contact: Scott Mullane mullanes@unimelb.edu.au
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
Generalized ensemble Kalman inversion for a malaria transmission model
Ensemble Kalman inversion (EKI) methods are a fast approach for parameter inference, but they have some major limitations. In general, they can only be used for Gaussian likelihood models, and they require the covariance matrix of the likelihood to be known. While a Gaussian likelihood is appropriate in many contexts, the assumption of a fully specified covariance matrix is restrictive and unrealistic in practice. There is a recent extension of EKI that allows it to be applied to models with a general (non-Gaussian) likelihood. This method has not been widely explored, and a big question is whether it can be used for a Gaussian likelihood model where the covariance is not fully specified. This project will answer this question in the context of a malaria transmission model.
Contact: Imke Botha imke.botha@unimelb.edu.au and Jennifer Flegg jennifer.flegg@unimelb.edu.au
Projects in causal inference
Causal inference aims to explore the causal relationships between variables, moving beyond their mere statistical dependence or coincidence typically investigated in standard statistical literature. Existing methods in causal inference often rely on strong model assumptions or are limited to low-dimensional settings. In this project, students will explore causal inference methodologies in more practical scenarios, such as high-dimensional data and data measured with errors. Students will employ modern techniques, including adaptive statistical methods, machine learning, and deep learning, to advance the field and develop robust solutions for complex data environments.
Contact: Wei Huang wei.huang@unimelb.edu.au
Applying deep learning to problems in genetic epidemiology
In phylogenetics, we use genomic data from pathogens to study infectious diseases. 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
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
Tracking Physical Activity: How reliable is your smart watch?
Physical inactivity has been long associated with elevated risk of various cardiovascular diseases and cancer. Wearable physical activity (PA) trackers such as Fitbit and Xiaomi Smartband provide cost-effective methods for capturing physical activities patterns at high resolution. They are being increasingly used in research settings as intervention as well as providing health- related information to consumers. However, data from these wearable trackers are known to contain systematic biases. For example, these wrist-worn devices are known to overestimate steps for activities involving wrist movement such as folding laundry or playing video game.
In this project, we propose a Hidden Markov Model to improve estimation of physical activity using data from wrist-worn wearable device. The model will use step counts, heart rate and other relevant manifest variables. Models with different emission probability and sojourn time distributions will be compared and the best model will be selected using real datasets from a set of patients recruited as part of a clinical trial at the Baker Institute (https://baker.edu.au/research/clinical-trials/optimise-study). Once developed the best model will be further validated using another set of patients. Simulation studies for assessing the robustness of the model against missing data due to irregular wearing pattern will also be investigated.
Contact: Agus Salim salim.a@unimelb.edu.au, www.salimlab.org
Rank-dependent branching processes
Many animal populations feature a social structure with dominance hierarchy. This is common in monkey populations such as baboons and vervets, as well as in wolves, birds and fishes. In such populations, individuals compete for resources and mating opportunities within their social group; relationships and interactions between the individuals result in a relative rank (social order) being created. An individual’s rank may depend on the rank of their parent or their siblings, as well as their age, experience, and physical fitness; the social rank of individuals is subject to change when a dominant animal (high rank) is challenged by a subordinate one (lower rank) through a conflict. The rank can affect access to food, territory, mates, and even survival. There are benefits and costs to being a dominant individual: those individuals have better foraging and reproductive success, however they suffer from more stress which may lead to a loss of fitness and lower life expectancy. Insights into the behaviour of individuals in these hierarchical societies can lead to better management of threatened and endangered populations.
Inspired by real ecological issues concerning the vervet monkeys, this project aims to model dominance hierarchy in populations using particular multitype branching processes, called rank-dependent branching processes.
Contact: Sophie Hautphenne sophiemh@unimelb.edu.au
Stochastic Process
Non-Brownian Motion on Matrix Groups
In this project, you can enjoy the beauty of the wonderful interplay of algebraic structures and probabilistic analysis which nowadays is coined Integrable Probability. The project will be in the subtopic of Random Matrix Theory. We want to generalise compact results of multiplicative stochastic processes on the unitary group to one of the other two classical groups (the orthogonal and unitary symplectic one). You will learn how various mathematical fields such as Group Theory, Harmonic Analysis and Probability interact with each other. You are not expected to know all about these fields. The necessary parts will be taught to you in the project. This, hopefully, helps you to see mathematics from a new perspective in an inspiring symphony of many areas.
Contact: Mario Kieburg m.kieburg@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
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
Rank-dependent branching processes
Many animal populations feature a social structure with dominance hierarchy. This is common in monkey populations such as baboons and vervets, as well as in wolves, birds and fishes. In such populations, individuals compete for resources and mating opportunities within their social group; relationships and interactions between the individuals result in a relative rank (social order) being created. An individual’s rank may depend on the rank of their parent or their siblings, as well as their age, experience, and physical fitness; the social rank of individuals is subject to change when a dominant animal (high rank) is challenged by a subordinate one (lower rank) through a conflict. The rank can affect access to food, territory, mates, and even survival. There are benefits and costs to being a dominant individual: those individuals have better foraging and reproductive success, however they suffer from more stress which may lead to a loss of fitness and lower life expectancy. Insights into the behaviour of individuals in these hierarchical societies can lead to better management of threatened and endangered populations.
Inspired by real ecological issues concerning the vervet monkeys, this project aims to model dominance hierarchy in populations using particular multitype branching processes, called rank-dependent branching processes.
Contact: Sophie Hautphenne sophiemh@unimelb.edu.au