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
There is no current research project in this area being offered for the Vacation Scholarship Program.
For more information on this research group see: Applied Mathematics
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
Counting pattern-avoiding permutations and other combinatorial objects
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, binary trees, and inversion sequences. 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
Learning and Teaching Innovation
Listed on this page are current research projects being offered for the Vacation Scholarship Program.
For more information on this research group see: Learning and Teaching Innovation
Exploring Indigenous Knowledges in Mathematics and Statistics
Exploring Indigenous Knowledges as they relate to tertiary mathematics and statistics. This topic is flexible and will be aligned to the scholar’s interests and background.
Contact: Paul Fijn paul.fijn@unimelb.edu.au
Designing Inclusive Teaching Resources
Evaluating and documenting a range of implementations and innovations in resourcees for teaching mathematics and statistics incorporating aspects of universal design for learning. Will consider resources, class materials and delivery, communication/organisation (e.g. LMS), and assessment. Suitable for students with an interest in equity, disability, and/or neurodiversity.
Contact: Paul Fijn paul.fijn@unimelb.edu.au
Qualitative studies in Mathematics Education
Mathematics and statistics graduates are familiar with quantitative data. However, in research, there is often a need to understand and make sense of qualitative data. This project seeks to develop an understanding of, and build an interpretation of qualitative data arising in mathematics education contexts.
Contact: TriThang Tran trithang.tran@unimelb.edu.au
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
Distinguishing heterogeneity from model misspecification
Deterministic mathematical models are widely applied to describe phenomena throughout the natural sciences and beyond. Allowing model parameters to vary according to probability distributions enables deterministic models to capture the sometimes significant variability between individuals. Given a specific mathematical model, it is, in many cases, possible to infer these individual-level distributions from population-level data. Many models are, however, likely to be misspecified (i.e., not an accurate representation of reality) to some extent. This project will explore whether it is ever possible to distinguish variability between individuals from model misspecification. Depending on student interests, this project can involve either or both of analytical work (based on calculus, analysis, and Taylor series expansions) or computational work.
Contact: Alexander Browning alex.browning@unimelb.edu.au and Adriana Zanca adriana.zanca@unimelb.edu.au
Multiscale modelling in biology
Many biological processes evolve over multiple scales, from molecular interactions within a cell to the dynamics of entire populations. Modelling such systems, and calibrating these models to data, is an open challenge. Various projects are available in this area that will use stochastic dynamical modelling, Bayesian inference and machine learning methods.
Contact: Tom Kimpson tom.kimpson@unimelb.edu.au and Jennifer Flegg jennifer.flegg@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
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, James Osborne jmosborne@unimelb.edu.au, and Isobel Abell isobel.abell@unimelb.edu.au
Mathematical Physics
Listed on this page are current research projects being offered for the Vacation Scholarship Program.
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
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
Mathematical Quantum Information: From Quantum Chaos to Black Holes
My research is at the interface of quantum information and fundamental theory, which includes the study of correlations and chaos in many-body quantum systems and of quantum processing in relativistic settings, such as curved spacetime. Potential projects for a vacation scholar could study what typical quantum correlations can teach us about the dynamics of a quantum system, how Hawking radiation seems to imply that information is lost in a black hole and other questions on mathematical quantum information. Such a project may provide an opportunity to learn some techniques from differential geometry, Lie theory, random matrix theory and representation theory. While some physics background could be helpful, it is not required. Please contact me if you want to discuss specific directions.
Contact: Lucas Hackl lucas.hackl@unimelb.edu.au
Operations Research
Listed on this page are current research projects being offered for the Vacation Scholarship Program.
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
Fast Second-Order Optimisation for Machine Learning
Optimisation algorithms are at the core of machine learning tasks, but the size of the models and datasets being used mean that simple 'first-order' algorithms are usually used. In this project, we will investigate if blending ideas from first-order and second-order methods can improve training speed for machine learning problems.
Contact: Lindon Roberts lindon.roberts@unimelb.edu.au
Pure Mathematics
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
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
Knot theory
How can we tell whether the trefoil knot is “the same” as a trivial round circle? There are a variety of knot invariants that are up to the task – the simplest ones use basic linear algebra, while more complicated invariants exist which draw on a range of mathematical areas. The precise topic will be chosen based on the student’s interests and background knowledge.
Contact: Arunima Ray aru.ray@unimelb.edu.au
Introduction to Kirby diagrams
This project concerns manifolds: topological spaces which locally resemble Euclidean space. Among these spaces, four-dimensional manifolds are especially mysterious. While they seem highly abstract, they can nevertheless be described concretely using diagrams, the so-called “Kirby diagrams”. This project will include an introduction to manifolds, handle decompositions of them, and a first look at describing 4-dimensional manifolds and surfaces within them. Some background knowledge in geometry or topology will be helpful but not strictly necessary.
Contact: Arunima Ray aru.ray@unimelb.edu.au
Understanding Topological Data Analysis
Topological data analysis (TDA) is a technique incorporating algebraic topology, namely notions of homology to gain greater insight into the shape of large data sets. This project seeks to understand the mathematics and techniques underlying TDA, with an eye towards applications.
Contact: TriThang Tran trithang.tran@unimelb.edu.au and Paul Fijn paul.fijn@unimelb.edu.au
Statistics and Data Science
Listed on this page are current research projects being offered for the Vacation Scholarship Program.
For more information on this research group see: Statistics
Applying and Interpreting Topological Data Analysis
Topological data analysis (TDA) is a technique incorporating algebraic topology to gain greater insight from large data sets. This project will explore applying TDA, generating graphical representations to explore and understand the relationships identified, interpreting the analysis, and communicating results. Requires knowledge of some statistical software (R preferred, python also suitable).
Contact: Paul Fijn paul.fijn@unimelb.edu.au and TriThang Tran trithang.tran@unimelb.edu.au
Projects in Generative Models Learning from Imprecisely Measured Data
Most statistical methods are designed for perfectly observed data, while lots of real data is measured with errors. For example, the ANHS and the U.S. NHANES surveys rely heavily on participants’ self-reported behaviours, such as what they ate in the last 24 hours. Studies have shown that this type of data can contain up to 75% noise. To illustrate, let X represent a participant’s true food intake. However, what they actually report is W = X + E, where E is an unknown random error. In some cases, the error E can be so significant that it accounts for as much as 75% of the total variability in the reported value W across the population. Analysing such data "as is" can lead to incorrect conclusions, which can negatively affect health policies and treatment guidelines. The project aims to solve this problem by developing a new generative model-based method that is theoretically justified and robust to handle large, complex datasets contaminated by measurement errors. Students are expected to learn the problem, related literature and methods, run some simulation experiments, and help develop R or Python packages under supervision.
Contact: Wei Huang wei.huang@unimelb.edu.au
Stochastic Processes
There is no current research project in this area being offered for the Vacation Scholarship Program.
For more information on this research group see: Stochastic Processes