# 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

### Algebra and calculus of random matrix integrations

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

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

### Density matrices, quantum entanglement and random matrices

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

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

### Enumerative geometry and physics

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

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

### Topological invariants in quantum systems

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

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

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

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

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

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

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

### Wave propagation in flat space

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

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

### Topics in general relativity

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