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

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.wuella@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