Due to holidays I will NOT be offering student projects for the summer break 2025-2026. Please see the School's website for details of the projects and how to apply for the Vacation Scholarship. The Australian Mathematical Sciences Institute (AMSI) also offers Vacation Research Scholarships. Please note that I am also offering to supervise projects outside of the vacation scholarship scheme. In order to ensure enough time commitment this would need to take place outside of regular semester time though.
Entanglement of completely symmetrised quantum states (2024-2025), Damien Kim (poster)
Implementing symmetries by topological defects (2023-2024), Connie Gardiner (poster)
Classification of monoidal categories with one 'non-invertible' object (2023-2024), Xing Hang (poster)
Topological invariants of quantum systems (2021/2022), Barnaby Russell (poster)
Matrix Product States (2021/2022), Nate Cross
Geodesics on cones (2020/2021), Miles Koumouris (poster and report)
Topological invariants in quantum systems (2020/2021), Yuhan Gai (poster)
Topological phases in quantum systems with quantum group symmetries (2020/2021), Michael Law (AMSI page with report)
Quantum Boltzmann Machines: An investigation (2020/2021), Saleh Naghdi (AMSI page with report)
Topological phases and quantum computation (2019/2020), Zoe Schwerkolt (poster)
Topological invariants in quantum systems (2019/2020), Brae Vaughan-Hankinson (poster)
Geometry and topology in quantum mechanics (2018/2019), Max West (poster)
Geometry and topology of quantum systems (2017/2018), Miaohan Long (poster)
Topological invariants in quantum systems (2017/2018), Lexin Ding (poster)
Topological invariants in quantum systems (2016/2017), Robert Dusanovic (AMSI page with report)
Time-reversal in quantum theory (2016/2017), UoM
MS Vacation Scholarships AMSI Vacation Research Scholarships
Here is a brief description of projects that I have offered in the past
ENTANGLEMENT OF COMPLETELY SYMMETRIZED QUANTUM STATES (Posted for 2024) - [Mathematical Physics]
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.
HIGHER SYMMETRIES AND THEIR REALIZATION IN PHYSICS (Posted for 2023) - [Pure Mathematics and Mathematical 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.
NEURAL NETWORK QUANTUM STATES - [Data Science and Mathematical Physics]
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 concepts 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.
GEODESICS ON CONES (Posted for 2020-2021) - [Geometry and Topology]
Geodesics are curves between two points on a manifold that extremize the length functional. In this project the student will learn about the basics of variational calculus which is concerned with the extremization of functions of functions (such as curves) and apply the techniques to the surprisingly rich question of how to find and describe geodesics on cones. If time permits the project will also involve having a closer look at the properties of the singularity at the tip of the cone.
CLASSIFYING PHASES OF MANY-BODY SYSTEMS USING MACHINE LEARNING (Posted for 2019-2020) [Mathematical Physics, Data Science]
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.
TOPOLOGICAL INVARIANTS IN QUANTUM SYSTEMS (Posted for 2019-2020) [Mathematical Physics]
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
TIME-REVERSAL IN QUANTUM THEORY AND GROUP REPRESENTATIONS BEYOND UNITARY OPERATORS (Posted for 2016-2017) [Mathematical Physics, Algebra]
According to a famous theorem by E. Wigner (1931), symmetries in quantum mechanics need to be implemented by unitary or anti-unitary operators acting on Hilbert space. While known for a long time and responsible for some of the astonishing properties of certain physical models, the implications for some more modern areas of mathematics and mathematical physics seemingly have not been analysed so far.
In this project, the summer vacation scholar will first of all reproduce the statement and the proof of Wigner's Theorem before trying to understand its consequences in various physical situations and making an attempt to incorporate the newly gained freedom in order to generalise established higher algebraic structures such as Hopf algebras and quantum groups.