The studentship will develop novel tools in three overlapping concurrent research directions, determining matrix factorisations from solutions to recently discovered “intrinsic” quadratic forms, by making use of cutting-edge research in Number Theory, Analysis, Geometry of Numbers and Linear Algebra. In a sense the research focus has its origins in the roots of Spectral Theory (Analysis), attributed to David Hilbert, whereby the underpinning focus is to establish conditions under which an operator can be expressed in terms of simpler operators.
The overarching aim is to further comprehend the recently discovered interconnectedness known to exist between the factorisation of n x n number squares (integer matrices) such as a 9x9 Sudoku square, a particular type of quadratic equation and matrix decompositions over superalgebras. For numbers such as 77 say, we can write the number as a product of two other numbers 77=7x11, and the same is sometimes true for integer matrices. An integer matrix factorisation enables a positive definite integer quadratic form to be expressed as a sum of squares of integral linear forms, a classical problem studied by Minkowski, Mordell, Conway and many other famous mathematicians.
The assumption of matrix decompositions in a matrix factorisation leads to a quadratic equation, solutions to which are necessary for a factorisation of the matrix to exist. The size of the set of integer solutions to the quadratic form gives an immediate upper bound for the total number of matrix factorisations. For quadratic forms of smaller dimension much is known about the integers they can represent. The key question is: Can you determine the matrix factorisation from that information?
The approach that will be pursued relies on recent developments on the bridge between Number Theory, Analysis, Geometry of Numbers and Linear Algebra, developed by the first supervisor (MCL) in collaboration with leading experts in these fields. The second local supervisor Dr Iskander Aliev (IA) complements this expertise in Integer Optimisation and Discrete Geometry. Dr Julian Sheuer (JS), a specialist in Analysis and Geometry, completes the supervisory team.
Cardiff University provides a highly inspiring research environment for postgraduates. Its Analysis group comprises internationally leading experts in pure and applied mathematics. The student will automatically join Cardiff’s Doctoral Academy with opportunities to interact with fellow postgraduate students from other disciplines, build networks and receive further training to attain professional skills.
This studentship is an outstanding opportunity for someone with a first degree at MMath level or an MSc to perform research within a team of established top scientists. BSc students may be considered if they are of truly exceptional calibre. A successful PhD student emerging from this project will be research trained and excellently situated to pursue a career in academia or industry. Industrial demand for pure-mathematics skills has exploded in the last decade, partly due to security applications, as demonstrated by the UK government’s investments in Heilbronn Fellowships directed by GCHQ.
For academic development, the supervisory team provides at least weekly meetings to support the student’s path. MAGIC courses offer relevant broadening skills training. The School’s seminars and the Welsh Mathematics Colloquium are excellent opportunities for academic exchange, while Cardiff’s Doctoral Academy provides peer-guidance and further professional skills training. The student will learn how to disseminate achieved results in papers and at conferences, for which the international connections to the Bohemian-Matrices research community of MCL is an excellent start.
The student’s professional skills development will be enhanced by emphasising applications in industry alongside opportunities to gain teaching practice, relevant for several career pathways post PhD.
For personal development, the student will grow into research independence and will learn to take ownership of their project. The breadth and scope of the project will enable the student to participate in a range of conferences, enabling the student to build personal research networks, leading to the development of leadership and interdisciplinary skills.
HOW TO APPLY
Applicants should apply through the Cardiff University online application portal, for a Doctor of Philosophy in Mathematics with an entry point of October 2021
In the research proposal section of your application, please specify the project title and supervisors of this project.
There is no requirement to submit a research proposal
In the funding section, please select "I will be applying for a scholarship / grant" and specify that you are applying for advertised funding from EPRSC Maths DTP.
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