Accepting Postgraduate Research Students

PhD projects

In general, I am interested in supervising projects in computational algebra or research software engineering, as well as projects on the frontier or intersection of these areas. Below is a description of a specific project which I am currently advertising.

**Enumerating algebraic structures: constraint programming for a working mathematician**

Combinatorial search problems emerge in many areas of modern mathematical research. Defining an algebraic structure by binary operations and their axioms, it would be interesting to determine the number of non-equivalent (with respect to certain definition) structures of a given order, and enumerate them, i.e. provide their complete and non-redundant list (for example, to use it to search for examples and counterexamples). For example, constraint-based approach proved to be useful for enumeration of small semigroups and monoids.

Still, there are several obstacles to the use of constraint programming techniques by a "working mathematician", i.e. a domain expert in mathematics, not possessing specialised expertise in constraint programming, and wiling to use it in their research. Among these obstacles are:
- lack of accessible ways to express mathematical definitions in constraint modelling terms;
- lack of robust and user-friendly interfaces to constraint programming tools from mathematical software packages;
- limitations of the constraint programming "model and run" paradigm, which assumes that the user treats the solver as a "black box" and makes it difficult to orchestrate solutions of multiple interrelated subproblems, inherent for many problems from the computational algebra domain.

This project lies on the intersection of discrete computational algebra and constraint programming. The researcher will apply constraint programming techniques to model algebraic structures, classify them and study their properties. The toolkit will include the constraint modelling pipeline, being developed in St Andrews, the computational algebra system GAP, and possibly other research software.

Depending on the applicant’s background, skills and interests, the work may include (but will not be limited to), a combination of the following research activities:
a) Improving the usability of constraint modelling tools for mathematicians
b) Developing and implementing new constraint modelling techniques
c) Applying constraint modelling tools and other software for large scale combinatorial search problems in abstract algebra
d) Compiling and distributing datasets containing collections of algebraic structures of certain types and some fixed orders, up to equivalence
e) Integrating the datasets for the redistribution with mathematical software systems
f) Computer-aided research in pure mathematics

As a primary application domain, we suggest the recently emerged theory of skew braces. A skew brace is a triple (A, +, *), where (A, +) and (A, *) are (not necessarily abelian) groups, and the equality

a * (b + c) = a * b - a + a * c

holds for all elements of A.

Skew braces draw a significant interest due to their connection with combinatorial solutions of the Yang-Baxter equation, that plays an important role in physics and pure mathematics, and by now already found applications in many areas of abstract algebra and recently in physics. The theory of skew braces has a lot of open questions, including enumerative ones (see e.g. https://arxiv.org/abs/2311.07112). The research is however not limited to skew braces, and may be generalised onto other algebraic structures, such as e.g. near-rings, magmas with various additional conditions on operations, etc.

For the details of the application procedure, please see
- https://www.st-andrews.ac.uk/computer-science/prospective/pgr/
- https://blogs.cs.st-andrews.ac.uk/csblog/2024/10/24/phd-studentships-available-for-2025-entry/