Our Projects ____________

Here we (PhD students supervised or co-supervised by Paul) keep some notes explaining to our sponsors some of the reasons why and how their sponsorship is supporting interesting work, and so why they should continue to sponsor us.
We hope that these very brief notes may be of interest not only to our sponsors, but also to our fellow students and co-workers. Perhaps to see some of the ways in which our various activities, while very different, can nonetheless fit beautifully together.

Nouf: -----   The Kadar-Yu algebras are a heirarchy of sequences of finite-dimensional algebras motivated by problems in computational statistical mechanics. For each element of the heirarchy there is a parameter, and for each value a sequence of algebras, whose representation theory is controlled by a limiting category. It is a major open problem to determine the parameter values for which the limit category is semisimple (an analogue of the famous root-of-unity criterion for classical Temperley-Lieb algebras). This is determined in principle by the singularity or otherwise of a set of Gram matrices. However the spectrum of these matrices is not easy to compute in practice. The Theorems in my thesis yield several key (and beautiful) partial results in this direction. I also draw attention to a collection of related interesting problems at the intersection of algebra and analysis (Tchebychev recursions, fourier transforms and so on). [PhD completed.]

Manar: ______ The power set of the unit square is closed under a suitable stack-shrink operation, as are the power sets of many similar sets. I call the square case `passport photograph magma'. The composition is not associative, but contains many subquotients that are algebraic structures of a diagramatic nature. My project is to study interesting subsets and interesting quotients - for example associative quotients. Specifically I am interested in the emergence of braid-group-like structures. My motivation is pure, but can also be taken as arising from problems of error-protection in quantum computing, where novel equivalences such as topological equivalences are both potentially useful in principle, and potentially physically implementable.

Basmah: _______ The general area of my project is topologies on finite sets. So far I have investigated in two specific directions. One is to see if there is any version of constructions appearing in low-dimensional topology (such as braids) that can be realised in finite topology. This is hard because there is nothing like the real line in finite topology, but it is well-motivated because such phenomena would have applications in quantum computing. My other direction starts from the observation that a finite topology on X is given by a suitable element of the power set of the power set; and that the symmetric group on X acts on P(P(X)) in such a way as to preserve the topology property. My first step in this direction is to study the representations of S_X on P(X) and P(P(X)) and so on, as |X| varies. The cases P^0(X) and P(X) were known to Gordon James, and one sees that the irreducible content of representations can be expressed in closed form for all |X|. Such stability properties are interesting with a number of potential applications, for example in statistical mechanics, and in representation theory itself. [PhD completed.]

Sarah: ______ Match categories are interesting examples of `natural' strict monoidal categories, that have been successfully used as the target categories in representation theory, replacing the usual target categories Vect and Mat. In fact it is not so easy to find categories with the beautiful properties of Match categories. But my project is to search for generalisations of Match categories. I am guided in part by consideration of how the choice of target impacts the properties of representation theory, and in this sense I am partly also studying representation theory itself, from a novel perspective.

Ben H: ______ One of my aims is to make the construction of extended cobordism categories more rigorous. So I am working in metric/low-dimensional topology and category theory. There are potential applications in quantum computing, but I am not close to a direct connection with this problem yet. More To follow. [PhD completed.]

Jean: _____ Integer partitions occur in many areas of mathematics and physics. Under certain cirumstances and in a suitable statistical sense there is an asymptotic shape for partitions of large n. I have been looking at the circumstances under which such shapes arise, and what shapes then arise. This kind of problem is naturally related to the problem of algorithms for generating combinatorial sets - and the resultant probability distributions, so I am looking at this as well. More To follow. [PhD completed.]

Nura: _____
Deguchi algebras are the special case of Martin-Rittenberg's heirarchy of `physical' quotients of the ordinary Hecke algebras corresponding to the (2,2) q-symmetriser idempotent (or equivalently to the Schur-Weyl dual of the sl(1|1) quantum supergroup). There is a very intriguing combinatorial isomorphism (at the set level) between these algebras and certain blob algebras. I am investigating this. It lifts in principle to an isomorphism of algebras in generic cases, but probably not in general.
More To follow. [PhD completed.]

Jack: _____ To follow (optional).

Ben M: ______ I am working on more than one project. One concerns generalisations of Temperley--Lieb categories to other ambient surfaces, including with cross-caps (motivated by a talk of Ibarra in our seminar series). Another concerns the progression of the big open project (previously considered by Nouf for example) on the representation theory of Kadar-Yu algebras. In particular I have been computing gram matrices of standard modules. More To follow.

Yue: _____ The PBR (partitioned binary relation) constructions sit at the top of a lattice of algebras and semigroups, including many famous and useful algebras. I am preparing to study the PBR construction. This can be in the sense of ideal theory, but recently I have also been looking at algorithmic constructions, which would make a parallel with Jean's work.

Mia: ______ I am working on the construction of a variant of the Kadar-Yu algebra. There are several motivations, but one of them is to understand some general properties of representation theory of these kinds of categories and towers of algebras. More to follow.

Ibby: _____ To be decided! I am gearing up to read Torzewska's paper with Faria Martins et al on magmoids.