Possible MPhil/PhD Projects in Pure Mathematics

Moment problems are classical in nature and intertwine many different topics in mathematics, eg, spectral theory of self-adjoint operators, orthogonal polynomials, probability theory and optimisation theory. In this project, we will investigate noncommutative/multidimensional analogues of moment problems, in the sense that the given sequence are matrices, bound linear operators, quaternions or Clifford numbers. Connections with algebraic geometry will also be emphasised.

Supervisor:

In the past 50 years, a major trend in Operator Theory focuses on the use of operator algebras for encoding geometrical and topological objects. Operator algebras may be considered as algebras of (bounded in norm) innite matrices with complex entries. A central aspect of the program is to explore the passage from intrinsic properties of the object into properties of the associated operator algebras, and use invariants of the latter to classify the former. There are two interrelated questions that orient our study:

1. Which (desirable) features of the object determine the operator algebra?
2. What is the (desirable) level of equivalence for classifying objects?

Examples of examined objects so far include tilings, tangles, graphs, dynamical systems, groups, semigroups, varieties, homogeneous ideals, factorial languages and stochastic matrices. The research so far incorporates operator algebras (both selfadjoint and non-selfadjoint) in terms of: representation theory, dilation theory, ideal structure, KMS-states theory, C*-envelopes, re exivity, and hyperrigidity; for objects related to: C*correspondences, product systems, subproduct systems, semigroup actions on operator algebras, and homogeneous ideals.

For this project we will examine operator algebras arising from such geometric structures in this endeavour.

Supervisor:

Operator semigroups are families of bounded linear operators which are parameterised by a time variable and describe the solutions of certain differential equations, such as the heat equation or the (damped) wave equation. There is a rich and beautiful abstract theory, motivated primarily by questions about the rate of energy decay in damped waves, which makes it possible to predict precisely the quantitative large-time asymptotic behaviour of operator semigroups given suitable information about the spectral properties of the so-called generator of the semigroup. This theory has recently reached a state of perfection in the case where the underlying Banach space is a Hilbert space, which is to say that its norm is particularly well behaved, and it is known that without any restrictions at all on the underlying space one in general obtains less sharp results.

The more theoretical part of this project would involve investigating the quantitative asymptotic behaviour of operator semigroups on Banach spaces which are not necessarily Hilbert spaces but nevertheless have some reasonably nice geometric properties (such as non-trivial (Fourier) type). However, there would also be scope to consider applications of the theory to important concrete problems, and in particular to certain coupled systems arising in the study of damped waves, in mathematical control theory and elsewhere. Depending on the student's background it may also be possible to investigate some aspects of the theory numerically.

This project will combine aspects of functional analysis, harmonic analysis and complex analysis, so a reasonably strong background in analysis is essential. Ideally, candidates will also have had some exposure to the modern theory of partial differential equations (and in particular to Sobolev spaces), or at the very least they will be willing to pick this up quickly.

Supervisor:

Representation theory and quantum groups is a vast area of active research. A student interested in pursuing research in this area should choose a focus area from among the following suggestions:

Focus 1: Rational Cherednik algebras and their representations in positive characteristic

Rational Cherednik algebras are a family of associative algebras parametrized by a reflection group and some numerical parameters. Since their introduction in 2002 they have been a very active area of research with links to integrable systems, symplectic geometry and combinatorics. Of particular interest is their representation theory, where a certain category O of ``nice" representations has been defined and studied by many authors, but is still not completely described in many cases.

The aim of this project is to study rational Cherednik algebras, and in particular their representations, over algebraically closed fields of positive characteristic. Typical questions would be describing category O for all rational Cherednik algebras associated to a particular reflection group, describing the block decomposition of category O, describing the centers of rational Cherednik algebras or its spherical subalgebra, and describing the polynomial representation of the algebra. We propose to work in a case where the characteristic of the field divides the order of the group, creating more involved phenomena. Early stages of the work might include programming.

Students should have basic knowledge of algebra and a firm grasp of linear algebra. Knowledge of some representation theory and basic programming is desirable but not necessary. Knowledge of rational Cherednik algebras or finite characteristic phenomena is not necessary.

                                              Primary supervisor: Dr Martina Balagovic

Focus 2: Quantum Lie superalgebras

Lie algebras are classical mathematical objects from the early 20th century, which can be thought of as infinitesimal symmetries. Their representations play a role in particle physics, and are well understood from a mathematical point of view. Quantum groups are certain deformations of Lie algebras depending on a parameter q, encoding quantum symmetries, and their representation is closely related to the representation theory of the underlying Lie algebras. Another generalisation of Lie algebras is to Lie superalgebras, where a Z2 grading (parity) encodes supersymmetries from physics.

This project is about combining those generalisations, and considering quantum Lie superalgebras. In particular, it will focus on types of Lie superalgebras where the quantisation has only recently been defined. The aim will be to describe these algebras and construct and study their representations (classify them and describe them combinatorially).

Students should have basic knowledge of algebra and a firm grasp of linear algebra. Knowledge of some representation theory and Lie algebras is desirable but not necessary. Knowledge of quantum groups or Lie superalgebras is not necessary.

                                              Primary Supervisor: Dr Martina Balagovic”

Focus 3: Representation theory of quantum symmetric pairs’

Classical representation theory studies how groups and rings act on vector spaces by linear maps. Representation theory hence encodes symmetry, and as such it appears in many areas of mathematics and mathematical physics. In a groundbreaking development in the 1980s mathematical physicists combined the notions of symmetry and quantisation and invented quantum groups. Quantum groups have played a fundamental role in representation theory ever since.

The theory of quantum groups has a younger sibling, the theory of quantum symmetric pairs, which is as rich in structure theory as quantum groups themselves. This theory is much less developed. The finite dimensional simple representation of these algebras, for example, have only been classified in a few special cases. The aim of this project is to develop aspects of a general representation theory of quantum symmetric pairs.

Supervisor: Dr Stefan Kolb

My research focuses on representation/Lie theory and the theory of tensor categories. Typically the category of finite dimensional representations of a group, or an affine group scheme, a supergroup or a quantum group, is a tensor category. One would like to understand the fusion rules that describe the decomposition of tensor products or one seeks qualitative information like the classification of tensor ideals, the connection to other categories via tensor functors or the use of the tensor structure to perform constructions such as the Reshetikhin-Turaev 3-manifold invariants built from modular tensor categories.  While the most important examples of tensor categories, or more generally, monoidal categories, arise from representation theory, the theory has outgrown its origins and has emerged into a vast and complex theory on its own, providing a unified language for many phenomena in different fields. Monoidal categories are now ubiquitious in areas such as representation theory, invariants of links and $3$-manifolds, algebraic geometry, quantum computing and mathematical physics.

I am particularly interested in the following two research areas.

1) Algebraic supergroups (such as $GL(m|n)$ and $OSp(m|2n)$) generalize algebraic groups. First motivated by questions in mathematical physics, they have become a thriving area in mathematics. I am interested in tensor product decompositions, the Duflo-Serganova cohomology functor, character/dimension formulae, quantized versions of Lie superalgebras and possible applications to mathematical physics. In a project in this area the successful candidate would study various ways to understand the tensor structures via categorical techniques, character formulas or homotopical algebra.

2) Deligne categories are families of universal tensor categories that interpolate other representation categories (e.g. representations of the symmetric group $S_n$ can be extended to complex parameters $S_t$, $t \in \mathbb{C}$). Quantized versions of these categories can be found in the works of Turaev. Other  mathematicians - Knop, Etingof, Flake, Maassen, Meir, Khovanov, Kononov, Ostrik  - have  defined a vast number of interpolating categories which capture the phenomenon of stabilization with respect to rank. In a project in this area the successful candidate would further develop this theory, in particular by comparing them to other categories which capture the phenomenon of stabilization with respect to rank.

Primary Supervisor: Thorsten Heidersdorf