# PhD projects

## Algebraic semigroup theory (Victoria Gould)

Victoria Gould works in algebraic semigroup theory, focussing on a number of inter-related areas: (1) the theory of $S$-acts over a monoid $S$, that is, representations of monoids by transformations; she has also considered partial actions of monoids and related structures, and when these can be extended to global actions (2) structure theory for semigroups, mainly in the little explored cases where the semigroups in question are not necessarily regular (3) semigroups of quotients and (4) endomorphism monoids of certain universal algebras.

Victoria has PhD students working on the above topics, but is happy to consider potential students in any area of semigroup theory.

Recent PhD: Yanhui Wang. Title of thesis Beyond regular semigroups'.
The thesis provides an in-depth examination of the relationship between wide classes of semigroups possessing a regular biordered set of idempotents, and ordered categories. In particular it gives a completely new way of approaching the classical connections between regular semigroups and Nambooripad's inductive categories, using what Wang calls generalised categories'.

## Analytic Number Theory and Random Matrix Theory (Chris Hughes)

Christopher Hughes works at the interface of number theory and random matrix theory, primarily in using random matrix models to understand the distribution of the Riemann zeta function. He is happy to supervise PhD projects on the Riemann zeta function (such as how big it can get on the critical line, and the general distribution of its values) and in random matrix theory (such as understanding the distribution of extreme values of characteristic polynomials of random matrices).

## Metric Number Theory and Diophantine approximation (Victor Beresnevich and Sanju Velani)

Victor Beresnevich and Sanju Velani work on a variety of problems in metric number theory and Diophanitne approximation that involve a range of techniques from Diophantine aproximation, analytic number theory, the geometry of numbers, probability theory, fractals and ergodic theory. Some examples include the Duffin-Scheffer problem on rational approximations to real numbers, problems on approximation by algebraic numbers, problems on badly approximable vectors, problems on Diophantine approximation on manifolds, etc. They are currently running a large scale research programme (see here) and any PhD student would become an integral part of the larger research group. If interested, please, contact either or both of them for possible PhD research projects.

## Theory of Algebraic Independence (Evgeniy Zorin)

Evgeniy Zorin works on the theory of transcendental numbers and of algebraic independence. This area of research deals with problems if some complicated objects are linked by a relation of a relatively simple nature. Classical questions of this type are whether some given real (or complex) numbers are linked by a polynomial relation with integer coefficients. Such studies depend usually on Diophantine approximation properties.

Evgeniy would be happy to supervise PhD projects within his area of competence.

## Linear Algebraic Groups (Michael Bate)

Linear algebraic groups are affine algebraic varieties which carry the structure of a group, so studying them mixes algebraic geometry - the geometry of polynomials - with group theory. Michael Bate is interested in the structure of these algebraic groups, especially over fields of positive characteristic. He uses a wide range of tools from group theory (representation theory; buildings; group actions) and algebraic geometry (invariant theory; instability; quotients). Michael has various PhD project ideas which would suit students with a strong interest in algebra and, in particular, group theory and representation theory.

## Representation Theory (Stephen Donkin)

A natural problem in group theory is to detemine all possible ways of representing a given group as matrices. Working over the complex numbers this was essentially done for the symmetric groups by Frobenius at the end of the 19th century and, building on the work of Frobenius, was done for the general linear groups by Schur at the beginning of the 20th century. So it is rather surprising that if the base field is changed to one of positive characteristic, the representation theory of the symmetric and general linear groups is still not well understood. The modern study of these and related problems use techniques from several different areas of algebra and algebraic geometry and the problems interact in interesting ways with other parts of mathematics. There are many very concrete problems in this area suitable for study for a PhD. So far 14 students have completed a PhD in representation theory with Stephen Donkin.

## (Co)Homology With Local Coefficients (Brent Everitt)

Brent Everitt currently works on a number of problems in algebraic topology and homological algebra with particular emphasis on applications to topology, group theory and combinatorics. He is particularly interested in the cohomology of algebraic/combinatorial structures with “local coefficients”: for example, the cohomology of a small category (or even a poset) with coefficients lying in a presheaf of modules. He would be interested in supervising PhD projects in knot homology theories (especially Khovanov homology), group cohomology and the cohomology of hyperplane arrangements.

Older research areas are in the geometry of discrete groups and geometric group theory. He is happy to hear from students who are interested in working on representations of groups acting discretely on hyperbolic spaces and on the structure of free groups.

## Differential Geometry (Ian McIntosh)

Ian McIntosh studies minimal surfaces and harmonic maps from surfaces into symmetric spaces and related homogeneous spaces. His particular expertise is in the construction of these using integrable systems methods. These methods have given a great deal of information about minimal tori, but there are still many natural questions which have not been answered and there are PhD projects available in this area. To some extent these methods can also be used to study conformal immersions of surfaces into the 3 or 4-sphere, using the ideas of quaternionic holomorphic geometry, and this is a possible area for interesting research. He is also very interested in the application of minimal surface theory to the problem of parameterising "good" representations of a surface group (fundamental group of a surface) into the isometry groups of real or complex hyperbolic space: this involves studying minimal surfaces of genus at least 2.

## Stochastic Partial Differential Equations (Zdzislaw Brzezniak)

Zdzislaw Brzezniak is interested in the theory of ferromagnetism which was initiated by Weiss, see Brown (1978) and references therein,  and further developed by Landau and Lifshitz (1935) and Gilbert (1955). According to this theory the orientation of the magnetic moment $M$ of a ferromagnetic material occupying  a 3-dimensional region $D$ at temperatures below the critical (so-called Curie) temperature is of constant length and satisfies a certain system of degenerate parabolic equations called now Landau-Lifshitz-Gilbert (LLG) equation. The stationary solutions of this equation correspond to the equilibrium states of the ferromagnet and are not unique in general. An important problem in the theory of ferromagnetism is to describe the phase transitions between different equilibrium states induced by thermal fluctuations. Therefore, the LLG equation needs to be modified in order to incorporate random fluctuations of the field $H$ into the dynamics of the magnetization $M$ and to describe noise-induced transitions between equilibrium states of the ferromagnet. The program to analyze noise induced transitions was initiated by Neel (1946) and further developed in  Brown (1963), Kamppeter (1999) and others. Recent mathematical publications include  Kohn,  Reznikoff and  Vanden-Eijnden, Magnetic elements at finite temperature and large deviation theory,  J. Nonlinear Sci. 15 (2005), Brzezniak, Goldys and Jegaraj, Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation,  Appl. Math. Res. Express. (2012)  and Large deviations for a stochastic Landau-Lifshitz equation, extended version, arXiv:1202.0370, Banas, Brzezniak and Prohl, Convergent finite element discretization of the stochastic Landau-Lifshitz-Gilbert equation, (preprint).

## Configuration Space Analysis and Multicomponent Interacting Systems (Alexei Daletskii)

Alexei Daletskii is interested in study of multicomponent interecting particle systems. This research is motivated by applications in statistical mechanics and biology. An infinite system of interacting particles living in a Euclidean space or a Riemannian manifold $X$ is modeled by a measure $\mu$ of the space $\Gamma_X$ of countable subsets of $X$ (the configuration space). There are many exciting problems which can be studied in this framework:

• study of the relationship between properties of Gibbs measures on configuration spaces of Riemannian manifolds and geometry and topology of underlying manifolds
• study of the (stochastic) dynamics in concrete statistical mechanical or biological models (in particular, described by Poisson and Gibbs cluster point processes)
• study of phase transitions in particular models of statistical mechanics
• applications of measures on $\Gamma_X$ to the representation theory of  diffeomorphism groups of $X$

## Ergodic Theory and Dynamical Systems (Zaq Coelho)

Zaq Coelho is interested in the dynamics of maps and flows from the point of view of Ergodic Theory, i.e. the study of invariant measures for these maps or flows. One aspect of the theory that has not been investigated is the dynamics of piecewise algebraic maps. The simplest of these maps are transformations of a group into itself that acts as an endomorphism on half of the space and acts as a translation on the other half of the space. A number of questions can be asked like: (a) is there a dense orbit? (b) what is the topological entropy of such a map? (c) is there an invariant probability measure which maximises entropy? Some simple examples exist from maps of the interval $[0,1]$ (with addition modulo 1), where these questions can be answered, but more general groups, they are open.

## Representation Theory of Cherednik Algebras (Maxim Nazarov)

Many people delight in the formal elegance the symmetry can bring to a design. Less well known is the 19th century mathematical discovery that symmetry forms the framework for Geometry. Later, 20th century physicists discovered that these same ideas lie at the heart of our understanding of the Universe. There is a special branch of Mathematics, called Representation Theory, which studies symmetries revealed (or sometimes concealed) by objects and phenomena in Mathematics and other fields of Science including Biology, Chemistry and Physics. That branch is the area of my research.

The aim of my current project is to study particular objects, called Cherednik algebras or double affine Hecke algebras. They appeared about 20 years ago as a result of blending ideas from the branch of modern Theoretical Physics called Conformal Field Theory, and from a more traditional branch of Mathematics, called the Theory of Special Functions. Since then the Cherednik algebras were found to play major roles in several other branches of Mathematics. The aim is to advance the theory of these algebras, and to find more links between Mathematics and Theoretical Physics.

## Operators on Ordered Banach Spaces; Integral Operators (Simon Eveson)

Simon Eveson works on partially ordered Banach spaces - in particular, on real function spaces - and linear operators which respect the order relation - in particular, integral operators with non-negative kernels. For such operators, we seek information about the eigenvalues, eigenvectors, singular values, asymptotic behaviour of iterates, etc., exploiting the order-preserving property of the operator. These techniques have applications in mathematical biology, quantum physics and mathematical economics. Mathematical models of economies naturally involve ordered Banach spaces (consumption streams are usually represented by non-negative vectors, and prices by non-negative linear functionals); recent work on equilibria in a new type of infinite-horizon economical model could offer an interesting opportunity for a PhD project applying functional-analytic techniques to economical models.

## Differential Geometry; Geometric Structures (Chris Wood)

Chris Wood is interested in differential geometric structures on Riemannian manifolds; in particular, criteria that can be used to determine whether one structure is in some way "better" than others of the same type, what such "optimal" structures look like, and where and how to find them.

The underlying ideas can be traced back to Klein's Erlangen Program, in which geometric objects reveal themselves through their symmetry groups, and the Lagrange-Hamilton principle of least action, which characterizes the configurations adopted by a (mechanical) system as those that are stationary for an associated "total energy" functional. More precisely, the idea is to apply the Eells-Sampson theory of harmonic maps to geometric symmetry reduction. Another, more pragmatic, reason for this is to relax the somewhat stronger condition of holonomy-invariance, which in certain topological environments may be impossible to satisfy. Some well-known geometric structures, that have already been studied to a greater or lesser extent from this point of view, are: Hermitian and almost-Hermitian structures, foliations and almost-product structures, metric almost contact structures, metric $f$-structures etc. Chris currently has research students working on harmonic vector fields, and energy functionals of "higher power".

## PhD projects

Overview

Projects in individual research areas: