A labelled oriented graph is a connected, finite, directed graph in which each edge is labelled by a vertex. A labelled oriented graph gives rise to a group presentation whose generating set is the vertex set and whose defining relations say that the initial vertex of an edge is conjugated to its terminal vertex by its label. A group G is a LOG group if it has such a presentation. A LOT group is a LOG group for which the graph is a tree. Every classical knot group is a LOT group. In fact LOT groups are characterised as the fundamental groups of ribbon n-discs in Dn+2.The most significant outstanding question on the topology of ribbon discs is: are they aspherical? The expected answer is yes. Howie has shown that it is sufficient to prove that LOT groups are locally indicable. An interesting research project would be to study the following two questions: is every LOT group locally indicable; is every LOT group an HNN extension of a finitely presented group?

• J. Howie, On the asphericity of ribbon disc complements, Trans Amer Math Soc, 289, 281-302, 1985

Let G be a group. An expression of the form g₁ t … g_k t=1 where each g_i is an element of G and the unknown t is distinct from G is called an equation over G. The equation is said to have a solution if G embeds in a group H containing an element h for which the equation holds. There are two unsettled conjectures here. The first states that if G is torsion-free then any equation over G has a solution. The second due to Kervaire and Laudenbach states that if the sum of the exponents of t is non-zero then the equation has a solution. There have been many papers published in this area. The methods are geometric making use of diagrams over groups and curvature. This subject is related to questions of asphericity of groups which could also be studied.

Dr. Pumplün currently studies forms of higher degree over fields, i.e. homogeneous polynomials of degree d greater than two (mostly over fields of characteristic zero or greater than d). The theory of these forms is much more complex than the theory of homogeneous polynomials of degree two (also called quadratic forms). Partly this can be explained by the fact that not every form of degree greater than two can be “diagonalized”, as it is the case for quadratic forms over fields of characteristic not two. (Every quadratic form over a field of characteristic not two can be represented by a matrix which only has non-zero entries on its diagonal, i.e. is diagonal.)

A modern uniform theory for these forms like it exists for quadratic and symmetric bilinear forms (cf. the standard reference books by Scharlau or Lam) seems to be missing, or only exists to some extent. Many questions which have been settled for quadratic forms quite some time ago are still open as as soon as one looks at forms of higher degree. It would be desirable to obtain a better understanding of the behaviour of these forms. First results have been obtained. Another related problem would be if one can describe forms of higher degree over algebraic varieties, for instance over curves of genus zero or one.

Dr. Pumplün is also studying nonassociative algebras over rings, fields, or algebraic varieties. Over rings, as modules these algebras are finitely generated over the base ring. Their algebra structure, i.e. the multiplication, is given by any bilinear map, such that the distributive laws are satisfied. In other words, the multiplication is not required to be associative any more, as it is usually the case when one talks about algebras. Her techniques for investigating certain classes of nonassociative algebras (e.g. octonion algebras) include elementary algebraic geometry. One of her next projects will be the investigation of octonion algebras and of exceptional simple Jordan algebras (also called Albert algebras) over curves of genus zero or one. Results on these algebras would also imply new insights on certain algebraic groups related to them.

Another interesting area is the study of quadratic or bilinear forms over algebraic varieties. There are only few varieties of dimension greater than one where the Witt ring is known. One well-known result is due to Arason (1980). It says that the Witt ring of projective space is always isomorphic to the Witt ring of the base field.

If you want to investigate algebras or forms over algebraic varieties, this will always involve the study of vector bundles of that variety. However, even for algebraically closed base fields it is usually very rare to have an explicit classification of the vector bundles. Hence, most known results on quadratic (or symmetric bilinear) forms are about the Witt ring of quadratic forms, e.g. the Witt ring of affine space, the projective space, of elliptic or hyperelliptic curves. An explicit classification of symmetric bilinear spaces is in general impossible because it would involve an explicit classification of the corresponding vector bundles (which admit a form). There are still lots of interesting open problems in this area, both easier and very difficult ones.

After the groundbreaking works of V. Voevodsky, it became possible to work with algebraic varieties by completely topological methods. An important role in this context is played by the so-called Generalized Cohomology Theories. This includes classical algebraic K-theory, but also a rather modern (and more universal) Algebraic Cobordism theory. The study of such theories and cohomological operations on them is a fascinating subject. It has many applications to the classical questions from algebraic geometry, quadratic form theory, and other areas. One can mention, for example: the Rost degree formula, the problem of smoothing algebraic cycles, and u-invariants of fields. This is a new and rapidly developing area that offers many promising directions of research.

From the beginning of the 20th century it was observed that quadratic forms over a given field carry a lot of information about that field. This led to the creation of rich and beautiful Algebraic Theory of Quadratic Forms that gave rise to many interesting problems. But it became apparent that quite a few of these problems can hardly be approached by means of the theory itself. In many cases, solutions were obtained by invoking arguments of a geometric nature. It was observed that one of the central questions on which quadratic form theory depends is the so-called "Milnor Conjecture". This conjecture, as we now understand it, relates quadratic forms over a field to the so-called motivic cohomology of this field. Once proven, this would provide a lot of information about quadratic forms and about motives (algebro-geometric analogues of topological objects) as well. The Milnor Conjecture was finally settled affirmatively by V. Voevodsky in 1996 by means of creating a completely new world, where one can work with algebraic varieties with the same flexibility as with topological spaces. Later, this was enhanced by F. Morel, and now we know that quadratic forms compute not just the cohomology of a point in the "algebro geometric homotopic world", but also the so-called stable homotopy groups of spheres as well. It is thus no wonder that these objects indeed have nice properties.

Therefore, by studying quadratic forms, one actually studies the stable homotopy groups of spheres, which should shed light on the classical problem of computing such groups (one of the central questions in mathematics as a whole). So it is fair to say that the modern theory of quadratic forms relies heavily on the application of motivic topological methods. On the other hand, the Algebraic Theory of Quadratic Forms provides a possibility to view and approach the motivic world from a rather elementary point of view, and to test the new techniques developed there. This makes quadratic form theory an invaluable and easy access point to the forefront of modern mathematics.

Banach function algebras are complete normed algebras of bounded, continuous, complex-valued functions defined on topological spaces. There are very many different examples with a huge variety of properties. Two contrasting examples are the algebra of all continuous complex-valued functions on the closed unit disc, and the subalgebra of this algebra consisting of those functions which are continuous on the closed disc and analytic on the interior of the disc. In the second of these algebras, any function which is zero throughout some non-empty open set must be constantly zero. This is very much not the case in the bigger algebra: indeed Urysohn’s lemma shows that for any two disjoint closed subsets of the closed disc, there is a continuous, complex-valued function defined on the disc which is constantly 0 on one closed set and constantly 1 on the other (algebras of this type are called regular algebras).

Most Banach function algebras have some features in common with one or the other of these two algebras. The aim of this project is to investigate a variety of conditions, especially regularity conditions, for Banach function algebras, and to relate these conditions to each other, and to other important conditions that Banach function algebras may satisfy.

Regularity conditions have important applications in several areas of functional analysis, including automatic continuity theory and the theory of Wedderburn decompositions. There is also a close connection between regularity and the theory of decomposable operators on Banach spaces.

Banach function algebras are complete normed algebras of bounded continuous, complex-valued functions defined on topological spaces. There are very many different examples with a huge variety of properties. Two contrasting examples are the algebra of all continuous complex-valued functions on the closed unit disc, and the subalgebra of this algebra consisting of those functions which are continuous on the closed disc and analytic on the interior of the disc. In the second of these algebras, any function which is zero throughout some non-empty open set must be constantly zero. This is very much not the case in the bigger algebra: indeed Urysohn’s lemma shows that for any two disjoint closed subsets of the closed disc, there is a continuous, complex-valued function defined on the disc which is constantly 0 on one closed set and constantly 1 on the other (algebras of this type are called regular algebras).

Most Banach function algebras have some features in common with one or the other of these two algebras. The aim of this project is to investigate a variety of conditions (including regularity conditions) for Banach function algebras, to relate these conditions to each other, and to other important conditions that Banach function algebras may satisfy, and to investigate the preservation or introduction of these conditions when you form various types of extension of the algebras (especially ‘algebraic’ extensions such as Arens-Hoffman or Cole extensions).

A meromorphic function is basically one convergent power series divided by another: such functions arise in many branches of pure and applied mathematics. Professor Langley has supervised ten PhD students to date, and specific areas covered by his research include the following.

• Zeros of derivatives. There is a long history of work on these questions which can be viewed as natural generalizations of the classical Picard theorem to the effect that an entire function which omits two values is a constant. Langley’s best known result is probably his 1993 proof of Hayman’s 1959 conjecture on the zeros of a meromorphic function and its second derivative. More recently, Langley, Bergweiler and Eremenko  settled the last outstanding case of a conjecture of Wiman from 1911 concerning the derivatives of real entire functions, and Langley has extended much of this work to meromorphic functions.
• Differential equations in the complex domain, in particular the growth and oscillation of solutions of linear differential equations. This has been an active area in recent years and there are potential applications.
• Properties of compositions of entire functions. This area has connections to complex dynamics, in which the famous Julia and Mandelbrot sets arise, and which has been a very successful field of research in recent years.
• Critical points of potentials associated with distributions of charged points and wires, and zeros of related meromorphic functions. This research was awarded an EPSRC Project Studentship in 2003-4.

This aim of the project is to further develop the theory and numerical methods for compensated convex transforms introduced by the proposer and to apply these tools to approximations, interpolations, reconstructions, image processing and singularity extraction problems arising from applied sciences and engineering.

Compact endomorphisms of commutative, semisimple Banach algebras have been extensively studied since the seminal work of Kamowitz dating back to 1978. More recently the theory has expanded to include power compact, Riesz and quasicompact endomorphisms of commutative, semiprime Banach algebras.

This project concerns the classification of the various types of endomorphism for specific algebras, with the aid of the general theory. The algebras studied will include algebras of differentiable functions on compact plane sets, and related algebras such as Lipschitz algebras. 

Complex dynamics is the study of iteration of analytic functions on the complex plane. This has been a remarkably active area of research in recent years. A rich mathematical structure is seen to emerge amidst the chaotic behaviour. Its appeal is enhanced by the intricate nature of the Julia sets that arise, and fascinating images of these fractal sets are widely admired.

Complex dynamics exploits many beautiful results from classical complex analysis. Among these is Nevanlinna's theory of value distribution, which offers a powerful generalisation of Picard's theorem that an entire function omitting two values must be constant.

Quasiregular mappings of n-dimensional real space generalise analytic functions on the complex plane. One can therefore attempt to develop a theory of quasiregular iteration parallel to the results of complex dynamics. Such a theory is just beginning to emerge, lying between the well-studied analytic case (where many powerful tools are available) and general iteration in several real variables, which is much less well-understood.

Many quantum physical systems (for example superconductors, superfluids, Bose-Einstein condensates) exhibit vortex states that can be described by Ginzburg-Landau type functionals. For various equations of motion for the physical systems, the dynamical behaviour of finite numbers of vortices has been rigorously established. We are interested in studying systems with many vortices (this is the typical situation in a superconductor). In the hydrodynamic limit, one obtains an evolution equation for the vortex density. Typically, these equations are relatives of the Euler equations of incompressible fluids: for the Gross-Pitaevskii equation (a nonlinear Schrödinger equation), one obtains Euler, for the time-dependent Ginzburg-Landau equation (a nonlinear parabolic equation), one obtains a dissipative variant of the Euler equations.

There are at least two interesting directions to pursue here. One is to extend recent analytical progress for the Euler equation to the dissipative case. Another one is to obtain hydrodynamic limits for other motion laws (for example, mixed or wave-type motions).

This is a project mostly about analysis of PDEs, possibly with some numerical simulation involved.

• M Kurzke, D Spirn: Vortex liquids and the Ginzburg-Landau equation. arXiv:1105.4781
• R Jerrard, D Spirn: Hydrodynamic limit of the Gross-Pitaevskii equation. arXiv:1310.4558

Some physical problems can be modelled by a function or vector field with a near discontinuity at a point. Specific examples include boundary vortices in thin magnetic films, and some types of dislocations in crystals. Typical static configurations can be found by minimizing certain energy functionals. As the core size of the singularity tends to zero, these energy functionals are usually well described by a limiting functional defined on point singularities.

This project investigates how to obtain dynamical laws for singularities (typically in the form of ordinary differential equations) from the partial differential equations that describe the evolution of the vector field. For some such problems, results for interior singularities are known, but their boundary counterparts are still lacking.

This project requires some background in the calculus of variations and the theory of partial differential equations.

• M Kurzke: The gradient flow motion of boundary vortices Ann. Inst. H. Poincaré Anal. Non Linéaire. 24(1), 91-112

Many problems in image analysis and data analysis can be represented mathematically as a network based
problems. Examples are image segmentation and data clustering. Traditionally these kind of graph based
problems have been tackled by combinatorial algorithms, but in recent years interest has grown in
applying ideas from variational methods and nonlinear partial differential equations (PDEs) to such
problems.

This has lead to an interesting mix of theoretical questions (what is the dynamics on the network induced
by these methods? which correspondences between PDE quantities and graph quantities can we find, e.g.
total variation, graph cut, curvature...?) and practical applications.

This project will investigate graph curvature and related quantities and make links to established
results in PDE theory.

Many problems in image analysis and data analysis, such as image segmentation or data clustering, require
the computation of pairwise similarity scores between nodes of a network. When the image or data set is
large, the computational cost is very prohibitive. One way to deal with this problem is to compute only a
subset of all the pairwise comparisons and use those to approximate the others.

Recent developments in the theory of (dynamics on) graph limits offer the hope that this subset can be
chosen in such a way that the resulting dynamics of the segmentation or clustering algorithm is a very
good approximation of the dynamics on the full graph.

This project will investigate this possibility and can be taken in a theoretical and/or application
oriented direction.

This aim of the project is to solve problems in vectorial calculus of variations, forward-backward diffusion equations, partial differential inclusions and coercivity problems for elliptic systems. These problems are motivated from the variational models for material microstructure, image processing and elasticity theory. Methods involve quasiconvex functions, quasiconvex envelope, quasiconvex hull, Young measure, weak convergence in Sobolev spaces, elliptic and parabolic partial differential equations, and other analytic and geometric tools.

Foundations of adaptive finite element methods for PDEs
(Or- Why do adaptive methods work so well?) 

Adaptive finite element methods allow the computation of solutions to partial differential equations (PDEs) in the most optimal manner that is possible. In particular, these methods require the least amount of degrees-of-freedom to obtain a solution up to a desired accuracy! In recent years a theory has emerged that explains this behaviour. It relies on classical a posteriori error estimation, Banach contraction, and nonlinear approximation theory. Unfortunately, the theory so far applies only to specific model problems.

Challenges for students:
* How can the theory be extended to, for example, nonsymmetric problems, nonlinear problems, or time-dependent problems?
* What about nonstandard discretisation techniques such as, discontinuous Galerkin, isogeometric analysis, or virtual element methods? 

Depending on the interest of the student, several of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group! 

• J.M. Cascon, C. Kreuzer, R.H. Nochetto, and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 26 (2008), pp. 2524-2550