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Title Equations over groups Algebra and Analysis Dr Martin Edjvet Let G be a group. An expression of the form g1 t … gk t=1 where each gi 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.
Title Quadratic forms and forms of higher degree, nonassociative algebras Algebra and Analysis Dr Susanne Pumpluen 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.
Title Cohomology Theories for Algebraic Varieties Algebra and Analysis Dr Alexander Vishik 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.
Title Quadratic Forms: Interaction of Algebra, Geometry and Topology Algebra and Analysis Dr Alexander Vishik 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.
Title Regularity conditions for Banach function algebras Algebra and Analysis Dr Joel Feinstein 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.
Title Properties of Banach function algebras and their extensions Algebra and Analysis Dr Joel Feinstein 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).
Title Meromorphic Function Theory Algebra and Analysis Prof James Langley 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, and specific areas covered by his and their research include those in the list below. Please note that since Professor Langley is close to retirement age he may not be available to supervise further students, and potential applicants are advised to contact him before applying.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. 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. Properties of compositions of entire functions. This area has connections to complex dynamics, in which the famous Julia and Mandelbrot sets arise.Critical points of potentials associated with distributions of charged points and wires, and zeros of related meromorphic functions.
Title Relative presentations and asphericity Algebra and Analysis Dr Martin Edjvet The subject area is Combinatorial Group Theory. The objects of study are relative pesentations which can be viewed as one-relator products. In particular we consider the questions: when is a relative presentation aspherical; what is the order of the group defined by a given relative presentation? This has applications to cyclically presented groups for example. The methods are geometric (borrowing ideas from algebraic topology); algebraic; and may involve computational group theory via GAP or automatic groups. R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer 1977. (Book.)Martin Edjvet and Arye Juhasz, The infinite Fibonacci groups and relative asphericity, Trans London Math Soc 4 (2017) 148-218W.A. Bogley, M. Edjvet and G. Williams, Aspherical relative presentations all over again, to appear in Groups St Andrews (conference proceedings)
Title Compensated convex transforms and their applications Algebra and Analysis Prof Kewei Zhang 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.
Title Endomorphisms of Banach algebras Algebra and Analysis Dr Joel Feinstein 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.
Title Iteration of quasiregular mappings Algebra and Analysis Dr Daniel Nicks Complex dynamics is the study of iteration of analytic functions on the complex plane. 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. Quasiregular mappings of n-dimensional real space generalise analytic functions on the complex plane. Roughly, a mapping is called quasiregular if it locally distorts space by only a bounded amount, so that small spheres are mapped to small ellipsoids. This is more flexible than the situation with analytic functions, where the Cauchy-Riemann equations tell us that infinitesimally small circles are mapped to small circles. There are many similarities between the behaviour of analytic functions and quasiregular mappings. 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 from complex analysis are available) and general iteration in several real variables, which is much less well-understood. The problems studied will be inspired and guided by existing results in complex dynamics. For example, we can ask questions about the ‘escaping set’ of a function – this is the set of all starting points from which the sequence of iterates tends to infinity. One of the challenges we encounter is that as we increase the number of iterations of a quasiregular mapping, the amount of local distortion may become increasingly large. This is very much a pure mathematics project and will appeal to someone who enjoys topics such as real analysis, complex analysis, metric spaces or discrete dynamical systems.
Title Mean field limits of Ginzburg-Landau equations Algebra and Analysis Dr Matthias Kurzke 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.The goal of the project is to study the dissipative equations and to understand instabilities and low regularity solutions. M Kurzke, D Spirn: Vortex liquids and the Ginzburg-Landau equation. arXiv:1105.4781S Serfaty: Mean Field Limits of the Gross-Pitaevskii and Parabolic Ginzburg-Landau Equations. arXiv:1507.03821M Duerinckx, S Serfaty: Mean-field dynamics for Ginzburg-Landau vortices with pinning and forcing. arXiv:1702.01919
Title Dynamics of boundary singularities Algebra and Analysis Dr Matthias Kurzke 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. PoincarĂ© Anal. Non LinĂ©aire. 24(1), 91-112
Title Where graphs and partial differential equations meet Algebra and Analysis Dr Yves van Gennip 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.
Title Graph limits for faster computations Algebra and Analysis Dr Yves van Gennip 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.
Title Vectorial Calculus of Variations, Material Microstructure, Forward-Backward Diffusion Equations and Coercivity Problems Algebra and Analysis, Algebra and Analysis Prof Kewei Zhang 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.
Title Dynamics of vortex singularities in nonlinear PDE Algebra and Analysis, Algebra and Analysis Dr Matthias Kurzke Nonlinear PDE of Ginzburg-Landau type can be used to model a wide range of phenomena, from ferromagnetic materials and superconductors to quantum field theory. For certain ranges of the Ginzburg-Landau parameter (for self-duality and for point vortices), the equations can be reduced to ODEs. The focus of the present project will be to study the equations of motion in a wider setting, with the aim to extend the range of validity of the reductions and to compare almost singular solutions of the PDEs with simplified ODEs that describe the motion of the singularities. The project will use mostly rigorous analysis and possibly numerical simulation.
Title Foundations of adaptive finite element methods for PDEs Scientific Computation, Algebra and Analysis Dr Kris van der Zee 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