You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.

Title Geometry and analysis of Schubert varieties
Group(s) Number Theory and Geometry
Proposer(s) Dr Sergey Oblezin
Description

Schubert varieties is a basic tool of classical algebraic and enumerative geometry. In modern mathematics these geometric object arise widely in representation theory, theory of automorphic forms and in harmonic analysis. In particular, it appears that the classical geometric structures can be naturally extended to infinite-dimensional setting (loop groups and, more generally, Kac-Moody groups), and such generalizations provide new constructions in infinite-dimensional geometry. Moreover, many of the arising constructions are supported by (hidden) symmetries and dualities of quantum (inverse) scattering theory and quantum integrability. Possible PhD projects will be devoted to extensive development of harmonic analysis on Schubert varieties with further applications to automorphic forms, arithmetic goemetry and number theory.

Relevant Publications
  • A.Gerasimov; D.Lebedev; S.Oblezin, Parabolic Whittaker functions and topological field theories I. Commun. Number Theory Phys. 5 (2011), no. 1, 135–201
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Title Orthogonal polynomials in probability, representation theory and number theory
Group(s) Number Theory and Geometry
Proposer(s) Dr Sergey Oblezin
Description

Orthogonal ensembles play a crucial role in many areas including random matrix theory, probability and harmonic analysis. In the recent decades a new striking connections with the theory of automorphic forms and number theory appeared. However, there is a definite lack of general results and implementations at present. Possible PhD projects will be aiming at developing these recent interactions among the group theory, harmonic analysis and number theory.

Relevant Publications
  • I.G. Macdonald, Symmetric functions and Hall polynomials. Second edition.
  • I.G.Macdonald, Affine Hecke algebras and orthogonal polynomials.
  • A.Gerasimov; D.Lebedev; S.Oblezin, Baxter operator formalism for Macdonald polynomials. Lett. Math. Phys. 104 (2014), no. 2, 115–139
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Title Number theory in a broad context
Group(s) Number Theory and Geometry
Proposer(s) Prof Ivan Fesenko
Description

Ivan Fesenko studies zeta functions in number theory using zeta integrals. These integrals are better to operate with than the zeta functions, they translate various properties of zeta functions into properties of adelic objects. This is a very powerful tool to understand and prove fundamental properties of zeta functions in number theory. In the case of elliptic curves over global fields, associated zeta functions are those of regular models of the curve, i.e. the zeta function of a two dimensional object. Most of the classical work has studied arithmetic of elliptic curves over number fields treating them as one dimensional objects and working with with generally noncommutative Galois groups over the number field, such as the one generated by all torsion points of the curve. The zeta integral gadget works with adelic objects associated to the two dimensional field of functions of the curve over a global field and using commutative Galois groups. The latter has already been investigated in two dimensional abelian class field theory and it is this theory which supplies adelic objects on which the zeta integral lives. For example, Fourier duality on adelic spaces associated to the model of the curve explains the functional equation of the zeta function (and of the L-function of the curve).  

The theory uses many parts of mathematics: class field theory, higher local fields and several different adelic structures, translation invariant measure and integration on higher local fields (arithmetic loop spaces), functional analysis and harmonic analysis on such large spaces, groups endowed with sequential topologies, parts of algebraic K-theory, algebraic geometry. This results in a beautiful conceptual theory. 

There are many associated research problems and directions at various levels of difficulty and opportunities to discover new objects, structures and laws.

Relevant Publications
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Title Computational methods for elliptic curves and modular forms
Group(s) Number Theory and Geometry
Proposer(s) Dr Christian Wuthrich
Description

Computational Number Theory is a fairly recent part of pure mathematics even if computations in number theory are a very old subject. But over the last few decades this has changed dramatically with the modern, powerful and cheap computers. In the area of explicit computations on elliptic curves, there are two subjects that underwent a great development recently: elliptic curves over finite fields (which are used for cryptography) and 'descent' methods on elliptic curves over global fields, such as the field of rational numbers.

It is a difficult question for a given elliptic curve over a number field to decide if there are infinitely many solutions over this field, and if so, to determine the rank of the Mordell-Weil group. Currently, there are only two algorithms implemented for finding this rank, one is the descent method that goes back to Mordell, Selmer, Cassels,... and the other is based on the work of Gross, Zagier, Kolyvagin... using the link of elliptic curves to modular forms. While the first approach works very well over number fields of small degree, it becomes almost impossible to determine the rank of elliptic curves over number fields of larger degree. The second method unfortunately is not always applicable, especially the field must be either the field of rational numbers or a quadratic extension thereof.

There is another way of exploiting the relation between elliptic curves and modular forms by using the p-adic theory of modular forms and the so-called Iwasawa theory for elliptic curves. Results by Kato, Urban, Skinner give us a completely new algorithm for computing the rank and other invariants of the elliptic curve, but not much of this has actually been implemented. Possible PhD projects could concern the further development of these new methods and their implementation.

Relevant Publications
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Title Variants of automorphic forms and their L-functions
Group(s) Number Theory and Geometry
Proposer(s) Dr Nikolaos Diamantis
Description

Classical automorphic forms are a powerful tool for handling difficult number theoretic problems. They provide links between analytic, algebraic and geometric aspects of the study of arithmetic problems and, as such, they are at the heart of the major research programmes in Number Theory, e.g. Langlands programme. Crucial for these links are certain functions associated to automorphic forms, called L-functions, which are the subject of some of the most important conjectures of Mathematics.

In recent years, investigations into the theory of automorphic forms have led into the study of variants of automorphic forms and of their L-functions, such as quasi-modular forms, harmonic Maass forms, mock modular forms, higher order modular forms and multiple Dirichlet series. In most cases, the motivation for introducing these objects was not just to generalize the classical automorphic forms and their L-functions, but to obtain novel tools to address already stated number theoretic problems. The techniques associated with these new objects in turn raise new interesting questions  and highlight connections beyond the original motivating problems. For example, the theory of harmonic Maass forms and modular forms has been used to resolve problems in partitions of numbers, and higher order modular forms have been applied to Percolation Theory problems in Physics.

As these techniques have only recently been discovered, they lead to a number of very interesting open questions, e.g. how to construct mock modular forms encoding specific partition functions, how to determine the arithmetic nature of high-order forms or how to use the theory of multiple Dirichlet series to bound moments of the Riemann zeta function. Questions of this type, are highly relevant both for the outstanding problems in classical automorphic forms and for the further development of the new subjects themselves. Therefore, many of these questions are very appropriate for a PhD project.
 

Relevant Publications
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