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Department of Mathematical Sciences

Department Research Colloquium

A series of colloquia, one per term, which will be of general interest and will give members of the department a chance to learn about important research topics in other areas of mathematical sciences. Talks will be followed by a wine reception in the coffee room. All are welcome!

Contact Paul Sutcliffe if you wish to propose a talk for this colloquium series, or nominate a colleague.

Michaelmas 2017

Jens Funke

"Theta Series in Arithmetic and Geometry"

Abstract:`In how many ways can an integer be represented as the sums of squares?’ This is a very classical question, which can be tackled by considering its associate generating series, that is, its`theta series'. The analytical properties of theta series (more generally associated to any positive definite integral quadratic form) naturally lead to the concept of `modular forms' which in recent decades have been playing an increasingly central role in number theory, most famously in Wiles's solution of Fermat Last Theorem. For indefinite quadratic forms, the naive question of representation numbers no longer makes sense, but one can associate using arithmetic and geometric data similar generating series, now called indefinite theta series. Furthermore, this led to the development of `mock modular forms' and `weak harmonic Maass forms' generalizing the classical notions of modular forms. In recent years these forms and indefinite theta series have played an important role in (combinatorial) number theory, e.g. by giving a modern understanding to Ramanujan's `mock theta functions', but also in mathematical physics, e.g. in string theory and various `moonshine' phenomena. The talk aims to give an accessible introduction to this topic and to present some recent developments.

3.00pm, Wednesday 22 November, CM101