# Number Theory and Geometry

Number theory is one of the oldest parts of mathematics. In its study of fundamental properties of numbers it uses every other part of mathematics and stimulates a variety of new developments in other areas. Number theory remains the most applicable part of pure mathematics through for example coding and cryptography and computer science.

The Nottingham number theory group includes six permanent members. In our work we use structures, methods and tools of arithmetical origin and from algebra, geometry, topology, K-theory, analysis and representation theory.

## Research Areas

- analytic number theory
- arithmetic, algebraic and anabelian geometry
- computational number theory
- geometric and categorical theories and correspondences
- mirror symmetry, Fano varieties, computational algebraic geometry
- higher class field theories, higher adelic analysis and geometry, higher automorphic forms
- local number theory, Iwasawa theory
- number theory, representation theory and quantum field theory
- zeta and L functions

## Group Activities

Members of the group are regularly involved in events, for example:

### Conferences

- Symmetries and correspondences: higher structures in number theory, July 3-4 2014, Nottingham.
- Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends, July 5-8 2014, Oxford, CMI
- RIMS and Symmetries and Correspondences RIMS workshop IUT Summit, July 18-27 2016, Kyoto
- The different faces of geometry, a workshop in honour of Fedor Bogomolov, September 12-14 2016, Nottingham

### Study Groups

The group runs several study groups and graduate courses for PhD students. Recent examples include:

- Higher number theory, higher class field theory, higher adelic structures and higher zeta integrals, 2014-2105
- Topological methods and motivic cohomology, 2015-2016
- Arakelov and Berkovich geometries, 2016

### Awards

The group has been well supported by EPSRC through the following awards:

- Topological field theories, Baxter operators and the Langlands programme, EPSRC experienced researcher fellowship for Sergey Oblezin, 2013-2017
- Symmetries and correspondences intra-disciplinary developments and applications, Nottingham-Oxford EPSRC programme grant, 2015-2021
- The Combinatorics of Mirror Symmetry, EPSRC fellowship for Alexander Kasprzyk, 2016-2020
- Explicit methods for Jacobi forms over number fields
**,**EPSRC first grant for Fredrik Stromberg, 2016-2017

## Group Members

### Staff:

Professor Fesenko works on a range of number theoretical aspects, including local number theory, higher dimensional number theory and arithmetic geometry, commutative and noncommutative class field theories, zeta functions studied via zeta integrals, arithmetic measure and integration theory on higher dimensional objects. He is also interested in interfaces of number theory with representation theory, algebraic geometry, functional analysis, measure and integration theory, infinite group theory, mathematical logic and quantum physics.

Dr Kasprzyk works in algebraic geometry and computational algebraic geometry. He is one the leaders in the ongoing classification work of Fano varieties using Mirror Symmetry.

Dr Oblezin is a current EPSRC Fellow and is working on the project 'Topological field theories, Baxter operators and the Langlands programme'. This aims to discover new geometric and analytic structures in number theory and in automorphic forms.

Dr Strömberg works particularly on the computation of Maass waveforms (L2 eigenforms of the Laplace-Beltrami operator on hyperbolic surfaces). This involves a lot of high-performance computing as well as a deep and rich theory. Maass waveforms are objects of spectral theory of hyperbolic surfaces, but they are connected to both Analytic Number Theory and Quantum chaos.

Dr Wuthrich works on the arithmetic of elliptic curves, in particular on self-points, (computational) Iwasawa theory for elliptic curves, p-adic height pairings, Euler systems and modular points on elliptic curves.

### Visitors:

Alexander Beilinson (Univ. Chicago), Dorian Goldfeld (Columbia Univ.), Laurent Lafforgue (IHES), Olivia Caramello (Paris 6), Go Yamashita (RIMS)

Number Theory and Geometry Seminars

Number Theory and Geometry PhD Projects

## Thinking of joining us?

We are keen to create an environment which supports all staff members by providing:

## Contact

Professor Ivan Fesenko

School of Mathematical Sciences

University of Nottingham

University Park

Nottingham

NG7 2RD

telephone: +44 (0) 115 95 14952

e: ivan.fesenko@nottingham.ac.uk