Pure Maths PhD topics

Dr Nigel Byott
Hopf orders in group algebras in characteristic $p$.

This project is at the interface between number theory and algebra. Hopf orders in group algebras are of interest in studying Galois module structure for rings of integers. If K is a complete field of characteristic $0$, and G is a group of order $p$ or $p^2$, the Hopf orders in the group algebra $K[G]$ can be described explicitly, but there is as yet no corresponding result even for cyclic groups of order $p^3$. The case where $K$ has characteristic $p$ has not been systemically studied, but the arithmetic in this situation, while less familiar, is in some ways simpler. The aim of this project would be to find analogues in characteristic p of the known results in characteristic 0, and then to see if they could be extended to larger groups.

Dr Robin Chapman
Constructing lattices via combinatorial matrices

Lattices in Euclidean spaces can be constructed in a systematic way using skew-Hadmard matrices and the arithmetic of imaginary quadratic fields [1]. For example the Leech lattice, central in the theory of finite simple groups, can be easily constructed using Paley's skew-Hadamard matrix of order 24. However more general classes of combinatorial matrices can be used in place of skew-Hadamard matrices.
Also quaternion algebras can take the place of imaginary quadratic fields. This project will consider these matrices, and the lattices they generate, from both a theoretical and computational viewpoint.

[1] Robin Chapman: Conference matrices and unimodular lattices. European J. Combin. 22 (2001), 1033--1045.

Professor Andreas Langer
Displays over ramified Witt vectors and p-divisible groups

Over a ring on which a prime p is nilpotent formal p-divisible groups can be described by the category of Displays which were invented by Thomas Zink. The project is about an extension of this result to Displays over ramified Wiit vectors and p-divisible groups with O_K-action.
The project requires a very good knowledge of local fields and some knowledge in algebraic geometry which can be acquired in MAGIC-courses.

[1] Thomas Zink: The display of a formal p-divisible group, Asterisque 278, 2002.
[2] Michiel Hazewinkel: Twisted Lubin-Tate Formal Group Laws, ramified Witt vectors and Artin-Hasse Exponentials, transactions of the AMS, Vol 259, No1, 1980 [3] Jean-Pierre Serre: Local Fields

Dr Mohamed Saidi
Rational points and arithmetic fundamental groups

A fundamental problem in number theory and Diophantine geometry is the study of rational points of algebraic varieties. Twenty five years ago Grothendieck formulated what is now known as the "section conjecture", which establishes a precise dictionary between rational points of hyperbolic curves over number fields and splittings of their arithmetic fundamental groups. If true, this conjecture would establish an unprecedented bridge between Diophantine geometry and profinite group theory. Recently, a lot of progress has been made on the p-adic local analogue, in characteristic zero, of the section conjecture by Pop, Tamagawa-Saidi, and Mochizuki-Hoshi.
The topic I propose is to investigate the p-adic local analogue of the section conjecture in characteristic p>o for non-isotrivial hyperbolic curves.
This project requires a good background in algebraic number theory and Galois theory. The prospective PhD student would have to learn basic Galois cohomology and algebraic geometry.

More possible PhD research topics.