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Number Theory

Number Theory

The Number Theory Group at Royal Holloway has a broad range of interests.

The research interests of the group include analytic number theory (circle method, sieve methods, exponential sums), arithmetic statistics (distribution of algebraic and integral points, class groups, special integers, etc), Diophantine geometry (abelian varieties, heights), Diophantine approximation, geometry of numbers, computational and combinatorial number theory.

To find out more about the specific research interests of the Number Theory  Group members see “Members and Interests” below.

The group has collaborations with many international research teams  in various countries, including USA, Germany, France, Austria, Denmark, and Sweden,  and we regularly host visiting researchers. Jointly with the Univ. of Reading, King’s College, Queen Mary and UCL, the Number Theory Group organises ERLASS (Egham-Reading-London Arithmetic Statistics Seminar), and it is also part of CANTA (Centre for Combinatorial Methods in Algebra, Number Theory and Applications) at Royal Holloway.

 

Professor Rainer Dietmann:

I work at the interface of Analytic Number Theory and Diophantine equations. My current research interests include probabilistic
Galois theory, applications of the circle method to systems of Diophantine equations, including investigating their solubility over
local fields, lines on rational hypersurfaces, gaps between arithmetically defined sequences and additive combinatorics.

 

Professor James McKee:

My research interests lie at the interface of number theory and combinatorics, with computational leanings. My recent work has mostly been connected with Pisot numbers, Salem numbers, other algebraic numbers whose Galois conjugates are geometrically constrained, and associating algebraic numbers to certain combinatorial objects.

 

Dr Eira Scourfield (Emeritus):

Arithmetic functions such as Euler’s function and the sum-of-divisors function have been studied for centuries. My current research aims to derive asymptotics for the number of divisors and exact divisors up to y of these functions at primes up to x with y large in terms of x.

 

Professor Martin Widmer:

I work on the theory of height functions, on problems related or motivated by Lehmer’s conjecture. For instance, I am trying to understand which fields of algebraic numbers have infinitely many elements of bounded arithmetic complexity (Weil height), and the implications to other subjects such as logic (decidability), Diophantine geometry, and “exotic” Diophantine approximation. I am also interested in counting various algebraically defined objects of arithmetic interest, and, more generally, to study their distribution (arithmetic statistics). Currently, I am focussing on problems related to the distribution of  class groups of number fields. 

More details about my research can be found on my personal webpage.

Most of our publications can be found here.

 

All members of the number theory group are experienced PhD supervisors and active members accept new PhD students. Please contact the members of staff you are interested in working with and apply to our PhD programme.

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