Royal Holloway - Number Theory Seminar

The NT seminar takes place on Wednesday at 4pm in Horton HLT1.
Here is a campus plan.

January 10: No seminar!

January 17: Martin Widmer (Royal Holloway)

"Average bounds for the $\ell$-torsion in class groups"

January 24: Sam Chow (York)

"Lonely runners in function fields"

Abstract. The lonely runner conjecture (1967) asserts that if m runners with distinct constant speeds run around a unit circle starting at a common time and place, then each runner will at some time be separated by a distance of at least 1/m from the others. We formulate the conjecture in a function field setting, and obtain some partial results. We interpret the question as a covering problem involving partial circulant matrices, and analyse the extremal problem using a combinatorial structure known as a sunflower. Joint with Luka Rimanić.

January 31: Christopher Frei (Manchester)

"A three primes theorem with Artin primes"

Abstract: Vinogradov's celebrated theorem states that every sufficiently large odd integer is the sum of three primes. A conjecture of Artin, proved under GRH by Hooley, asserts that any given integer which is neither -1 nor a perfect square is a primitive root for infinitely many primes p. In this talk, we discuss recent work that combines the results of Vinogradov and Hooley, studying representations of odd integers as sums of three primes, which all have prescribed primitive roots. This is joint work with E. Sofos and P. Koymans (Leiden).

February 7: Reynold Fregoli (Royal Holloway)

"Sums of reciprocals of fractional parts"

Abstract: Sums of reciprocals of fractional parts have played a central role in number theory since the early 1900s. In this talk we present some results obtained by Le and Vaaler in 2013 for generalised sums of reciprocals of fractional parts. These results though, are conditional, since a strong generalised version of bad approximability is  assumed for the linear forms involved. We shall show how to weaken this constraint and to make their result unconditional in a certain number of cases.

February 14: Sofia Lindqvist (Oxford)

"Monochromatic solutions to $x+y=z^2$"

Abstract: We show that given any 2-colouring of the natural numbers, there are infinitely many monochromatic solutions to x+y=z^2. The proof uses various results from additive combinatorics and analytic number theory. In particular we make use of the arithmetic regularity lemma of Green to find a long monochromatic arithmetic progression. By assuming that we have no monochromatic solutions to x+y=z^2 we can then apply an iterative argument to find longer and longer monochromatic arithmetic progressions, until we eventually reach a contradiction. This is joint work with Ben Green.

February 21: Djordjo Milovic (UCL) 

"Spins of prime ideals and 16-ranks of class groups in thin families"

Abstract: We will associate a "spin" to each ideal in certain quadratic number rings, and we will show how a number-field version of Vinogradov's method (a sieve involving "sums of type I" and "sums of type II") can be used to prove that spins of prime ideals oscillate. Such equidistribution results have applications to the distribution of 16-ranks of class groups in certain one-prime-parameter families of quadratic number fields.

February 28: No seminar!

March 7: No seminar!

March 14: No seminar!

March 21: 
No seminar!

April 26: Victor Beresnevich (York)

April 30: Philip Dittmann (Oxford)

Autumn 2017

Spring 2017

Summer 2016

Spring 2016

Autumn 2015

Spring 2015

Autumn 2014