University of Reading: Friday 4th April 2025

Directions: please see www.reading.ac.uk/about/visit-us

Lecture room: Ditchburn Lecture Theatre, JJ Thomson Building — west end of building 3 on this map — right next to the lecture theatre used for the last GWNT at Reading.


Register here by 21 March.

 

Schedule


Speakers


Sugata Mondal (Reading)


Title: Exceptional eigenvalue of hyperbolic surface.


Abstract: Selberg's 1/4 conjecture says that the first non-zero eigenvalue of the Laplacian of a congruence hyperbolic surface is at least one. The conjecture motivated the study of exceptional eigenvalues (i.e. eigenvalues below 1/4) for hyperbolic surfaces. It is not very difficult to construct hyperbolic surfaces (closed, finite or infinite volume) with many exceptional eigenvalues. In this talk, I will discuss the history and recent results around the counting of these eigenvalues.


Beth Romano (King's)


Title: Generalizing the LLL algorithm


Abstract: Given a lattice L in n-dimensional space, the Lenstra-Lenstra-Lovasz (LLL) algorithm gives a basis for L that is reduced, in a certain sense. I will talk about joint work with Jack Thorne in which we generalize this algorithm. More specifically, we can think about the problem of lattice reduction in terms of a symmetric space for the group SL_n. In our generalization we replace SL_n with an arbitrary split reductive group. In the talk, I will give some background about the classical LLL algorithm, and I will mention how the general setting appears in number-theoretic applications.


Chris Keyes (King's)


Title: Towards Artin's conjecture on p-adic quintic forms


Abstract: Let K be a p-adic field whose residue field has q elements and suppose f is a homogeneous polynomial of degree d in n+1 variables over K. A conjecture, originally due to Artin, states that when d is prime and n is at least d^2, f=0 has a nontrivial solution in K. This conjecture is known in degrees 2 and 3 due to Hasse and Lewis, respectively. It is also "asymptotically true," due to work of Ax and Kochen, in that it holds when q is sufficiently large with respect to d, though this is difficult to make effective. In this talk, we present recent joint work with Lea Beneish in which we prove the quintic version of the conjecture holds whenever q is at least 7. Our methods include both a refinement to a geometric approach of Leep and Yeomans (who showed q at least 47 suffices) and a significant computational component.


Min Lee (Bristol)


Title: Effective multiplicity one theorems


Abstract: The strong multiplicity one theorem (proved by Jacquet and Langlands for GL(2)) implies that if two Maass–Hecke cusp forms share the same Laplacian eigenvalue and the same Hecke eigenvalues for almost all primes, then the two forms must be equal up to a constant multiple. This naturally leads to the question: what is the minimal number of Hecke eigenvalues required to determine whether two Maass–Hecke cusp forms are proportional? Alternatively, one may ask: what is the dimension of the joint eigenspace for a given finite set of Hecke operators and the Laplace operator?


We address these two questions using two different methods. This is a joint work with Junehyuk Jung.