Lectures & Talks

[PMI Number Theory Seminar] - Numerical Computations with the Selberg trace formula

Date: 2017 Apr 10 18:00 - 19:00
Speaker: Min Lee (University of Bristol)
Place: Math. Bldg. 404
첨부파일: poster_Min Lee.pdf
PMI Number Theory seminar

Speaker: Dr Min Lee (University of Bristol)
Date and Time: April 10 (Monday)
6pm-7pm Math build room 404


Title: Numerical Computations with the Selberg trace formula

Abstract:
Among the most well-known examples of L-functions are the Riemann zeta
function and the L-functions associated to classical modular forms.
Less well known, but equally important, are the L-functions associated
to Maass forms, which are eigenfunctions of the Laplace-Beltrami
operator on a hyperbolic surface. Named after H. Maass, who discovered
some examples in the 1940s, Maass forms remain largely mysterious.

Fortunately, there are concrete tools to study Maass forms: trace
formulas, which relate the spectrum of the Laplace operator on a
hyperbolic surface to its geometry. After Selberg introduced his
famous trace formula in 1956, his ideas were generalised, and various
trace formulas have been constructed and studied. However, there are
few numerical results from trace formulas, the main obstacle being
their complexity. Various types of trace formulas are investigated,
constructed and used to understand automorphic representations and
their L-functions from a theoretical point of view, but most are not
explicit enough to implement in computer code.

Having explicit computations of trace formulas makes many potential
applications accessible. In this talk, I will explain the
computational aspects of the Selberg trace formula for GL(2) for
general levels and applications towards the Selberg eigenvalue
conjecture and classification of 2-dimensional Artin representations
of small conductor.