Project Details
Description
The research of the PI (Miller) focuses on the analytic theory of
automorphic forms. The proposed research concentrates on two analytic
tools in the subject. The first is the trace formula. The PI hopes to
sharpen Arthur's trace formula for GL(n) so that it can be applied to
analytic questions in the same fashion that Selberg's original trace
formula for GL(2) has been. One of the expected applications is to
counting various types of automorphic forms, for example a general Weyl
law for arithmetic quotients of reductive Lie groups. The second tool is
that of automorphic L-functions. The PI and his coworker Wilfried Schmid
(Harvard University) are engaged in a research program to study
automorphic forms using the boundary distributions of eigenfunctions on
symmetric spaces. These boundary techniques allow new constructions of
L-functions, and offer a new way to investigate many problems in
automorphic forms. The proposed research involves developing this
technique and its applications.
The study of automorphic forms slices across many important areas of
modern mathematical research, including number theory, representation
theory, geometry, analysis, and mathematical physics. Through
L-functions, Langlands has conjectured many deep and interesting
structural relationships between automorphic forms which have implications
in the above areas. As an example, the work of Wiles et al demonstrates
the link between certain automorphic forms and the ancient problem of
solving equations between squares and cubes. The proposed research aims
to apply and develop new tools for automorphic forms and L-functions from
analysis, which is the branch of mathematics expanding calculus, and
representation theory, the concrete study of symmetry. Current
applications of automorphic forms and L-functions are manifest in
constructing the sophisticated codes which enable high-speed and secure
transactions over the internet.
Status | Finished |
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Effective start/end date | 7/15/01 → 6/30/04 |
Funding
- National Science Foundation: $69,190.00