Stephen Rallis's work presents two methods to determine functoriality from a classical group to Gl(n). This is done by applying the Converse Theorem coupled with an effective theory of good L functions. The second method is by use of the relative trace formula. The ideas are based on an earlier theory of automorphic descent. He develops with the descent formalism ways to construct Cap modules for classical groups. Also the aforementioned theory of L functions allows a procedure to relate nonvanishing of Bessel periods to the nonvanishing of tensor L functions on classical groups.
2. For the general public
Stephen Rallis's research uses systematically the basic principles of symmetry to investigate fundamental problems in number theory. Incredibly basic problems such as the celebrated Fermat Last Conjecture require such ideas. Stephen Rallis develops various types of symmetry (in a mathematical way) that make up this subject. The main emphasis is to find new ways to show the nonnegativity of self dual automorphic L functions for Gl(n) and other classical groups at the center of symmetry of the functional equation. Also he develops new period conditions to analyze these special values.This is important for the various arithmetic, analytic and geometric ideas tied to nonvanishing of L functions at specific points. This work trains graduate students in mathematical research.
