Seminar Global Analysis

Dr. Jasmin Matz and Dr. Boris Vertman

Winter semester 2013

Talks

31.10.2013	Jasmin Matz Title: Weyl law for Hecke operators of imaginary quadratic number fields
	Abstract: The classical Weyl law gives an asymptotic for the function counting the Laplacian eigenvalues on a compact manifold ordered according to their size. Lapid and Müller studied the refined question of the asymptotic distribution of the infinitesimal characters of the cuspidal representations of GL(n) over the rationals \Q. They also proved an upper bound on the remainder term. A natural generalisation of this is to consider an operator acting on the cusp forms of GL(n) which commutes with all invariant differential operators (these are the Hecke operators). The question then is to find an asymptotic of the distribution of the infinitesimal characters weighted by the eigenvalues of this operator, and to establish an upper bound for the remainder depending on the operator norm in an explicit way. In this and the forthcoming talks I want to explain the recent progress on this problem for the case of GL(n) over imaginary quadratic number fields. In the first talk I want to explain the number theoretic motivation behind such a quantitative Weyl law for Hecke operators, namely its connection to the theory of low-lying zeros of families of automorphic L-functions, and to state the result.
7.11.2013 14:00 s.t.!	Jasmin Matz Title: Weyl law for Hecke operators of imaginary quadratic number fields II
	Abstract: This is a continuation of the last talk in which I want to explain the main ingredients of the proof of the Weyl law for Hecke operators over imaginary quadratic number fields. In particular, I will recall Arthur's trace formula and explain how it is used in the proof.
14.11.2013	Jasmin Matz Title: Weyl law for Hecke operators of imaginary quadratic number fields III
	Abstract: I want to explain what the objects and concepts used in the previous talks mean in classical (non-adelic) language for SL(2) over the rational numbers Q.
21.11.2013	Boris Vertman Title: Elliptic theory of edge degenerate differential operators
	Abstract: We develop a theory for solving boundary value problems for certain edge degenerate elliptic differential operators. In the first lecture we will review the classical theory of elliptic boundary value problems which will then motivate the constructions in the degenerate case. This will be the content of the second and third lectures.
5.12.2013	Boris Vertman Title: Elliptic theory of edge degenerate differential operators (II)
	Abstract: We develop a theory for solving boundary value problems for certain edge degenerate elliptic differential operators. This second lecture focuses on elements of the b-calculus, while the third lecture will evolve around theory of edge operators.
12.12.2013	Boris Vertman Title: Elliptic theory of edge degenerate differential operators (III)
	Abstract: We develop a theory for solving boundary value problems for certain edge degenerate elliptic differential operators. This third and last lecture focuses on elements of the edge-calculus and boundary value problems.

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