# Research Summary

My work pursues a new approach to identification of causal mechanisms for locations of zeros of certain automorphic L-functions. As quipped by Hilbert and Polya, the Riemann Hypothesis would follow from existence of a self-adjoint operator with eigenvalues s(s-1) for zeros s of zeta(s), since s(s-1) real implies s is on the critical line (or real). The obvious difficulty being that the situation is lacking in candidate operators. A spark of hope appeared when Haas (1977) miscalculated eigenvalues for the invariant Laplacian on Maass' waveforms and obtained zeros of zeta and another L-function in his list of s-parameters. Hejhal (1981) diagnosed the numerical procedural error and observed that the actual operator involved was not quite self-adjoint. Colin de Verdiere (1982-3) showed how to make a genuinely self-adjoint operator plausibly related to the problem. The status of Colin de Verdiere's speculation was unclear until recent work of Bombieri and Garrett gave a precise formulation in terms of distributions and Sobolev spaces -- traditionally used in analysis and PDEs. Very recently, they seem to have shown that the original speculation definitively fails but that the mechanism of this failure sheds light on the behavior of self-dual L-functions.

# Papers

*Differential equations in automorphic forms* (to appear in Dec 2018 in Communitcations in Number Theory and Physics arXiv 1801.00838)

*A spectral interpretation of zeros of certain functions* (under revision at Journal of Number Theory: arXiv 1706.08552)

*Self-Adjoint Operators and Zeros of L-functions* (in preparation)

Thoughts on this matter: Self-Adjoint Operators and Zeros of L-functions

# Talks

32nd Automorphic Forms Workshop (March 2018) A Spectral interpretation of Zeros of Certain Functions

31st Automorphic Forms Workshop (March 2017) A Spectral interpretation of Zeros of Certain Functions

30th Automorphic Forms Workshop (March 2016) Self-Adjoint Operators and Zeros of L-functions Talk

# Notes

Notes on Spectral Theory: Spectral Theory