#math.nt

[arsphysica]

Geometrization of Trace Formulas. (arXiv:1004.5323 [math.RT])

Following our joint work arXiv:1003.4578 with Robert Langlands, we make the first steps toward developing geometric methods for analyzing trace formulas in the case of the function field of a curve defined over a finite field. We also suggest a conjectural framework of geometric trace formulas for curves defined over the complex field, which exploits the categorical version of the geometric Langlands correspondence.

arXiv:1004.5323 [math.RT]

[arsphysica]

The Berry-Keating Hamiltonian and the Local Riemann Hypothesis (arXiv:1104.1850 [math-ph])

The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2 projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.

arXiv:1104.1850 [math-ph]

[arsphysica]

The Work of Ngo Bao Chau. (arXiv:1012.0382 [math.NT])

In August 2010, Ngo Bao Chau was awarded a Fields Medal for his deep work relating the Hitchin fibration to the Arthur-Selberg trace formula, and in particular for his proof of the Fundamental Lemma for Lie algebras. This article gives a brief introduction to his work for a general mathematical audience.

arXiv:1012.0382 [math.NT]

[arsphysica]

> "In his paper "Beyond Endoscopy," Langlands tries to understand functoriality via poles of L-functions. The following paper further investigates the analytic continuation of a L-function associated to a $GL_2$ automorphic form through the trace formula. Though the usual way to obtain the analytic continuation of an L-function is through its functional equation, this paper shows that by simply assuming the trace formula, the functional equation of the L-function may be recovered. This paper is a step towards understanding the analytic continuation of the L-function at the same time as capturing information about functoriality. > From an analytic number theory perspective, obtaining the functional equation from the trace formula implies that Voronoi summation should in general be also a consequence of the trace formula."