L-functions, monoids and Bessel functions

Freydoon Shahidi
Purdue University
October 1, 2016
This is a survey of how the existing theories of L-functions are in agreement with Braverman-Kazhdan/Ngo's construction of L-functions which generalizes that of Godement-Jacquet. One hopes that such insights may play a role in understanding the corresponding Fourier transform and Poisson formula which are of particular interest in functoriality.

Asymptotics for Hecke eigenvalues with improved error term

Jasmin Matz
Universität Leipzig
October 1, 2016
Asymptotics for the distribution of Hecke eigenvalues in families of automorphic forms are useful to study families of L-functions, provided one has a sufficiently good estimate for the error term of the asymptotic. I want to present ongoing joint work with T. Finis in which we give an upper bound for the error term with an explicit dependence on the Hecke operator. For certain applications an improvement of this bound would be necessary, but to go beyond our bound it seems necessary to remove part of the geometric side of the trace formula.

Beyond endoscopy and geometric terms

James Arthur
University of Toronto
October 1, 2016
Beyond Endoscopy is the proposal by Langlands for using the trace formula to attack the Principle of Functoriality. Many of the problems at this (early) stage concern the geometric terms in the trace formula. Most immediate perhaps is to extend Altug's application of Poisson summation from $\mathrm{GL}(2)$ to more general groups. One could then try to identify the contribution of the nontempered representations on the spectral side to the resulting Fourier transforms of geometric terms.

Packaging the construction of Kuranishi structure on the moduli space of pseudo-holomorphic curve

Kenji Fukaya
Stonybrook University
October 4, 2016
This is a part of my joint work with Oh-Ohta-Ono and is a part of project to rewrite the whole story of virtual fundamental chain in a way easier to use. In general we can construct virtual fundamental chain on (basically all) the moduli space of pseudo-holomorphic curve. It depends on the choices. In this talk I want to provide a statement to clarify which is the data we need to start with and in which sense the resulting structure is well defined. A purpose of writing such statement is then it can be a black box and can be used without looking the proof.

Derivation of the Vlasov equation

Peter Pickl
Ludwig-Maximilians-Universität München
October 4, 2016
The rigorous derivation of the Vlasov equation from Newtonian mechanics of $N$ Coulomb-interacting particles is still an open problem. In the talk I will present recent results, where an $N$-dependent cutoff is used to make the derivation possible. The cutoff is removed as the particle number goes to zero. Our result holds for typical initial conditions, only. This is, however, not a technical assumption: one can in fact prove deviation from the Vlasov equation for special initial conditions for the system we consider.