Title: Some Geometry Aspects of Non-Abelian Zeta Functions
1. Fokker-Planck Equations and Non-Abelian Zeta Functions
Abstract: Over the moduli space of rank $n$ semi-stable lattices $\La$ is a universal family of toruses. Along the fiber $\R^n/\La^\vee$ over $[\La]$, there are natural differential operators and differential equations, particularly, the heat equations, the Fokker-Planck equations in statistical mechanics, the Hamiltonians in quantum mechanics, and quantum harmonic oscillators. In this talk, we explain why, by taking averages over the moduli spaces, all these are connected with the zeros of rank $n$ non-abelian zeta functions of the field of rationals. Since a weak Riemann hypothesis is proved for these non-abelian zeta functions, all but finitely many non-abelian zeta zeros lie on the central line. We expect this would give some implications in both physics and mathematics.
2. Volumes of Moduli Spaces of Semi-Stable Lattices
Abstract: By a result of Siegel, the total volume of the moduli space of rank $n$ lattices is $\whz(2)\cdots\whz(n)$. In this talk, we investigate the volume of the submoduli space parametrizing semi-stable lattices. This is achieved via an analytic interpretation of parabolic reduction for the stability condition, an analogue of a result of Lafforgue for number fields. We will give an explicit formula for such a volume in terms of the Riemann zeta function $\whz(s)$, and the structures of parabolic subgroups of $\SL_n$. At the end of this talk, we will mention the application of this result to the establishment of a weak Riemann Hypothesis for non-abelian zeta functions.
报 告 人：翁 林 教授 日本九州大学 数理学研究院