报告人：Nicolas Templier (Cornell University)
地点：Room 1114, Sciences Building No. 1
Abstract: Gauss sums are the finite fields analogs of the Gamma function. In 1805, Gauss determined the
sign of quadratic Gauss sums, which is an analog of the well-known evaluation of Gamma(1/2) as the square
root of pi. The equidistribution of Gauss sums has only been proved in the 1970s by Deligne and Katz. The
proof uses hyper-Kloosterman sums which are the finite fields analogs of the confluent hypergeometric
functions, such as Bessel functions. In 2010, Heinloth-Ngo-Yun have defined generalized Kloosterman sums, which go beyond the already broad class of hypergeometric functions.
In a different area of mathematics, mirror symmetry has had spectacular applications to enumerative
geometry since the 1990s and is nowadays under active development at the crossroad of symplectic
and complex geometry. Rietsch's conjecture concerns mirror symmetry for flag manifolds, which
are an important class of Fano manifolds. It says that the quantum connection of a flag manifold should be isomorphic to the character D-module of a geometric crystal.
With Thomas Lam in 2017, we have proved Rietsch's conjecture for all Grassmannians Gr(k,n)
using generalized Kloosterman sums. Our work relates mirror symmetry and automorphic forms in a new way, giving more evidence for a program envisioned by Witten about twenty years ago.
The talk will keep the prerequisite knowledge to a minimum by introducing all of the above concepts with
the example of projective spaces. In fact we'll spend a significant portion of the talk discussing properties of Bessel functions because it gives insights on how our proof goes.