Representation Theory and Moduli Spaces

这项研究是数学几个领域的交叉，包括表示理论、代数几何、数论和数学物理。特别是，将探索朗兰兹程序（起源于数论-可以说是数学中最古老的学科）与量子场论之间的新联系。在更技术性的方面，PI将研究附着在仿射Kac-Moody群上的各种模空间（如仿射Grassmannian、某些代数曲面上主束模空间的Uhlenbeck紧化等），以解决以下问题：1) p进群表示的局部l函数的构造，2)泛函域上仿射Kac-Moody群的Hecke代数和Eisenstein级数，3)四维规范理论中的瞬时计数等问题。

This research lies in the intersection of several fields of mathematics, including representation theory, algebraic geometry, number theory, and mathematical physics. In particular, new connections between the Langlands program (which originates in number theory - arguably the oldest subject in mathematics) and quantum field theory will be explored.On the more technical side the PI will study various moduli spaces attached to affine Kac-Moody groups (such as the affine Grassmannian, Uhlenbeck compactifications of moduli spaces of principal bundles on some algebraic surfaces, and some others) in order to attack the following problems: 1) a construction of local L-functions for representations of p-adic groups, 2) Hecke algebras and Eisenstein series for affine Kac-Moody groups over functional fields, and 3) instanton counting and other problems in 4-dimensional gauge theory.