Geometric Methods in Representation Theory

首席研究员将在“测谎论和几何学”的一般领域开展五个数学研究项目。第一个项目涉及几何复杂性理论中著名的Valiant猜想的一个解决方案，该猜想断言如果一个n阶方矩阵的恒等式可以实现为一个最小尺寸m的方阵行列式的线性投影，则函数m的增长速度比n中的任何多项式都快。第二个项目旨在解决Hitchin关于限制在半单李代数的某些子空间上的本原伴随不变形式不为零的猜想。这对光滑射影曲线上的半稳定主丛的模空间的几何具有有趣的意义。第三个项目(与俄罗斯的E.Feigin和德国的P.Littelmann合作)旨在研究半单李代数的任何有限维不可约表示(更广泛地说是Demazure模)的分次特征。第四个项目的目的是研究可对称化Kac-Moody李代数的饱和张量锥。这个项目是早先由PI和P.Belkale共同完成的关于半单李代数的项目的推广。第五个项目涉及完成一本关于Verlinde公式及其完整证明和结果的研究生教科书。Verlinde公式的证明(最初是由数学物理学家E.Verlinde猜想给出的)以及各种应用散布在文献中，没有包含这些的单一来源；这样的一本书填补了现有文献的空白。这个数学研究项目汇集了几个数学领域，包括拓扑学、组合学、代数几何和表示论，预计将在数学和计算机科学方面产生重大影响。更具体地说，几何复杂性理论中关于Valiant猜想的第一个项目对理论计算机科学中的一个重大悬而未决的问题具有重要的影响，该问题通常被称为Cook的`P vs NP‘问题。非正式地，它询问是否每一个能被计算机快速验证其解的问题也能被计算机快速解决；它被认为是该领域最重要的公开问题之一。此外，预计该解决方案将在该地区催生大量活动。由首席调查员撰写的两本书(其中一本是与来自法国的M.Brion合著)已成为这一主题的标准文本。国际和平研究所正在努力完成一本关于弗林德公式及其完整证明和结果的新的研究生教科书。这将是关于这一主题的第一本书；预计它将成为研究生以及专业数学家和数学物理学家的基本资料来源，从而大大促进教和学。他已经成功指导了6名博士生和1名硕士生；目前他正在指导2名博士生。他在德国和美国共同组织了几个享有盛誉的国际会议。此外，他还在犹他州共同组织了一个跨学科暑期班，主要面向数学和计算机科学的博士后和博士后。

The principal investigator will work on five mathematics research projects in the general area of `Lie Theory and Geometry.' The first project concerns a solution of the celebrated Valiant's conjecture in Geometric Complexity Theory asserting that if the permanent of a square matrix of size n can be realized as a linear projection of the determinant of a square matrix of smallest size m, then the function m grows faster than any polynomial in n. The second project is aimed at solving a conjecture of Hitchin on the nonvanishing of primitive adjoint invariant forms restricted to certain subspaces of a semisimple Lie algebra. This has interesting implications on the geometry of the moduli spaces of semistable principal bundles on smooth projective curves. The third project (jointly with E. Feigin from Russia and P. Littelmann from Germany) aims at studying the graded character of any finite-dimensional irreducible representation (more generally of a Demazure module) of a semisimple Lie algebra. The aim of the fourth project is to study the saturated tensor cone for symmetrizable Kac-Moody Lie algebras. This project is an extension of an earlier project for the semisimple Lie algebras completed by the PI jointly with P. Belkale. The fifth project concerns the completion of a graduate textbook on ``Verlinde formula, its complete proof and consequences." The proof of the Verlinde formula (which was initially conjecturally given by the mathematical physicist E. Verlinde) as well as various applications are scattered through the literature and there is no single source containing these; such a book fill a void in the existing literature. This mathematics research project brings together several areas of mathematics including Topology, Combinatorics, Algebraic Geometry and Representation Theory, and is expected to have significant consequences in Mathematics and in Computer Science. More specifically, the first project on the Valiant's conjecture in Geometric Complexity Theory has significant implications towards a major unsolved problem in Theoretical Computer Science commonly referred to as Cook's ``P versus NP" problem. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer; it is considered to be one the most important open problems in the field. In addition, it is expected that the solution will spawn a lot of activity in the area. Two books authored by the principal investigator (one of these coauthored with M. Brion from France) have become standard texts on the subject. The PI is working towards the completion of a new graduate textbook on ``Verlinde formula, its complete proof and consequences." This would be the very first book on the subject; it is expected that it will serve as a basic source for graduate students and professional mathematicians and mathematical physicists alike thus substantially promoting teaching and learning. The PI has successfully supervised six Ph.D. students and one Master's student; he is currently supervising two Ph.D. students. He has coorganized several prestigious international conferences in Germany and USA. In addition, he is coorganizing an interdisciplinary summer school in Utah aimed mainly at Ph.D. students and postdoctoral fellows in Mathematics and Computer Science.