Geometric Methods in Representation Theory

首席研究员将致力于五个数学研究项目在一般领域的“李学理论和几何。第一个项目涉及解决著名的Valiant的猜想在几何复杂性理论断言，如果永久的一个方阵的大小n可以实现为一个线性投影的行列式的最小大小m的方阵，那么函数m增长速度比任何多项式n。第二个项目旨在解决Hitchin关于半单李代数的某些子空间上的原始伴随不变形式的非零性的猜想。 这对光滑射影曲线上半稳定主丛的模空间的几何有着有趣的影响。第三个项目（与E. Feigin（来自俄罗斯）和P. Littelmann（来自德国）的目的是研究半单李代数的任何有限维不可约表示（更一般地是Demazure模）的分次特征。第四个项目的目标是研究可对称化Kac-Moody李代数的饱和张量锥。这个项目是一个早期的项目的扩展，由PI与P. Belkale共同完成的半单李代数。第五个项目涉及完成关于``Verlinde公式的研究生教科书，其完整的证明和后果。“Verlinde公式的证明（最初由数学物理学家E. Verlinde）以及各种应用程序分散在文献中，没有一个单一的来源包含这些;这样一本书填补了现有文献中的空白。 这个数学研究项目汇集了数学的几个领域，包括拓扑学，组合数学，代数几何和表示论，预计将在数学和计算机科学中产生重大影响。更具体地说，第一个关于几何复杂性理论中Valiant猜想的项目对理论计算机科学中一个主要的未解决问题有着重要的影响，通常被称为库克的“P对NP”问题。非正式地，它问是否每个问题的解决方案可以快速验证的计算机也可以快速解决的计算机;它被认为是一个最重要的开放问题在该领域。此外，预计该解决方案将在该地区产生大量活动。主要研究者撰写的两本书（其中一本与M. Brion从法国）已成为标准文本的主题。PI正在努力完成一本关于Verlinde公式及其完整证明和后果的新研究生教科书。“这将是关于这个问题的第一本书;预计它将作为研究生和专业数学家和数学物理学家的基本来源，从而大大促进教学和学习。PI已经成功指导了六个博士学位。学生和一个硕士生;他目前正在指导两个博士。学生他在德国和美国共同组织了几次著名的国际会议。此外，他还在犹他州合作组织了一个跨学科的暑期学校，主要针对博士生。学生和博士后研究员在数学和计算机科学。

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.