Geometric Methods in Representation Theory

PI打算继续在“李论与几何”的一般领域开展工作。他提出了以下四个项目。设g是一个简单的李代数，设E是g的两个副本上的外部代数。然后，在伴随作用下，E有三个g的副本，它们的总次数是E的二项。设a是E的商代数除以由这三个g副本生成的理想。Cachazo-Douglas-Seiberg-Witten推测（并证明了经典g） A中不变量的逆变换代数是由A的唯一的二阶（1,1）不变元素产生的，它的对偶Coxeter数的幂为零。PI与麻省理工学院的P. Etingof教授共同开展的第一个项目的目的是全面证明这一猜想。他们在这个猜想和循环同源性之间提出了一个有趣的联系。与P. Belkale教授合作的第二个项目是由Klyachko， Fulton, Knutson-Tao， Berenstein-Sjamaar和Belkale教授的工作所激发的，这些工作与著名的霍恩猜想关于埃米特特征值问题及其推广到任何半单群有关。PI提出了一个问题，即在任意的部分标志群中，三个或三个以上的舒伯特环何时非平凡相交。他们提出了两种完全不同的可能解决方案；一个是抛物线列维分量L的表示理论，另一个是一个归纳公式（类似于霍恩对他的不等式集的归纳定义），它将确定三个舒伯特循环何时非平凡相交的问题简化为L群的相应问题，等等。他们已经取得了一些积极的成果。第三个课题是在主幂零对轨道闭包上的函数环上证明V. Ginzburg的一个猜想。利用几何不变理论，PI与来自Arhus的J. Thomsen教授共同将猜想简化为研究通过“关联半单对”的线的轨道闭合。此外，它们在决定这一关闭方面取得了相当大的进展。与来自罗马的C. Procesi教授合作的第四个项目涉及施普林格纤维的上同代数。提出将这些变量的等变上同调化为“显式”仿射变量的坐标环，然后通过特化得到奇异（非等变）上同调。在特殊线性群的情况下，它们已经解决了施普林格纤维等变上同调的仿射模型的求解问题。拟议的项目强调了数学统一的主题，因为他们预计将从几个数学领域获得思想，包括拓扑学，代数几何，表示理论和组合学。此外，第一个项目起源于数学物理，预计它的最终解决方案将以一种关键的方式使用数学物理的思想。PI建议的所有这些项目都代表了当前感兴趣的一些非常困难和重要的问题，如果这些问题得到解决，也将有助于澄清现有已知案例的结果。此外，预计这些解决方案将在该地区产生大量活动。PI和他的合作者在他提议使用的技术方面拥有相当多的专业知识，并且他们已经成功地使用了其中的一些技术来解决其他问题。所有的项目都是与来自美国和国外的其他数学家合作完成的。PI最近写了两本书。这些书应该作为研究生和专业数学家的基本资料。PI希望这两本书（都是各自领域的第一本）成为高级研究生教材，从而实质性地促进教与学。目前，PI有一名学生Arzu Boysal正在攻读博士学位。此前，他已成功指导了4名博士生。

ABSTRACTThe PI intends to continue work in the general area of `Lie Theory and Geometry.' He proposes the following four projects. Let g be a simple Lie algebra and let E be the exterior algebra on two copies of g. Then, under the adjoint action, E has three copies of g sitting in total degree two terms of E. Let A be the quotient algebra of E divided by the ideal generated by these three copies of g. Motivated from Supersymmetric Gauge Theory, Cachazo-Douglas-Seiberg-Witten conjectured (and proved for classical g) that the bigraded algebra of invariants in A is generated by the unique invariant element of A of bidegree (1,1) subject to the only relation that its power by the dual Coxeter number is zero. The aim of the first project undertaken by the PI jointly with Prof. P. Etingof from MIT is to prove this conjecture in full generality. They have come up with an interesting link between this conjecture and Cyclic Homology. The second project in collaboration with Prof. P. Belkale is motivated by the works of Klyachko, Fulton, Knutson-Tao, Berenstein-Sjamaar and Belkale related to the celebrated Horn's conjecture on the Hermitian eigen value problem and its generalization to any semisimple group. The PI proposes to study the problem of deciding when three or more Schubert cycles in an arbitrary partial flag variety intersect nontrivially. They have come up with two entirely different formulations of possible solution; one in terms of representation theory of the Levi component L of the parabolic, while the other is an inductive recipe (similar to Horn's inductive definition of his set of inequalities) which reduces the problem of deciding when three Schubert cycles intersect nontrivially to the corresponding problem for the group L and so on. Already some positive results have been obtained by them. The third project concerns proving a conjecture of V. Ginzburg on the ring of functions on the closure of the orbit of a principal nilpotent pair. By using Geometric Invariant Theory, the PI jointly with Prof. J. Thomsen from Arhus has reduced the conjecture to the study of orbit closure of the line passing through an `associated semisimple pair.' Moreover, they have made a considerable progress towards the determination of this closure. The fourth project in collaboration with Prof. C. Procesi from Rome deals with the cohomology algebra of Springer fibers. It is proposed to realize the equivariant cohomology of these varieties as the coordinate ring of an `explicit' affine variety and then obtain the singular (nonequivariant) cohomology by specialization. The problem of finding affine models for the equivariant cohomology of Springer fibers has already been solved by them for the case of special linear groups.The proposed projects underline the theme of unity in mathematics as they are expected to derive ideas from several areas of mathematics including Topology, Algebraic Geometry, Representation Theory and Combinatorics. In addition, the first project owes its origins in Mathematical Physics and it is expected that its final solution will have to use ideas from Mathematical Physics in a crucial manner. All of these projects suggested by the PI represent some of the very difficult and important problems of current interest which, if solved, should also help clarify the results in the existing known cases. In addition, it is expected that the solutions will spawn a lot of activity in the area. The PI and his collaborators have considerable expertise in the techniques he is proposing to use and they have successfully used some of these techniques earlier in solving other problems. All the projects are in collaboration with other mathematicians from within USA and abroad. The PI has recently written two books. These books should serve as a basic source for graduate students and professional mathematicians alike. The PI expects these books (both being the first in their areas) to become advanced graduate texts thus substantially promoting teaching and learning. Currently, PI has one student Arzu Boysal working for her PhD. He has successfully supervised four PhD students earlier.