Mathematical Sciences: Geometry of Flag Varieties and Representation Theory

摘要库马尔9622887库马尔将继续在李理论和几何的一般领域工作。第一个项目是研究仿射Kac-Moody李代数的BGG分解的抛物型模拟所产生的一个复形。该复的同调中某些不可约分支的不存在将导致仿射Kac-Moody代数的正水平可积表示的融合积的一个有趣的几何定义。此外，这也将导致对任意半单单连通群(目前已知的经典群和G2)的Verlinde的显式维数公式的证明。第二个项目涉及证明三维射影空间中曲线的完全交集的一个众所周知的问题，该问题转化为与SL(2)有关的仿射标志簇问题。设C为光滑仿射代数曲线，G为半单群。第三个项目是证明从C到G的所有代数映射的空间Alg(C，G)与从C到G的所有连续映射的对应空间Cont(C，G)之间的弱同伦等价。特别地，由于Cont(C，G)的有理同伦群是已知的，这个等价性将决定Alg(C，G)的有理同伦群。最后，库马尔正在为“仿射Kac-Moody群、它们的旗帜变种和表示理论”这门课写讲稿，他打算把这门课扩展成一本涵盖基本理论的书。在一些数学和物理现象中(类似地，在许多生物和化学过程中)，对称性扮演着重要的角色。首先，球体的例子很有启发性。任何球体的一个基本属性是它是一个完全对称的物体，换句话说，它上的任何两个点看起来都是一样的。这种对称性在一段时间内被用来推导出球体的一些相当复杂的性质。同样，让我们看看另一个来自物理学的例子(更准确地说，来自爱因斯坦的狭义相对论)。爱因斯坦从基本的“不变性”假设中推导出了他的基本方程式，即“物理定律在所有相对于彼此匀速运动的坐标系中都是相同的”(连同光速的不变性)。库马尔提议开展的各种项目都有一个共同的主题：利用手头问题背后的对称性来找到问题的解决方案。更准确地说，库马尔正在试图使用“自同构群”来解决不同的数学问题，这反过来将在理论物理中产生重要的结果，特别是量子场论。

Abstract Kumar 9622887 Kumar will continue work in the general area of lie theory and geometry. The first project involves studying a certain complex which arises from the parabolic analog of the BGG resolution for affine Kac-Moody Lie algebras. Non-existence of certain irreducible components in the homology of this complex will lead to an interesting geometric definition of the fusion product for positive-level integrable representations of affine Kac-Moody algebras. Also, this will lead to a proof of the explicit dimension formula of Verlinde for arbitrary semisimple simply-connected groups (known so far for the classical groups and G2). The second project involves proving a well known problem on complete intersections of curves in the three dimensional projective space, which translates into a problem about the affine flag variety associated to SL(2). Let C be a smooth affine algebraic curve and G a semisimple group. The third project is concerned with proving a weak homotopy equivalence between the space Alg(C,G) of all the algebraic maps from C to G and the corresponding space Cont(C,G) of all the continuous maps (from C to G). Since the rational homotopy groups of Cont(C,G) are known, in particular, this equivalence would determine the rational homotopy groups of Alg(C,G). Finally Kumar is writing lecture notes for a course on "affine Kac-Moody groups, their flag varieties and representation theory," which he intends to expand into a book covering the basic theory. In several mathematical and physical phenomena (and similarly in many biological and chemical processes), symmetry plays an important role. To start with, the example of a sphere is illuminating. One basic property of any sphere is that it is a perfectly symmetrical object, in other words any two points on it "look" the same. This symmetry was exploited over a period of time to derive some rather intricate properties of spheres. Similarly let us look at another example coming from physics (more precisely from Einstein's Special Theory of Relativity). Einstein derived his very fundamental equations from the basic "invariance" postulate that "the laws of physics are the same in all coordinate systems which move uniformly relative to one another" (together with the invariance of the velocity of light). Various projects, which Kumar is proposing to work on, have one common theme: Exploit the symmetry underlying the problem at hand to find the solution of the problem. More precisely expressed, Kumar is trying to use the "automorphism groups" to solve different mathematical problems, which in turn will have important consequences in theoretical physics, particularly Quantum Field Theory.