Mathematical Sciences: Geometry of Flag Varieties and Representation Theory

摘要Kumar 9622887 库马尔将继续工作在一般领域的谎言理论和几何。第一个项目涉及研究一个特定的复杂的BGG决议仿射Kac-Moody李代数的抛物模拟。 不存在的某些不可约组件在这个复杂的同源性将导致一个有趣的几何定义的融合产品的正水平可积表示的仿射卡茨-穆迪代数。此外，这将导致一个证明的显式维数公式Verlinde任意半单单连通群（已知迄今为经典群和G2）。第二个项目涉及证明三维射影空间中曲线的完全相交的一个众所周知的问题，这转化为与SL（2）相关的仿射标志簇的问题。设C是光滑仿射代数曲线，G是半单群.第三个项目是证明所有从C到G的代数映射的空间Alg（C，G）与所有连续映射（从C到G）的相应空间Cont（C，G）之间的弱同伦等价。由于Cont（C，G）的有理同伦群是已知的，特别地，这种等价将确定Alg（C，G）的有理同伦群。最后，库马尔正在为一门关于“仿射卡茨-穆迪群，它们的旗形变种和表示理论”的课程写讲义，他打算将其扩展成一本涵盖基本理论的书。 在许多数学和物理现象中（类似地在许多生物和化学过程中），对称性起着重要的作用。首先，球体的例子很有启发性。任何球体的一个基本性质是它是一个完全对称的物体，换句话说，它上面的任何两点“看起来”都是一样的。这种对称性在一段时间内被用来推导球体的一些相当复杂的性质。 同样，让我们看另一个来自物理学的例子（更确切地说，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，即“物理定律在所有相对于彼此均匀运动的坐标系中都是相同的”（以及光速的不变性）。Kumar提出的各种项目都有一个共同的主题：利用手头问题的对称性来找到问题的解决方案。更确切地说，库马尔试图使用“自同构群”来解决不同的数学问题，这反过来又会对理论物理学产生重要影响，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.