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.
