The PI (Shrawan Kumar) intends to continue work in the general area of `Lie Theory and Geometry.' He proposes to work on four projects. The first project aims at settling the following problem on complete intersections raised by Kollar and Peskine in 1988. Let C(t) be a family of complex smooth curves in the three dimensional projective space such that the general member is a complete intersection. Then, is the special member also a complete intersection? The PI has converted this problem into a problem of the nonexistence of equivariant morphisms from the three dimensional affine space minus the origin to the instanton moduli space of SU(2) connections on a rank two bundle over the four dimensional Euclidean space modulo the based gauge equivalence. This moduli space has been extensively studied and the PI expects to use these results to solve the above problem. Let G be a semisimple algebraic group over the complex numbers and let s be a diagram automorphism of G with fixed subgroup L. The second project aims at determining the eigencone of L as well as the tensor product decomposition for the representations of L in terms of that of G. The PI and Belkale have formulated some precise conjectures to determine this. Once solved, it would give an optimal solution of the saturation problem for non-simplylaced groups provided the saturation holds for simplylaced groups. The third project (jointly with Arzu Boysal) aims at determining the fusion algebra explicitly. Let g be a simple Lie algebra. Fix a positive integer k (called the level). Let R(g) be the representation ring of g. The fusion ring R(g;k) (at level k) is a quotient ring of R(g) under a ring homomorphism f. Gepner determined its kernel for the special linear Lie algebra and it was subsequently determined by Bourdeau-Mlawer-Riggs-Schnitzer and Bouwknegt-Ridout for the symplectic Lie algebras. The PI and Boysal have come up with explicit conjectural description of the kernel of f for other classical groups as well as for the exceptional Lie algebra of type G(2). The PI proposes to prove this conjectural description and also find an analogous description of the kernel of f for all other exceptional groups. The fourth project aims at writing a book on ``Verlinde formula, its complete proof and consequences." E. Verlinde gave a conjectural formula in 1988 for the dimension of the space of conformal blocks. This space appears as a basic object in Rational Conformal Field Theory. In the special case of the Wess-Zumino-Witten model, the space of conformal blocks admits an interpretation as the space of generalized theta functions arising in the theory of moduli of vector bundles on projective curves. This interpretation opened a completely new horizon for the study of the moduli of vector bundles (more generally, principal G-bundles) on projective curves. These developments leading to the proof of the Verlinde formula as well as various applications are scattered through the literature and there is no single source containing these. The proposed book would fill this void in the literature.
The proposed projects underline the theme of unity in mathematics as they are expected to derive ideas from several areas of mathematics including Topology, Combinatorics, Algebraic Geometry, Representation Theory and Mathematical Physics. 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 has considerable expertise in the techniques he is proposing to use. The PI's proposed book on ``Verlinde formula, its complete proof and consequences" 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 alike thus substantially promoting teaching and learning. The two books written earlier by the PI (one coauthored with M. Brion) have become standard texts on the subject. The PI has successfully supervised six PhD students.
