This research plan is divided into three projects in algebraic combinatorics and its applications to other areas of mathematics. The first project is concerned with several concrete aspects of Schubert calculus on generalized flag varieties. Its goal is to find combinatorial formulas for expressing the product of two Schubert classes (in cohomology) in the basis of Schubert classes; this is equivalent to counting points in a suitable triple intersection of Schubert varieties. The emphasis will be on the multiplication rule in the cohomology of the variety of complete flags in complex space. Many different approaches have been used in this area, but the only manifestly positive formulas that exist are limited to some easier special cases. The second project is concerned with the development of a simple combinatorial model (recently introduced by the investigator in collaboration with A. Postnikov) for the representation theory of complex semisimple Lie groups, as well as for the Chevalley-type multiplication formula in the equivariant K-theory of the corresponding generalized flag variety. The construction is based on combinatorics of decompositions in the corresponding affine Weyl group and enumeration of saturated chains in the Bruhat order on the (nonaffine) Weyl group. This model has several advantages over other combinatorial structures in the area, such as various specializations of the Littelmann path model. The new model will be extended in several directions, such as: the representation theory of Kac-Moody algebras, standard monomial theory, and the quantum K-theory of flag varieties. The connections with other models and a deeper study of the combinatorics involved will also be pursued. The third project is concerned with a combinatorial study of certain formal group laws related to topology. The main application is to certain immersion problems for lens spaces and projective spaces.
A unifying theme of the projects outlined here is the emphasis on combinatorics and computation. During the last decades, computation has gained an important role in mathematical research. This stimulated the development of combinatorics, as it became clear that combinatorial structures are particularly well suited for encoding complex mathematical objects, while combinatorial methods are well suited for related computations. This research plan is part of the ongoing effort to perform concrete computations, based on combinatorial structures. One of the main objects of study are generalized flag varieties. Although these are classical varieties, they feature prominently in current mathematical research due to their remarkable combinatorial complexity and the subtle interplay between various areas related to them. Examples of such areas relevant to the projects in this plan are: enumerative geometry (concerned with problems such as counting the lines or planes satisfying a number of generic intersection conditions, which are equivalent to performing certain cohomology calculations), and the geometric construction of representations of Lie groups (and, more generally, Kac-Moody groups). Flag varieties also provide a useful testbed for the development of combinatorial models relevant to computations in various cohomology theories of certain projective varieties.
