Combinatorial Models in Algebra, Geometry, and Topology

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This research plan is divided into three projects in algebraic combinatorics andits applications to other areas of mathematics. The first project is concernedwith several concrete aspects of Schubert calculus on generalized flagvarieties. Its goal is to find combinatorial formulas for expressing the productof two Schubert classes (in cohomology) in the basis of Schubert classes; thisis equivalent to counting points in a suitable triple intersection of Schubertvarieties. The emphasis will be on the multiplication rule in the cohomology ofthe variety of complete flags in complex space. Many different approaches havebeen used in this area, but the only manifestly positive formulas that exist arelimited to some easier special cases. The second project is concerned with thedevelopment of a simple combinatorial model (recently introduced by theinvestigator in collaboration with A. Postnikov) for the representation theoryof complex semisimple Lie groups, as well as for the Chevalley-typemultiplication formula in the equivariant K-theory of the correspondinggeneralized flag variety. The construction is based on combinatorics ofdecompositions in the corresponding affine Weyl group and enumeration ofsaturated chains in the Bruhat order on the (nonaffine) Weyl group. This modelhas several advantages over other combinatorial structures in the area, such asvarious specializations of the Littelmann path model. The new model will beextended in several directions, such as: the representation theory of Kac-Moodyalgebras, standard monomial theory, and the quantum K-theory of flag varieties.The connections with other models and a deeper study of the combinatoricsinvolved will also be pursued. The third project is concerned with acombinatorial study of certain formal group laws related to topology. The mainapplication is to certain immersion problems for lens spaces and projectivespaces.A unifying theme of the projects outlined here is the emphasis on combinatoricsand computation. During the last decades, computation has gained an importantrole in mathematical research. This stimulated the development of combinatorics,as it became clear that combinatorial structures are particularly well suitedfor encoding complex mathematical objects, while combinatorial methods are wellsuited for related computations. This research plan is part of the ongoingeffort to perform concrete computations, based on combinatorial structures. Oneof the main objects of study are generalized flag varieties. Although these areclassical varieties, they feature prominently in current mathematical researchdue to their remarkable combinatorial complexity and the subtle interplaybetween various areas related to them. Examples of such areas relevant to theprojects in this plan are: enumerative geometry (concerned with problems such ascounting the lines or planes satisfying a number of generic intersectionconditions, which are equivalent to performing certain cohomology calculations),and the geometric construction of representations of Lie groups (and, moregenerally, Kac-Moody groups). Flag varieties also provide a useful testbed forthe development of combinatorial models relevant to computations in variouscohomology theories of certain projective varieties.