Many problems in enumerative geometry can be reduced to a computation in the cohomology ring of a flag manifold X = G/P. The structure constants of this ring are ruled by deep and beautiful combinatorics, and their study falls in the intersection of several mathematical disciplines. For example, when X is a Grassmann variety, these structure constants are the Littlewood-Richardson coefficients that also describe tensor products of representations of GL(n), products of symmetric polynomials, and play a role in numerous other areas ranging from linear algebra to statistics and complexity theory in computer science. The celebrated Littlewood-Richardson rule expresses any Littlewood-Richardson coefficient as the number of certain combinatorial objects called tableaux. A more general combinatorial formula, conjectured by Allen Knutson, states that the structure constants of a two-step flag variety are equal to the number triangular puzzles with specified integer labels on the sides. The investigator hopes to prove this conjecture. Earlier work of the investigator has established that the structure constants of two-step flag varieties specialize to the Gromov-Witten invariants of Grassmannians, and therefore count the number of rational curves of a fixed degree that meet three Schubert varieties in general position. The Gromov-Witten invariants also determine the structure of the(small) quantum cohomology ring, whose definition is inspired by physics and has relations to mirror symmetry. A proof of Knutson's conjecture will therefore establish the most precise description of this ring as a fact. The investigator will also study other questions concerning the K-theory and quantum K-theory of flag manifolds.
A typical question in classical algebraic geometry is to identify the complete list of geometric figures of some type that satisfy a list of conditions. While it can be difficult or impossible to identify the individual figures, it is in many cases possible to say how many there are. Enumerative geometry is the study of such counting problems as well as methods to solve them. Powerful techniques have been developed that can translate an enumerative geometric problem into an algebraic problem, so that the number of solution figures is the result of a computation. However, the combinatorial aspects of an enumerative problem are in most cases best understood in the presence of a formula that makes it clear that the number of solutions is non-negative. For example, such positive formulas are much more useful for proving general statements about which enumerative problems have any solutions at all. Surprisingly, positive formulas are significantly more difficult to discover and prove than non-positive formulas. In return the positive formulas tend to surround themselves with deep combinatorial structures and methods that provide even more insight into the geometric problem than the formulas themselves. The investigator will attempt to prove a number of positive formulas of this type. He also plans to write a computer program capable of computing the solutions of a large family of enumerative problems. Examples of this type are important for making progress in the field, and are at the same time very useful for students or others who would like to learn the subject. Finally, the investigator will continue to engage graduate and undergraduate students in his research.
