Enumeration in algebra and geometry

This thesis is devoted to solution of two classes of enumerative problems. The first class is related to enumeration of regions of hyperplane arrangements. We investigate deformations of Coxeter arrangements. In particular, we prove a conjecture of Stanley on the numbers of regions of Linial arrangements. These numbers have several additional combinatorial interpretations in terms of trees, partially ordered sets, and tournaments. We study a more general class of truncated affine arrangements, counting their regions, giving formulas for their Poincare polynomials, and proving a “Riemann hypothesis” on location of zeros of the latter. In addition, we find a couple of new interpretations for the Catalan numbers. The second class of problems comes from enumerative algebraic geometry and Schubert calculus and is related to Gromov-Witten invariants of complex flag manifolds. We present a method for their calculation using a new construction for the quantum cohomology ring of the flag manifold. This construction provides quantum analogues of results of Bernstein, Gelfand, and Gelfand on this subject and of the theory of Schubert polynomials of Lascoux and Schutzenberger. The quantum version of Monk’s formula is established, and a general Pieri-type formula is derived. While being remote from each other at first glance, both these subjects can be attacked with algebraic and combinatorial methods. Thesis Supervisor: Richard P. Stanley Title: Professor of Applied Mathematics

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