Papers and preprints

Written under the supervision of William Fulton, this includes the results of papers 7. and 5. below, as well as some additional material about triality and more general symmetry for degeneracy locus problems.  This document also includes tables of formulas referred to in the other papers.

This is part of my Ph. D. thesis.  A map from a vector bundle to its dual is called symmetric if it is locally given by symmetric matrices; these maps are never sufficiently general to apply the usual Giambelli--Thom--Porteous formula for degeneracy, but there is another determinantal formula in this case, related to type C flag varieties.  In this paper, a new type of symmetry for vector bundle maps is defined, corresponding to the triality symmetry of Spin₈.  The degeneracy locus formulas for maps with this symmetry can be deduced from formulas for Schubert loci in the G₂ flag variety.

Based on similar results for other cohomology theories and a wealth of evidence, Griffeth--Ram and Graham--Kumar formulated conjectures about (sign-alternating) positivity in the equivariant K-theory of flag varieties.  We prove these conjectures, using techniques from Michel Brion's proof of the corresponding fact in ordinary K-theory, Sue Sierra's generalization of Kleiman transversality, and the group action for mixing spaces I used in my previous "Positivity" paper (two items down from this one).

This is part of my Ph. D. thesis.  Using linear algebra and octonions, the paper gives a description of the type G₂ flag variety similar to those of classical types, and one that naturally works in bundles.  Cohomology rings of these bundles are computed, along with formulas for Schubert loci in them, which should be thought of as universal G₂ degeneracy loci.

William Graham showed that the structure constants of the equivariant cohomology of a (generalized) flag variety are positive in the roots.  This note gives a short, geometric proof, based on a transversality argument.

Hessenberg varieties are special subvarieties of the flag variety.  In this article, we compute their cohomology classes, and find that they are certain specializations of double Schubert polynomials.  This motivates the combinatorial problem of expressing such specialized double Schubert polynomials in the basis of (single) Schubert polynomials; we give formulas for this decomposition in many cases.

This note discusses the occurrence of 1's as coefficients in the quantum cohomology of a Grassmannian.  One motivation for this work comes from a (yet to be developed) theory of "mod 2 real quantum cohomology"; in this context, the main result implies an analogue of a theorem of Fulton and Woodward about non-vanishing of quantum products.  However, the heart of the paper is a construction in the classical combinatorics of Young tableaux.

A conjecture of W. Neumann and J. Wahl states that every Q-Gorenstein singularity whose link is a Q-homology sphere has a complete intersection universal abelian cover.  This paper exhibits a cusp (as classified by its resolution graph) which serves as a counterexample to the conjecture when the hypothesis that the link be a Q-homology sphere is lifted.

Begun in 2001 during my junior year at Columbia, and supervised by Prof. Walter Neumann, this represents my first foray into the world of mathematical research.  I received invaluable comments on the preparation of the manuscript from Prof. Neumann and from the anonymous referee.

This is my undergraduate thesis, submitted to the Columbia University Department of Mathematics, and supervised by Prof. Doug Zare.  The main result is an enumeration of the tilings of certain plane regions known as Aztec rectangles; it turned out (to my slight embarrassment) that this enumeration was known, but for what it's worth, I have not seen the interpretation in terms of ASM-row paths elsewhere.  (ASM-row paths are lattice paths whose steps are given by rows of alternating-sign matrices.)