Written things
Notes, etc.

Papers and preprints

20. "Chern class formulas for classical-type degeneracy loci," with William Fulton, available at arXiv:1504.03615.

After working on item 15. (below), we found a streamlined approach to the problem, which is simultaneously simpler and more general.  This article describes the results emphasizing a geometric perspective.

19. "Effective divisors on Bott-Samelson varieties," available at arxiv:1501.00034.

In general, it is not easy to describe the set of all line bundles on a projective variety which possess nonzero sections.  For Bott-Samelson varieties, which are certain towers of P1-bundles, the problem is non-trivial but tractable, as explained here.

18. "The Lie algebra of type G2 is rational over its quotient by the adjoint action," with Mathieu Florence and Zinovy Reichstein, C. R. Math. Acad. Sci. Paris 351 (2013), no. 23-24, 871-875.  Also available at arXiv:1308.5940.

Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein asked when the function field k(g) is generated by algebraically independent elements over the field of invariants k(g)G; they showed the answer is positive when G is a split simple group of type An or Cn and negative in all other types, except possibly G2.  Here we prove the assertion of the title, settling the remaining case.

17. "Operational K-theory," with Sam Payne, Documenta Math. 20 (2015) 357-399.  Also available at arXiv:1301.0425.

This project started with a very simple goal.  Motivated by discussions with our fellow participants at a HIM workshop, we wanted to show that the ring of piecewise exponential functions on an arbitrary fan is isomorphic to the "operational equivariant K-theory" of the corresponding toric variety.  (It was known that for smooth toric varieties, this ring computes the equivariant K-theory of vector bundles.)  After developing the foundational aspects of this operational theory, we found that it turned out to have many nice geometric properties for general varieties and schemes.

16. "Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians," with Ed Richmond and Alex Yong, Compositio Math. 149 (2013), no. 9, 1569-1582.  Also available at arXiv:1210.5003.

We extend the saturation property of Littlewood-Richardson coefficients to the structure constants of the equivariant cohomology of the Grassmannian, and connect the result to an eigenvalue problem --- thus giving a kind of equivariant extension of the Horn conjecture.

15. "Degeneracy loci, Pfaffians, and vexillary signed permutations in types B, C, and D," with William Fulton, available at arXiv:1210.2066.

Building on work of Kazarian and Ikeda-Mihalcea-Naruse, we identify a class of signed permutations whose corresponding double Schubert polynomials have simple Pfaffian formulas.  We also give simplified, geometric proofs of the degeneracy locus formulas in classical types.

14. "Okounkov bodies of finitely generated divisors," with Alex Kuronya and Victor Lozovanu, Int. Math. Res. Not. 2014, no. 9, 2343-2355.  Also available at arXiv:1206.2499.

We show that for a wide class of big divisors on a variety X, one can choose a flag so that the associated Okounkov body is a rational simplex.  Under more restrictive hypotheses, we show that the corresponding semigroup is finitely generated, as well.

13. "Schubert varieties are log Fano over the integers," with Alan Stapledon, Proc. Amer. Math. Soc. 142 (2014), no. 2, 409-411.  Also available at PDF.

The title sums it up.  We did this in response to a question of Karl Schwede and Karen Smith.

12. "Positivity of equivariant Gromov-Witten invariants," with Linda Chen, Math. Res. Lett. 22 (2015) 1-9.  Also available at arXiv:1110.5900.

Using the same equivariant moving lemma as in item 11., we show that equivariant Gromov-Witten invariants exhibit Graham-positivity.  (Here we work with homogeneous spaces for general reductive groups, while our other paper deals only with type A.)

11. "Equivariant quantum Schubert polynomials," with Linda Chen, Adv. Math. 254 (2014) 300-330. Also available at arXiv:1110.5896.

In the 1990's, Ciocan-Fontanine and Chen showed that certain specializations of Fulton's universal double Schubert polynomials represent Schubert classes in the quantum cohomology of a partial flag variety.  We generalize those results to equivariant quantum cohomology; the key new tool is an equivariant moving lemma using the group action introduced in item 4. below. 

10. "Okounkov bodies and toric degenerations," Math. Ann. 356 (2013), no. 3, 1183-1202.  Also available at arXiv:1001.4566.

Consider a smooth projective variety X and a divisor D.  Building on work of Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata showed how to associate to this setup a convex body that reflects much of the geometry.  This paper gives a criterion for this "Okounkov body" to be polyhedral; when the criterion is satisfied, X admits a flat degeneration to the corresponding toric variety.  Examples include Schubert varieties and their Bott-Samelson desingularizations.

9. "Arc spaces and equivariant cohomology," with Alan Stapledon, Transformation Groups 18, no. 4 (2013), 931-969.  Older versions available here PDF and here arXiv:0910.2316.

The arc space of a smooth variety X is an infinite-dimensional space parametrizing germs of maps of a curve into X.  The construction is functorial, so if a group acts on X, it acts naturally on the arc space as well.  The arc space comes with a projection to X which is a homotopy equivalence, and this project stems from the simple observation that one can replace the equivariant cohomology of X with that of its arc space.  Remarkably, one gets a lot of mileage out of this point of view.

Alan and I got started on these ideas while sharing office 1852 as graduate students at the University of Michigan.  Thanks 1852!

8. "Degeneracy loci and G2 flags," Ph. D. thesis, University of Michigan, 2009. PDF   (This re-formatted version saves some paper.)

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.

7. "Degeneracy of triality-symmetric morphisms," Algebra & Number Theory 6 (2012), 689-706.  Also available at arXiv:0901.1347.

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 loci, 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.

6. "Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces," with Steve Griffeth and Ezra Miller, J. Eur. Math. Soc. 13 (2011), 57-84.  Also available at arXiv:0808.2785.

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).

5. "Chern class formulas for G2 Schubert loci" PDF, Trans. Amer. Math. Soc. 363 (2011), 6615-6646.  Also available at arXiv:0712.2641.

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.

4. "Positivity in the cohomology of flag bundles (after Graham)" PDF or arXiv:0711.0983.

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.

3. "Schubert polynomials and classes of Hessenberg varieties," with Julianna Tymoczko, J. Algebra 323 (2010), 2605-2623.  Also available at arXiv:0710.3182.

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.

2. "A note on quantum products of Schubert classes in a Grassmannian," J. Algebr. Comb. 25 (2007) 349-356.  Available from SpringerLink or at math.CO/0608546.

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.

1. "A cusp singularity with no Galois cover by a complete intersection," Proc. Amer. Math. Soc. 132 (2004), 2517-2527.  PDF

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.

0. "Alternating sign matrices and tilings of Aztec rectangles"  PDF

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.)