Department of Mathematics
University of California
Berkeley, CA 94720-3840
I received my Ph.D. in May 2011 from the math department at UC Berkeley. My advisor was David Eisenbud. My thesis was titled Secant Varieties of Segre-Veronese Varieties. I am interested in algebraic geometry, commutative algebra and their computational aspects. Here is my CV.
- The GSS Conjecture,
We prove a conjecture of Garcia, Stillman and Sturmfels, which states that the ideal of the first secant variety of a Segre product of projective spaces is generated by 3x3 minors of flattenings. We introduce techniques similar to the ones in ``3x3 Minors of Catalecticants'', providing a new strategy for analyzing the ideals and coordinate rings of secant varieties to Segre varieties.
- 3x3 Minors of Catalecticants,
We answer a question of Geramita, by proving the equality of the ideals of 3x3 minors of the ``middle'' catalecticant matrices. This was predicted by Macaulay's theorem on the growth of the Hilbert function of an Artin algebra, and was known to hold for the special case of Hankel matrices. The techniques we introduce give a new perspective on the problem of determining the ideals of secant varieties to Veronese varieties.
- Affine Toric Equivalence Relations are Effective,
(Proc. Amer. Math. Soc. 138: 3835-3847, 2010. PDF or
We prove that if X is an affine toric variety, and R is an equivalence relation on X, preserved by the diagonal torus action, then R comes from a toric map X->Y. Moreover, if R is finite then there exists a geometric quotient X/R. We also show that the Amitsur complex associated to a map of monoid rings (defined at the monoid level) is exact. This gives a new class of ring extensions, besides the augmented and faithfully flat ones, for which the Amitsur complex has no nontrivial cohomology.
- Appendix to János Kollár, Quotients by Finite Equivalence Relations,
We answer a question of Kollár, by providing an example of a noneffective finite equivalence relation R on a two-dimensional affine space X. In other words, R is not constructed as the fiber product of X with itself over a finite base. We construct our example by employing an interesting relationship between equivalence relations and the cohomology of Amitsur complexes.
M2 code that verifies the assertions made in this appendix.
Here is my custom version (with documentation) of the Macaulay2 SchurRings package. Note that this is not (yet) incorporated in Macaulay2.
I also wrote a package implementing the push forward functor for finite ring maps.
Posters and Slides
- Poster on the GSS conjecture, from WAGS, Nov 6-7, 2010 at University of Arizona, Tucson.
- Slides on finite equivalence relations, from the Eighth AMS-SMM International Meeting, June 4, 2010 at UC Berkeley.
- Poster on finite equivalence relations, from WAGS, May 1-2, 2010 at UBC, Vancouver.
Notes compiled together with Enric Nart during the ``Computing Integral Closure'' Workshop at MSRI, July 2010.
Mathematics Research Communities report on Boij-Söderberg Theory in the nonstandard graded case, with B. Barwick, J. Biermann, D. Cook II, W. F. Moore and D. Stamate, Snowbird, June 2010.
Spring 2011 : Math 16B
Fall 2010 : Math 54
Spring 2010 : Math 1B
Fall 2009 : Math 1B