Math 217 Linear Algebra

Fall 2012: Math 565 Combinatorics and Graph Theory
Winter 2012: Math 566 Combinatorial Theory
Fall 2011: Math 565 Combinatorics and Graph Theory

About Me

My research interests include algebraic combinatorics, representation theory, graph theory, complexity theory, and algebraic geometry.

I graduated from UC Berkeley under the direction of Mark Haiman. I did a one year postdoc at the University of Chicago with Ketan Mulmuley on complexity theory and the Kronecker problem. I am now in my third year of a three year postdoc at the University of Michigan.

I recently received an NSF grant for the project quantizing Schur functors.

I am now applying for tenure track positions.
Research Statement
Short Research Statement


Kronecker coefficients for one hook shape, September 2012.


Kronecker coefficients for one hook shape. Revised preprint, June 2013. PDF

Representation theory of the nonstandard Hecke algebra. Preprint, January 2012. PDF

(with K. Mulmuley and M. Sohoni) Geometric complexity theory IV: nonstandard quantum group for the Kronecker problem. Revised preprint, April 2013; accepted in Memoirs of the AMS. PDF

Quantum Schur-Weyl duality and projected canonical bases. J. Algebra. 402, (2014), 499--532. PDF

Nonstandard braid relations and Chebyshev polynomials. Revised preprint, May 2014; accepted in Journal of Algebra. PDF

An insertion algorithm for catabolizability. European J. Combin. 33, no. 2 (2012), 267--276. PDF

Cyclage, catabolism, and the affine Hecke algebra. Adv. Math. 228, no. 4 (2011), 2292--2351. PDF

W-graph versions of tensoring with the Sn defining representation. J. of Algebraic Combin. 34, no. 4 (2011), 545--585. PDF

A factorization theorem for affine Kazhdan-Lusztig basis elements. Preprint (2009). PDF

The toric ideal of a graphic matroid is generated by quadrics. Combinatorica 28, no. 3 (2008), 283--297. PDF

(with A. Berglund and P. Hersh) Combinatorics of multigraded Poincaré series for monomial rings. J. Algebra. 308, no. 1 (2007), 73--90. PS

A special case of Hadwiger's conjecture. J. Combin. Theory Ser. B 97, no. 6 (2007), 1056--1073. PDF
A longer version that was my senior thesis: PDF

(with R. Durrett) Random Oxford graphs. Stochastic Process. Appl. 115, no. 8 (2005), 1257--1278. PDF


Cyclage, catabolism, and the affine Hecke algebra. (2009), Advisor: Mark Haiman. PDF

Other Writings

Cohomology of the complex Grassmannian. An expository paper for the final in Hutchings' algebraic topology class. PDF

Longest common subsequences and the Bernoulli matching model: numerical work and analyses of the R-reach simplification. For my spring semester undergraduate junior paper. PDF

Magma Data

Kazhdan-Lusztig coefficients:
These files cbparabolic give certain canonical basis elements of $V^{\otimes r}$ in terms of the monomial basis. The file labeled by the partition $\nu$ contains all the canonical basis elements corresponding to the Yamanouchi words with content $\nu$. This data is in magma-readable format. All computations are done over the finite field $\mathbb{F}_{100003}$. The Kazhdan-Lusztig coefficients are given in several formats. See the end of the file for the most human-readable format.

Magma Code

combinatorics.txt   Lots of functions from algebraic combinatorics including cyclage, catabolizability, and the standardization map of Lascoux and Schützenberger.

affineHecke.txt  Some messy, slow, but quite general code to compute canonical bases for iterated restriction and inductions. Supports affine Weyl group computations in type A and was written to work for all types but it does not yet do so. Computes cells and the partial order on cells in terms of tableaux.

affineHeckeuserRes3H4.txt  An example using affineHecke.txt to compute the cells of
$\text{Res}_{H_K} \text{Ind}_{H_J}^H$ triv, where $K = \{s_1, s_2\}, J = \{s_2\}$, and $H$ is the Hecke algebra of type $A_3$.