Winter 2019: Math 533 Abstract Algebra I
Fall 2018: Math 201 Linear Algebra
Fall 2018: Math 222 Combinatorics
Winter 2018: Math 533 Abstract Algebra I
Fall 2017: Math 222 Combinatorics
Fall 2016: Math 533 Abstract Algebra I
Winter 2016: Math 221 Discrete Mathematics
Winter 2015: Math 201 Linear Algebra
Winter 2015: Math 533 Abstract Algebra I

Spring 2013: 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, complexity theory, graph 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 did a three year postdoc at the University of Michigan under the mentorship of John Stembridge. I was a visiting professor for a year at University of Southern California. I am now an assistant professor at Drexel University.


I was on the job market 2012-2013. Here are my research statements:   Research Statement   Short Research Statement

In 2012 I received an NSF grant for the project quantizing Schur functors.


Catalan functions and k-Schur positivity, November 2018.
Catalan functions and k-Schur positivity, April 2018.
Kronecker coefficients and noncommutative super Schur functions, September 2016.
The Rule of Three for commutation relations, May 2016.
Noncommutative Schur functions, October 2015.
Kronecker coefficients for one hook shape, September 2012.
version with less background   version with more background


(with J. Morse, A. Pun, and D. Summers) $k$-Schur expansions of Catalan functions. Preprint, November 2018. PDF

(with J. Morse, A. Pun, and D. Summers) Catalan functions and $k$-Schur positivity. Preprint, April 2018. PDF

(with R. I. Liu) Kronecker coefficients and noncommutative super Schur functions. J. Combin. Theory Ser. A. 158, (2018), 315-361. PDF

(with S. Fomin) Rules of Three for commutation relations. J. Algebra. 500, (2018), 193--220. PDF

(with T. Church, H. Cohn, J. A. Grochow, C. Umans) Which groups are amenable to proving exponent two for matrix multiplication?. Preprint, December 2017. arXiv

(with T. Church, H. Cohn, J. A. Grochow, E. Naslund, W. F. Sawin, C. Umans) On cap sets and the group-theoretic approach to matrix multiplication. Disc. Analysis, no. 3 (2017). arXiv

(with S. Fomin) Noncommutative Schur functions, switchboards, and Schur positivity. Sel. Math., (2016), 1--40. PDF

Haglund's conjecture on 3-column Macdonald polynomials. Math. Z. 283, (2016), 601--628. PDF

What makes a D0 graph Schur positive? J. Algebraic Combin. 44, no. 3 (2016), 677--727. PDF
Data files: vertexset.txt   involutions.txt   involutionswithzeros.txt   maplegraph.txt   checkaxioms.txt

(with R. I. Liu and K. Mészáros) Subalgebras of the Fomin-Kirillov algebra. J. Algebraic Combin. 44, no. 3 (2016), 785--829. PDF

Kronecker coefficients for one hook shape. Sem. Lothar. Combin. 77, (2016), 40 pp. PDF

Representation theory of the nonstandard Hecke algebra. Algebras and Representation Theory (2014), 1--27. PDF

(with K. Mulmuley and M. Sohoni) Geometric complexity theory IV: nonstandard quantum group for the Kronecker problem. Mem. Amer. Math. Soc. 235(1109), (2015), ix--160. PDF

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

Nonstandard braid relations and Chebyshev polynomials. J. Algebra 423, (2015), 375--404. 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 $A_3$.