## About Me

I am an Assistant Professor in the Departments of Computer Science and Mathematics at the University of Colorado at Boulder, where I am a member of the CS Theory Group. My research has two main thrusts (with deep underlying relations beneath):

- Interactions between theoretical computer science and mathematics (particularly algebraic geometry, representation theory, and group theory), and
- Developing the theory of complex systems and complex networks, and applying this theory with my collaborators in a variety of fields, such as ecology, evolutionary biology, economics, climate, and beyond. I'm always looking for new problems that need new theory!

I was previously an Omidyar Fellow at the interdisciplinary Santa Fe Institute for complex systems. Prior to SFI, I was a postdoc in the University of Toronto CS Theory Group, and prior to that I got my Ph.D. at the University of Chicago.

## What's new

(July 23-26, 2018) Invited speaker at the Clay Math Institute/Oxford Workshop on Complexity Theory! Student support available.

(May 1, 2018) Giving a talk in the Topology Seminar in the CU Math Department on Computational topology and the Unique Games Conjecture.

(Mar 20, 2018) New paper posted: "Computational topology and the Unique Games Conjecture", to appear at SoCG 2018. Joint with Jamie Tucker-Foltz.

(Mar 16, 2018) Gave a talk in the CU Boulder Applied Math Colloquium: "Computational Complexity, Dynamical Systems, and Non-Convex Optimization."

(Mar 2, 2018) Gave a talk at the Rocky Mountain Algebraic Combinatorics Seminar: "Combinatorial Polytopes in Algebraic and Geometric Complexity Theory," based partially on these two papers.

(Feb 28, 2018) Presented "Computational Complexity and Mathematics" at the CU Boulder Math Club. With computer science, math, pizza, and drinks, how can you go wrong?

(Feb 16, 2018) "Computational topology and the Unique Games Conjecture" accepted to SoCG 2018. Joint with Jamie Tucker-Foltz.~~Preprint coming soon~~ UPDATE 3/20/2018: now available!

(Due Jan 31, 2018) We solicited applications for a Postdoctoral Fellowship beginning 2018! Algebraic/geometric complexity, spectral algorithms, unique games, and more with me and Alex Kolla. Check out our theory group.

(Jan 31, 2018) Gave a talk on Ideal Proof Systems at this Dagstuhl Workshop on Proof Complexity.

(Jan 12, 2018) Gave an invited talk at the AMS Current Events Bulletin, "The Cap Set Conjecture, the polynomial method, and applications (after Croot-Lev-Pach, Ellenberg-Gijswijt, and others)." You can find a write-up associated with the talk in the 2018 Current Events Bulletin booklet.

(Dec 22, 2017) Monotone projection lower bounds from extended formulation lower bounds finally appeared in Theory of Computing.

(Due Dec 15, 2017) I was seeking Ph.D. students, in computational complexity and/or complex systems & networks.

(Dec 7, 2017) New paper posted: Which groups are amenable to proving exponent two for matrix multiplication? Joint with Jonah Blasiak, Thomas Church, Henry Cohn, and Chris Umans. In which, among other things, we give a nonabelian generalization of the Croot-Lev-Pach-Ellenberg-Gijswijt method.

(Dec 5, 2017) Presented in the CU Boulder Math Lie Theory seminar: Representation theory and additive combinatorics in algorithms for matrix multiplication.

(Nov 14, 2017) Gave CU Boulder Kempner Colloquium in the Math Department: "Wildness & geometry in representation theory & computational complexity."

(Oct 23, 2017) New paper posted: Minimum circuit size, graph isomorphism, and related problems! Joint with Eric Allender, Dieter van Melkebeek, Cris Moore, and Andrew Morgan.

(Oct 6, 2017) Gave a talk at the Rocky Mountain Algebraic Combinatorics Seminar in Fort Collins: Representation theory and additive combinatorics in algorithms for matrix multiplication.

(Sep 1, 2017) New paper posted: Designing Strassen's algorithm, showing a clean proof of Strassen's algorithm using only a simple unitary 2-design in two dimensions. Joint with Cris Moore.

(Jul 7, 2017) The final, open access version of Algorithms for group isomorphism via group extensions and cohomology has now appeared in SIAM Journal on Computation! Joint with Youming Qiao. Open access forever thanks to funding from the Santa Fe Institute.

(Jun 30, 2017) I am now officially, by courtesy, also an Assistant Professor in the Department of Mathematics!

(Jun 23, 2017) Comparing information-theoretic measures of complexity in Boltzmann machines accepted to

(May 11, 2017) New paper posted: On the records, joint with the Santa Fe Institute postdocs. This paper was the result of our second (annual?) "72 Hours of Science" event, in which we conceived, developed, executed, and wrote up an entire scientific project in only 72 hours (albeit with a team of 15 postdocs)! See the Appendix of the paper for a little more detail. Comments and suggestions are more than welcome.

(Apr 18, 2017) On cap sets and the group-theoretic approach to matrix multiplication invited as a plenary talk at the STOC TheoryFest, which highlights some of the best CS theory work from any venue in the past two years!

See all past updates

(May 1, 2018) Giving a talk in the Topology Seminar in the CU Math Department on Computational topology and the Unique Games Conjecture.

(Mar 20, 2018) New paper posted: "Computational topology and the Unique Games Conjecture", to appear at SoCG 2018. Joint with Jamie Tucker-Foltz.

(Mar 16, 2018) Gave a talk in the CU Boulder Applied Math Colloquium: "Computational Complexity, Dynamical Systems, and Non-Convex Optimization."

(Mar 2, 2018) Gave a talk at the Rocky Mountain Algebraic Combinatorics Seminar: "Combinatorial Polytopes in Algebraic and Geometric Complexity Theory," based partially on these two papers.

(Feb 28, 2018) Presented "Computational Complexity and Mathematics" at the CU Boulder Math Club. With computer science, math, pizza, and drinks, how can you go wrong?

(Feb 16, 2018) "Computational topology and the Unique Games Conjecture" accepted to SoCG 2018. Joint with Jamie Tucker-Foltz.

(Due Jan 31, 2018) We solicited applications for a Postdoctoral Fellowship beginning 2018! Algebraic/geometric complexity, spectral algorithms, unique games, and more with me and Alex Kolla. Check out our theory group.

(Jan 31, 2018) Gave a talk on Ideal Proof Systems at this Dagstuhl Workshop on Proof Complexity.

(Jan 12, 2018) Gave an invited talk at the AMS Current Events Bulletin, "The Cap Set Conjecture, the polynomial method, and applications (after Croot-Lev-Pach, Ellenberg-Gijswijt, and others)." You can find a write-up associated with the talk in the 2018 Current Events Bulletin booklet.

(Dec 22, 2017) Monotone projection lower bounds from extended formulation lower bounds finally appeared in Theory of Computing.

(Due Dec 15, 2017) I was seeking Ph.D. students, in computational complexity and/or complex systems & networks.

(Dec 7, 2017) New paper posted: Which groups are amenable to proving exponent two for matrix multiplication? Joint with Jonah Blasiak, Thomas Church, Henry Cohn, and Chris Umans. In which, among other things, we give a nonabelian generalization of the Croot-Lev-Pach-Ellenberg-Gijswijt method.

(Dec 5, 2017) Presented in the CU Boulder Math Lie Theory seminar: Representation theory and additive combinatorics in algorithms for matrix multiplication.

(Nov 14, 2017) Gave CU Boulder Kempner Colloquium in the Math Department: "Wildness & geometry in representation theory & computational complexity."

(Oct 23, 2017) New paper posted: Minimum circuit size, graph isomorphism, and related problems! Joint with Eric Allender, Dieter van Melkebeek, Cris Moore, and Andrew Morgan.

(Oct 6, 2017) Gave a talk at the Rocky Mountain Algebraic Combinatorics Seminar in Fort Collins: Representation theory and additive combinatorics in algorithms for matrix multiplication.

(Sep 1, 2017) New paper posted: Designing Strassen's algorithm, showing a clean proof of Strassen's algorithm using only a simple unitary 2-design in two dimensions. Joint with Cris Moore.

(Jul 7, 2017) The final, open access version of Algorithms for group isomorphism via group extensions and cohomology has now appeared in SIAM Journal on Computation! Joint with Youming Qiao. Open access forever thanks to funding from the Santa Fe Institute.

(Jun 30, 2017) I am now officially, by courtesy, also an Assistant Professor in the Department of Mathematics!

(Jun 23, 2017) Comparing information-theoretic measures of complexity in Boltzmann machines accepted to

*Entropy*! Joint with my former REU student from SFI (now at Berkeley), Max Kanwal, and Nihat Ay.(May 11, 2017) New paper posted: On the records, joint with the Santa Fe Institute postdocs. This paper was the result of our second (annual?) "72 Hours of Science" event, in which we conceived, developed, executed, and wrote up an entire scientific project in only 72 hours (albeit with a team of 15 postdocs)! See the Appendix of the paper for a little more detail. Comments and suggestions are more than welcome.

(Apr 18, 2017) On cap sets and the group-theoretic approach to matrix multiplication invited as a plenary talk at the STOC TheoryFest, which highlights some of the best CS theory work from any venue in the past two years!

See all past updates