Assistant Professor of Computer Science • Ph. D. University of Chicago, 2012
Research interests:
Theoretical computer science, group theory, representation theory, algebraic geometry
Research:
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). In particular, this currently involves studying orbit closures of algebraic groups, using representations of finite groups and additive combinatorics to study algorithms for matrix multiplication, and studying representation theory and group cohomology in the context of algorithms for finite group (and graph, and other structures) isomorphism.
- Developing the rigorous mathematical 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!
Select Publications:
- New applications of the polynomial method: The cap set conjecture and beyond. Bull. AMS 56(1):29-64, 2019.
- Circuit complexity, proof complexity, and polynomial identity testing: The ideal proof system. With T. Pitassi. Journal of the ACM, 65(6), Article No. 37, November 2018. Preliminary version in FOCS '14.
- Boundaries of VP and VNP. With K. D. Mulmuley and Y. Qiao. Appeared in ICALP '16, submitted for journal publication. Preprint arXiv:1605.02815
- Unifying known lower bounds via geometric complexity theory. Open access. Computational Complexity, Special Issue, 24(2):393-475, 2015.
- Polynomial-time isomorphism test of groups that are tame extensions. With Y. Qiao. Preprint arXiv:1507.01917
- Which groups are amenable to proving exponent two for matrix multiplication? With J. Blasiak, T. Church, H. Cohn, and C. Umans. Preprint arXiv:1712.02302
- On cap sets and the group-theoretic approach to matrix multiplication. With J. Blasiak, T. Church, H. Cohn, E. Naslund, W. Sawin, and C. Umans. Discrete Analysis 2017:3.
- Matrix multiplication algorithms from group orbits. With C. Moore. Preprint arXiv:1612.01527