Theoretical computer science, group theory, representation theory, algebraic geometry
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!
- Boundaries of VP and VNP. With K. D. Mulmuley and Y. Qiao. Appeared in ICALP '16, submitted for journal publication. Preprint arXiv:1605.02815
- 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.
- Polynomial-time isomorphism test of groups that are tame extensions. With Y. Qiao. Preprint arXiv:1507.01917
- Circuit complexity, proof complexity, and polynomial identity testing. With T. Pitassi. Appeared in FOCS '14, submitted for journal publication. Preprint arXiv:1404.3820.
- Unifying known lower bounds via geometric complexity theory. Open access. Computational Complexity, Special Issue, 24(2):393-475, 2015.
- Matrix multiplication algorithms from group orbits. With C. Moore. Preprint arXiv:1612.01527
- 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