Speaker: Josh Grochow, University of Chicago
Title: Matrix Lie Algebra Isomorphism and Geometric Complexity Theory
Testing whether two Lie algebras of matrices are isomorphic is a fundamental algorithmic problem in algebra with potential applications to several problems in complexity theory, such as matrix multiplication, formula size, and the Mulmuley--Sohoni Geometric Complexity Theory program, as well as problems in areas as diverse as differential equations, particle physics, and group theory. We start from the beginning, with the definition of Lie algebras, and through lots of examples will try to give a feel for what Lie algebras are and how they behave. We show that certain cases of Matrix Lie Algebra Isomorphism are equivalent to Graph Isomorphism, and that the problem can be viewed as a "non-abelian" generalization of the Linear Code Equivalence problem. We also show how to solve other cases in polynomial time, and then apply these algorithms towards a derandomization of a recent randomized algorithm of Kayal for testing when a function can be gotten from the determinant by a linear change of variables. We will also touch on how Matrix Lie Algebra Isomorphism fits into the larger Geometric Complexity Theory program. The talk will not require any prior background on Lie algebras.