Abstract:
Suppose two matrices $A,B$ almost commute, in the sense that their commutator $AB-BA$ has small rank. Can we perturb $A$ and $B$ by small-rank matrices to obtain two commuting matrices? We study this stability property for general systems of polynomial equations over matrices, in terms of the algebra defined by the equations. This leads to a rich theory of stable associative and Lie algebras, with connections to linear soficity, amenability, growth, and group stability.
Based on joint work with Tomer Bauer and Be'eri Greenfeld.