Ulam's stability problem asks if every "almost" solution of an equation (i.e. an approximate solution with respect to some metric) is "close" to an exact solution. Extending the thoroughly studied theory of group stability, we study Ulam stability type problems for associative and Lie algebras. Namely, we investigate obstacles to rank-approximation of matrix "almost" solutions by exact solutions for systems of non-commutative polynomial equations.
This leads to a rich theory of stable associative and Lie algebras, with connections to linear soficity, amenability, growth, and group stability. We develop rank-stability and instability criteria, examine the effect of algebraic constructions on rank-stability, and prove that while finite-dimensional associative algebras are rank-stable, "most" finite-dimensional Lie algebras are not.
Joint work with Guy Blachar and Be'eri Greenfeld.
