| Speaker: Jurij Volcic Affiliation: Texas A&M University Time: 14:00 Tuesday, 21 May, 2019 Location: 303-257 |
| Let f be a noncommutative polynomial (an element of the free algebra in noncommuting variables, eg. 1-xy+yx+x^2). The free locus of f is the set of all tuples of matrices X such that f(X) is singular. That is, the free locus is an infinite family of determinantal hypersurfaces (one for each size of matrices). The adjective "free" relates to quickly emerging free analysis and free real algebraic geometry, which study noncommutative functions on matrices in a certain dimension-free setting. Free loci naturally arise in (noncommutative) control theory and convex optimization. In this talk we will relate them to factorization in free algebra: roughly speaking, components of the free locus of f correspond to distinct irreducible factors of f, and irreducible polynomials (atoms) are determined by their free loci. Special attention will be given to the role of invariant theory for the general (and special) linear group in the proofs. The talk is based on joint work with Bill Helton and Igor Klep. |