UF Analysis Seminar
https://ufl.zoom.us/j/97491684396
9/10 James Eldred Pascoe 4PM USET
Title: Algebraic geometry and topology arising from tracial and determinantal free noncommutative functions
Abstract: We give an elementary method determining whether or not two expressions in noncommuting indeterminants have the same determinant when evaluated on matrices. Our method uses the canonical implementation of the Weil divisor given by the logarithmic derivative. The divisor of a free rational function is always the difference of two polynomial divisors, and hence, via factorization results of Helton-Klep-Volcic, the set of rational divisors form a free abelian group generated by the polynomials. Determining which functions arise as principal divisors, or more generally as gradients of tracial free functions, motivates a theory of tracial cohomology and fundamental groups. The natural fundamental group is given by the equivalence classes of “paths” taking some base point to itself which can be distinguished by analytic continuation. Suprisingly, the tracial fundamental group is abelian and isomorphic to a direct sum of copies of the rationals. Finally, time permitting, we will note that our methods can be adapted to a sheaf theoretic formulation and work on thin sets, such as noncommutative varieties, and hence bear insight into problems on domains of commuting tuples of matrices.
9/17 Kelly Bickel 5PM USET
Matrix Monotonicity in the Quasi-Rational Setting
In this talk, we will show quasi-rational functions on the bidisk D^2 (intuitively, inner functions that are rational in one of the variables) yield functions that preserve matrix inequalities on rectangles in R^2. This result provides a partial answer to a 2012 conjecture posed by Agler, McCarthy, and Young about local-to-global matrix monotonicity of two-variable functions. The main tools include (1) concrete realization formulas with nice boundary behavior, (2) formulas that allow one to change realization domains without adding unnecessary singularities, and (3) recent advances in non-commutative function theory. This is joint work with J.E. Pascoe and Ryan Tully-Doyle.