College of Liberal Arts and Sciences

# Department of Mathematics

## Analysis and Probability

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Title: The Inverse Function Theorem in the Noncommutative Setting
Speaker: Mark E. Mancuso (Washington University in St. Louis)
Time: Friday, October 19, 2018 at 1:30 pm
Place: MONT 313Abstract: Classically, the inverse function theorem says that a $C^1$ function is locally invertible around a point of nonsingularity of its derivative. In this talk, we introduce the theory of noncommutative functions, which in the matrix case, are functions on a graded domain of tuples of matrices that preserve direct sums and similarities. We present inverse function-type theorems in this setting. Unlike in the classical case, noncommutative inverse function theorems are global: the derivative of a noncommutative function $f$ is invertible everywhere if and only if $f$ is invertible on its domain. We also discuss recent work on the inverse function theorem for operator noncommutative functions defined on domains sitting inside of $B(\mathcal{H})^d,$ for an infinite dimensional Hilbert space $\mathcal{H}$.

Organizer: Iddo Ben-Ari