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Texas A&M University

Noncommutative Geometry Seminar

Date: November 22, 2019

Time: 3:00PM - 4:00PM

Location: BLOC 624

Speaker: Ilya Kachkovskiy, Michigan State University


Title: Almost commuting matrices

Abstract: Suppose that $X$ and $Y$ are two self-adjoint matrices with the commutator $[X,Y]$ of small operator norm. One would expect that $X$ and $Y$ are close to a pair of commuting matrices. Can one provide a distance estimate which only depends on $\|[X,Y]\|$ and not on the dimension? This question was asked by Paul Halmos in 1976 and answered positively by Huaxin Lin in 1993 by indirect C*-algebraic methods, which did not provide any explicit bounds. It was conjectured by Davidson and Szarek that the distance estimate would be of the form $C\|[X,Y]\|^{1/2}$. In the talk, I will explain some background on this and related problems, and the main ideas of the proof of this conjecture, obtained jointly with Yuri Safarov. If time permits, I will discuss some current work in progress.