Operator Algebras and Dynamics Seminar
All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.
Oct 10Diagonality and Idempotents with Applications to Problems in Operator Theory and Frame Theory Prof. Gary Weiss University of Cincinnati Time: 03:45 PM
View Abstract(Joint work with Jireh Loreaux.) We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis. Our proof depends on work of Fan-Fong-Herrero on diagonals of operators. We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent. We frame this work in a broader unifying context we call diagonality and explore historic and current work in this area. For instance, these characterizations extend to infinite dimensions the work of Giol, Kovalev, Larson, Nguyen and Tener on diagonals of idempotents.
Sep 15The Kadison-Singer Problem (part 2) Vrej Zarikian USNA Time: 03:45 PM
View AbstractWe continue to develop the tools required for Marcus, Spielman, and Srivastava's proof of the Kadison-Singer Problem. In particular, real stability and the mixed characteristic polynomial.
Sep 05The Kadison-Singer Problem (part 1) Vrej Zarikian USNA Time: 03:45 PM
View AbstractThe Kadison-Singer Problem was a famous open problem in the theory of operator algebras, posed in 1959. It was recently (and completely unexpectedly) solved (in the affirmative) by three computer scientists: Adam Marcus, Daniel Spielman, and Nikhil Srivastava. Their proof is both novel and (in a sense) elementary. In this series of lectures I will give a detailed account of their proof. The first lecture will explain the problem as well as some of its many reformulations. I will also introduce and explore the properties of "real stability" for complex polynomials, a key ingredient of the proof.
Aug 21On diagonal actions of branch groups and corresponding characters Artem Dudko Stony Brook Time: 03:45 PM Abstract