Operator Algebras and Dynamics Seminars

Fall 2019

All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.

• Nov

  04
• Norming for Discrete Crossed Products

  Vrej Zarikian

  Time: 03:45 PM

  Abstract

• Oct

  28
• Unique Extension Problems for C*-Inclusions

  Vrej Zarikian

  Time: 03:45 PM

  Abstract

• Oct

  21
• Mixing on random rank-one transformations

  Darren Creutz

  USNA

  Time: 03:45 PM

• Sep

  30
• Staircase transformations are mixing

  Darren Creutz

  USNA

  Time: 03:45 PM

  View Abstract

  Ergodic theory includes, among other things, the study of iterated maps of [0,1] to itself in terms of probability. I will introduce various mixing notions that such maps can possess and discuss a method of constructing examples of maps with various mixing properties, culminating in a presentation of my work, joint with C. Silva, that general staircase transformations are mixing. Two weeks later I will continue the presentation on more recent work in these directions.

• Sep

  16
• On the Koopman operator of algorithms

  Felix Dietrich

  Johns Hopkins University & Princeton University

  Time: 03:45 PM

  View Abstract

  Most numerical algorithms are acting iteratively on a state variable. In this case, it is possible to treat the algorithm as a dynamical system, where the variable parametrizing integral curves is the iteration number instead of the time. This conceptual shift allows us to employ tools usually used for the study of dynamical systems in the study of numerical algorithms. One powerful tool is the Koopman operator associated to the system, which encodes the evolution of observables on the state. The operator acts linearly on the function space of observables, and thus is amenable to spectral decomposition. Its spectrum allows insight into the qualitative behavior of the underlying system, and even quantitative predictions are possible if enough eigenfunctions of the operator are available. In this talk, we explore the qualitative properties of several important algorithms in numerical optimization through the spectrum of their associated Koopman operator. We provide analytical formulations of the spectrum and eigenfunctions for the Newton-Raphson algorithm on the complex plane, and discuss a numerical approximation of the continuous spectrum for chaotic algorithms. Literature: https://arxiv.org/abs/1907.10807