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Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of non- linear dynamic behavior (e.g. through normal forms). In this presentation we will argue that the use of the Koopman operator and its spectrum are particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven machine learning algorithmic developments. Recall that the Koopman operator describes the dynamics of observation functions along a flow or map, and it is formally the adjoint of the Frobenius-Perrron operator that describes evolution of densities of ensembles of initial conditions. The Koopman operator has a long theoretical tradition but it has recently become extremely popular through numerical methods such as dynamic mode decomposition (DMD) and variants, for applied problems such as coherence and also in control theory. We demonstrate through illustrative examples that we can nontrivially extend the applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards a systematic discovery of rectifying integrability coordinate transformations.
Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of non- linear dynamic behavior (e.g. through normal forms). In this presentation we will argue that the use of the Koopman operator and its spectrum are particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven machine learning algorithmic developments. Recall that the Koopman operator describes the dynamics of observation functions along a flow or map, and it is formally the adjoint of the Frobenius-Perrron operator that describes evolution of densities of ensembles of initial conditions. The Koopman operator has a long theoretical tradition but it has recently become extremely popular through numerical methods such as dynamic mode decomposition (DMD) and variants, for applied problems such as coherence and also in control theory. We demonstrate through illustrative examples that we can nontrivially extend the applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards a systematic discovery of rectifying integrability coordinate transformations.


[https://math.yale.edu/people/john-schotland John Schotland] (Yale University)
'''[https://math.yale.edu/people/john-schotland John Schotland] (Yale University)'''


Nonlocal PDEs and Quantum Optics
Nonlocal PDEs and Quantum Optics

Revision as of 06:32, 9 September 2023


Applied and Computational Mathematics Seminar


Fall 2023

date speaker title host(s)
Sep 8 Erik Bollt (Clarkson University) A New View on Integrability: On Matching Dynamical Systems through Koopman Operator Eigenfunctions Chen
Sep 15 4:00pm B239 John Schotland (Yale University) Nonlocal PDEs and Quantum Optics Li
Sep 22 Balazs Boros (U Vienna) Craciun
Sep 29 Peter Jan van Leeuwen (Colorado State University) Chen
Wed Oct 4 Edriss Titi (Cambridge/Texas A&M) Distringuished Lecture Series Smith, Stechmann
Oct 6 No Friday seminar Distinguished lecture this week on Wednesday
Oct 13 Da Yang (University of Chicago) Smith
Oct 20 Yuehaw Khoo (University of Chicago) Li
Oct 27 Shukai Du (UW) Stechmann
Nov 3 Lise-Marie Imbert-Gérard (University of Arizona) Rycroft
Nov 10 Timothy Atherton (Tufts) Chandler, Spagnolie
Nov 17 Daphne Klotsa Rycroft
Nov 24 Thanksgiving break
Dec 1
Dec 8
Pending Invite sent to Talea Mayo Fabien

Abstracts

Erik Bollt (Clarkson University)

A New View on Integrability: On Matching Dynamical Systems through Koopman Operator Eigenfunctions

Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of non- linear dynamic behavior (e.g. through normal forms). In this presentation we will argue that the use of the Koopman operator and its spectrum are particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven machine learning algorithmic developments. Recall that the Koopman operator describes the dynamics of observation functions along a flow or map, and it is formally the adjoint of the Frobenius-Perrron operator that describes evolution of densities of ensembles of initial conditions. The Koopman operator has a long theoretical tradition but it has recently become extremely popular through numerical methods such as dynamic mode decomposition (DMD) and variants, for applied problems such as coherence and also in control theory. We demonstrate through illustrative examples that we can nontrivially extend the applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards a systematic discovery of rectifying integrability coordinate transformations.

John Schotland (Yale University)

Nonlocal PDEs and Quantum Optics

Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.

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