Calendar

Posted April 9, 2021

Applied Analysis Seminar Questions or comments?

Rui Han, LSU
A Polynomial Roth Theorem for Corners in the Finite Field Setting

The investigation of polynomial extensions of the Roth''s theorem was started by Bourgain and Chang, and has seen a lot of recent advancements. The most striking of these are a series of results of Peluse and Prendiville which prove quantitative versions of the polynomial Roth and Szemerédi theorems in the integer setting. There is yet no corresponding result for corners, the two dimensional setting for polynomial Roth's Theorem, where one considers progressions of the form (x, y), (x+t, y), (x, y+t^2) in [1 ,..., N]^2, for example.
We will talk about a recent result on the corners version of the result of Bourgain and Chang, showing an effective bound for a three term polynomial Roth's theorem in the finite field setting. This is based on joint work with Michael Lacey and Fan Yang.

Posted March 22, 2021
Last modified March 26, 2021

Control and Optimization Seminar Questions or comments?

Jean-Michel Coron, Universite Pierre et Marie Curie, France
Boundary Stabilization of 1-D Hyperbolic Systems

Hyperbolic systems in one space dimension appear in various real life applications, such as navigable rivers and irrigation channels, heat exchangers, plug flow chemical reactors, gas pipe lines, chromatography, and traffic flow. This talk will focus on the stabilization of these systems by means of boundary controls. Stabilizing feedback laws will be constructed. This includes explicit feedback laws which have been implemented for the regulation of the rivers La Sambre and La Meuse. The talk will also deal with the more complicated case where there are source terms.

Posted April 7, 2021

Harmonic Analysis Seminar

Egor Maximenko, National Polytechnic Institute, Mexico
Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences

In this talk, based on a joint article with Kevin Esmeral (https://doi.org/10.1007/s11785-016-0557-0), we describe the C*-algebra generated by radial Toeplitz operators with bounded symbols acting on the Fock space.
We prove that this C*-algebra is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric $\rho(j,k)=|\sqrt(j)-\sqrt(k)|$.
More precisely, we show that the spectral sequences (i.e., the sequences of the eigenvalues) of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.
The main idea is to approximate the spectral sequences by convolutions and apply an appropriate version of Wiener''s density theorem.

Posted February 14, 2021
Last modified April 7, 2021

Applied Analysis Seminar Questions or comments?

Marcelo Disconzi, Department of Mathematics, Vanderbilt University
General-relativistic viscous fluids

The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical relativity simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and comprehensive theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem, discuss the mathematics behind it, and report on a new approach to relativistic viscous fluids that addresses these issues.

Posted March 18, 2021
Last modified March 28, 2021

Algebra and Number Theory Seminar Questions or comments?

Gene Kopp, University of Bristol
TBA

Posted March 12, 2021
Last modified March 30, 2021

Control and Optimization Seminar Questions or comments?

Sophie Tarboureich, Laboratoire d'Analyse et d'Architecture des Systemes (LAAS), France
Algorithms for Event-Triggered Control

Event-triggered control consists of devising event-triggering mechanisms leading to only seldom control updates. In the context of event-triggered control, two objectives that can be pursued are (1) emulation, whereby the controller is a priori predesigned and only the event-triggering rules have to be designed and (2) co-design, where the joint design of the control law and the event-triggering conditions has to be performed. It is important to keep in mind that control systems are connected to generic digital communication networks for implementation, transmission, coding, or decoding. Therefore, event-triggered control strategies have been developed to cope with communication, energy consumption, and computation constraints. The talk is within this scope. Considering linear systems, the design of event-triggering mechanisms using local information is described through linear matrix inequality (or LMI) conditions. From these conditions, the asymptotic stability of the closed loop system, together with the avoidance of Zeno behavior, are ensured. Convex optimization problems are studied to determine the parameters of the event-triggering rule with the goal of reducing the number of control updates.

Posted April 7, 2021

Harmonic Analysis Seminar

Egor Maximenko, National Polytechnic Institute, Mexico
Radial operators on polyanalytic weighted Bergman spaces

In this talk, based on a recent paper with Roberto Moises Barrera-Castelan and Gerardo Ramos-Vazquez, we describe the von Neumann algebra $\mathcal{R}_n$ of radial operators acting on the $n-$ analytic weighted Bergman space $\mathcal{A}_n^2$ on the unit disk. First, extending the results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and Wunsche in 2005) form an orthonormal basis of $\mathcal{A}_n^2$. Using this basis, we provide the Fourier decomposition of $\mathcal{A}_n^2$ into the orthogonal sum of the subspaces associated with different frequencies. This leads to the decomposition of the von Neumann algebra $\mathcal{R}_n$ into the direct sum of some matrix algebras.

In other words, all radial operators are represented as matrix sequences. In particular, we represent in this form the Toeplitz operators with bounded radial symbols, acting in $\mathcal{A}_n^2$. Moreover, using ideas by Englis (1996), we show that the set of all Toeplitz operators with bounded generating symbols is not weakly dense in the algebra of all bounded linear operators acting in $\mathcal{A}_n^2)$

Posted March 22, 2021
Last modified March 28, 2021

Algebra and Number Theory Seminar Questions or comments?

Marc Besson, University of North Carolina at Chapel Hill
TBA

Posted March 5, 2021
Last modified March 26, 2021

Control and Optimization Seminar Questions or comments?

Vincent Andrieu, CNRS
An Overview of Asymptotic Observer Design Methods

Dynamic observers are estimation algorithms allowing us to reconstruct missing data from a model of a dynamic system and information obtained from the measurements. In this presentation, we present the main methods allowing the synthesis of an asymptotic observer. Starting from necessary conditions inspired by Luenberger''s work, we show the importance of contraction properties. Then, we give different existing methods. Finally, we give an overview of open issues in the field.

Posted March 18, 2021
Last modified March 26, 2021

Control and Optimization Seminar Questions or comments?

Lars Gruene, University of Bayreuth, Germany
On Turnpike Properties and Sensitivities and their Use in Model Predictive Control

Model predictive control (MPC) is one of the most popular modern control techniques. It generates a feedback-like control input from the iterated solution of open-loop optimal control problems. In recent years, there was a lot of progress in answering the question when MPC yields approximately optimal solutions. In this talk we will highlight the role of the turnpike property for this analysis. Moreover, we will show that for PDE-goverened control problems the turnpike property can be seen as a particular instance of a more general sensitivity property. This can be used in order to obtain efficient discretization schemes for the numerical solution of the optimal control problems in the MPC algorithm.

Posted March 18, 2021
Last modified March 26, 2021

Control and Optimization Seminar Questions or comments?

George Avalos, University of Nebraska
TBA

Posted March 15, 2021
Last modified March 26, 2021

Control and Optimization Seminar Questions or comments?

Francesco Bullo, University of California, Santa Barbara IEEE, IFAC, and SIAM Fellow
Non-Euclidean Contraction Theory and Network Systems

In this talk we discuss recent work on contraction theory and its application to network systems. First, we introduce weak semi-inner products as an analysis tool for non-Euclidean norms and establish equivalent characterizations of contraction and incremental stability. We also review some robustness, ISS and network stability in this new setting. Second, we discuss the notion of weakly and semi-contracting systems. For weakly contracting systems we prove a dichotomy for asymptotic behavior of their trajectories and show asymptotic stability for certain non-Euclidean norms. For semi-contracting systems we study convergence to invariant subspaces and applications to networks of diffusively-coupled oscillators.