Posted January 16, 2024

Last modified February 28, 2024

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:20 pm 233 Lockett
Ben McCarty, University of Memphis

A new approach to the four color theorem

The Penrose polynomial of a graph, originally defined by Roger Penrose in an important 1971 paper, shares many similarities with Kauffman’s bracket and the Jones polynomial. In order to capitalize on these similarities, we first modify the definition of the Penrose polynomial to obtain a related family of polynomials, called the n-color polynomials. Each of the n-color polynomials may be thought of as an analog of the Jones polynomial, and is the graded Euler characteristic of a bigraded homology theory (analogous to Khovanov homology). We then show how to define a spectral sequence leading to a filtered homology theory (analogous to Lee homology) where coloring information becomes apparent. We will then discuss several applications of the theory to graph coloring and the four color problem. This is joint work with Scott Baldridge.

Posted February 20, 2024

Pasquale Porcelli Lecture Series Special Lecture Series

3:30 pm – 4:30 pm Atchafalaya Room, LSU Student Union
R. Tyrrell Rockafellar, University of Washington

Risk and Uncertainty in Optimization

Abstract: New mathematics is on the forefront in many emerging areas of technology, and its methods for sorting out ideas and testing for truth and shortcomings are as vital as ever. This talk aims to explain how that has worked in confronting “risk”. Problems of optimization are concerned with deciding things “optimally”. In many situations in management, finance, and engineering design, however, plans have to be fixed in the present without knowing fully how they will play out in the future. A future cost or hazard may depend on random variables with probability distributions that a present decision can only influence in a limited way. Should optimization then rely on average outcomes? Worst-case outcomes? High-probability avoidance of dangerous outcomes? Or what? This is a subject with a history of competing approaches that reached a turning point with the axiomatic development of a powerful theory of risk. The mathematical concepts and results from that have been overturning tradition in one important area of application after another.

Posted February 20, 2024

Pasquale Porcelli Lecture Series Special Lecture Series

2:30 pm – 3:30 pm Hill Memorial Library
R. Tyrrell Rockafellar, University of Washington

Variational Analysis and Geometry

The theory needed for problems of optimization has required vast developments of a kind of alternative calculus in which, for instance, discontinuous functions that might take on infinite values nonetheless have “subgradients” which are highly useful. In maximizing and minimizing, variables are often required to be nonnegative, or to have values not too high or too low. This leads to a fascinatingly different nonclassical kind of geometry.

Posted November 13, 2023

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Edmund Yik-Man Chiang, The Hong Kong University of Science and Technology

TBA

Posted January 18, 2024

Computational Mathematics Seminar

3:30 pm – 4:20 pm Digital Media Center: Room 1034
Soeren Bartels, University of Freiburg, Germany

Babuska's paradox in linear and nonlinear bending theories

The plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem.

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Justin Murray, Louisiana State University

TBA

Posted February 20, 2024

Pasquale Porcelli Lecture Series Special Lecture Series

2:30 pm – 3:30 pm Hill Memorial Library
R. Tyrrell Rockafellar, University of Washington

Variational Convexity and Local Optimality

For necessary and sufficient conditions for local optimality, the inherited ideal has been for them to be as close to each other as possible. In optimization, however, what’s more important is sufficient conditions that identify key common features in a problem which support algorithmic developments. Variational convexity, although only recently identified as such a condition, appears to be fundamentally important.

Posted January 9, 2024

Last modified February 27, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jone Lopez de Gamiz Zearra, Vanderbilt University

On subgroups of right-angled Artin groups

In this talk we will discuss subgroups of right-angled Artin groups (RAAGs for short). Although, in general, subgroups of RAAGs are known to have a wild structure and bad algorithmic behaviour, we will show that under certain conditions they have a tame structure. Firstly, we will discuss finitely generated normal subgroups of RAAGs and show that they are co-(virtually abelian). As a consequence, we deduce that they have decidable algorithmic problems. Secondly, we will recall results of Baumslag-Roseblade and Bridson-Howie-Miller-Short on subgroups of direct products of free groups and explain how they generalize to other classes of RAAGs.

Posted September 29, 2023

Last modified January 29, 2024

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Jacob Rasmussen, University of Illinois Urbana-Champaign

The L-space conjecture for 3-manifolds

The L-space conjecture of Boyer-Gordon-Watson and Juhasz relates three very different properties that a closed 3-manifold M can possess. One of these properties is algebraic: is \pi_1(M) left orderable? The second is geometric: does the M admit a coorientable taut foliation? The third is analytic: is the Heegaard Floer homology M as simple as it can be, given the size of H_1(M). If the conjecture is true, it would reveal the existence of a striking dichotomy for rational homology 3-spheres. In this talk, I'll explain what each of the three conditions appearing in the L-space conjecture mean, and then discuss efforts to prove and disprove it, and why we should care.

Posted January 22, 2024

Last modified February 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Dante Kalise, Imperial College

Feedback Control Synthesis for Interacting Particle Systems across Scales

This talk focuses on the computational synthesis of optimal feedback controllers for interacting particle systems operating at different scales. In the first part, we discuss the construction of control laws for large-scale microscopic dynamics by supervised learning methods, tackling the curse of dimensionality inherent in such systems. Moving forward, we integrate the microscopic feedback law into a Boltzmann-type equation, bridging controls at microscopic and mesoscopic scales, allowing for near-optimal control of high-dimensional densities. Finally, in the framework of mean field optimal control, we discuss the stabilization of nonlinear Fokker-Planck equations towards unstable steady states via model predictive control.

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Jake Murphy, LSU

TBA

Posted January 24, 2024

3:30 pm – 4:30 pm Lockett 232
Alan Chang, Washington University in St. Louis

TBA

Posted November 29, 2023

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233
Katherine Raoux, University of Arkansas

TBA

Posted January 20, 2024

Last modified February 5, 2024

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Chongying Dong, UC Santa Cruz

Monstrous moonshine and orbifold theory

This introductory talk will survey the recent development of the monstrous moonshine. Conjectured by McKay-Thompson-Conway-Norton and proved by Borcherds, the moonshine conjecture reveals a deep connection between the largest sporadic finite simple group Monster and genus zero functions. From the point of view of vertex operator algebra, moonshine is a connection among finite groups, vertex operator algebras and modular forms. This talk will explain how the moonshine phenomenon can be understood in terms of orbifold theory.

Posted February 12, 2024

Control and Optimization Seminar Questions or comments?

10:30 am – 11:20 am Note the Special Earlier Seminar Time For Only This Week. This is a Zoom Seminar. Click “Questions or Comments?” Above to Request a Zoom Link.
Antoine Girard, Laboratoire des Signaux et Systèmes
CNRS Bronze Medalist, IEEE Fellow, and George S. Axelby Outstanding Paper Awardee

Switched Systems with Omega-Regular Switching Sequences: Application to Switched Observer Design

In this talk, I will present recent results on discrete-time switched linear systems. We consider systems with constrained switching signals where the constraint is given by an omega-regular language. Omega-regular languages allow us to specify fairness properties (e.g., all modes have to be activated an infinite number of times) that cannot be captured by usual switching constraints given by dwell-times or graph constraints. By combining automata theoretic techniques and Lyapunov theory, we provide necessary and sufficient conditions for the stability of such switched systems. In the second part of the talk, I will present an application of our framework to observer design of switched systems that are unobservable for arbitrary switching. We establish a systematic and "almost universal" procedure to design observers for discrete-time switched linear systems. This is joint work with Georges Aazan, Luca Greco and Paolo Mason.

Posted February 17, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232
Samuel Punshon-Smith, Tulane University

TBA

Posted January 29, 2024

Computational Mathematics Seminar

3:30 pm – 4:20 pm Digital Media Center: Room 1034
Henrik Schumacher, University of Georgia

Repulsive Curves and Surfaces

Repulsive energies were originally constructed to simplify knots in $\mathbb{R}^3$. The driving idea was to design energies that blow up to infinity when a time-dependent family of knots develops a self-intersection. Thus, downward gradient flows should simplify a given knot without escaping its knot class. In this talk I will focus on a particular energy, the so-called \emph{tangent-point energy}. It can be defined for curves as well as for surfaces. After outlining its geometric motivation and some of the theoretical results (existence, regularity), I will discuss several hardships that one has to face if one attempts to numerically optimize this energy, in particular in the surface case. As we will see, a suitable choice of Riemannian metric on the infinite-dimensional space of embeddings can greatly help to deal with the ill-conditioning that arises in high-dimensional discretizations. I will also sketch briefly how techniques like the Barnes-Hut method can help to reduce the algorithmic complexity to an extent that allows for running nontrivial numerical experiments on consumer hardware. Finally (and most importantly), I will present a couple of videos that employ the gradient flows of the tangent-point energy to visualize some stunning facts from the field of topology. Although some high tier technicalities will be mentioned (e.g., fractional Sobolev spaces and fractional differential operators), the talk should be broadly accessible, also to undergrad students of mathematics and related fields.

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Peter Stringfield, Louisiana State University

TBA

Posted December 1, 2023

Last modified January 12, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Katherine Goldman, Ohio State University

TBA

Posted February 28, 2024

Computational Mathematics Seminar

3:30 pm Digital Media Center 1034
Yue Yu, Lehigh University

Nonlocal operator is all you need

During the last 20 years there has been a lot of progress in applying neural networks (NNs) to many machine learning tasks. However, their employment in scientific machine learning with the purpose of learning physics of complex system is less explored. Differs from the other machine learning tasks such as the computer vision and natural language processing problems where a large amount of unstructured data are available, physics-based machine learning tasks often feature scarce and structured measurements. In this talk, we will take the learning of heterogeneous material responses as an exemplar problem, to investigate the design of neural networks for physics-based machine learning. In particular, we propose to parameterize the mapping between loading conditions and the corresponding system responses in the form of nonlocal neural operators, and infer the neural network parameters from high-fidelity simulation or experimental measurements. As such, the model is built as mappings between infinite-dimensional function spaces, and the learnt network parameters are resolution-agnostic: no further modification or tuning will be required for different resolutions in order to achieve the same level of prediction accuracy. Moreover, the nonlocal operator architecture also allows the incorporation of intrinsic mathematical and physics knowledge, which improves the learning efficacy and robustness from scarce measurements. To demonstrate the applicability of our nonlocal operator learning framework, three typical scenarios in physics-based machine learning will be discussed: the learning of a material-specific constitutive law, the learning of an efficient PDE solution operator, and the development of a foundational constitutive law across multiple materials. As an application, we learn material models directly from digital image correlation (DIC) displacement tracking measurements on a porcine tricuspid valve leaflet tissue, and show that the learnt model substantially outperforms conventional constitutive models. https://www.cct.lsu.edu/lectures/nonlocal-operator-all-you-need

Posted January 22, 2024

Last modified January 24, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Boris Kramer, University of California San Diego

Scalable Computations for Nonlinear Balanced Truncation Model Reduction

Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems on nonlinear manifolds and preserves either open- or closed-loop observability and controllability aspects of the nonlinear system. Two computational challenges have so far prevented its deployment on large-scale systems: (a) the computation of Hamilton-Jacobi-(Bellman) equations that are needed for characterization of controllability and observability aspects, and (b) efficient model reduction and reduced-order model (ROM) simulation on the resulting nonlinear balanced manifolds. We present a novel unifying and scalable approach to balanced truncation for large-scale control-affine nonlinear systems that consider a Taylor-series based approach to solve a class of parametrized Hamilton-Jacobi-Bellman equations that are at the core of balancing. The specific tensor structure for the coefficients of the Taylor series (tensors themselves) allows for scalability up to thousands of states. Moreover, we will present a nonlinear balance-and-reduce approach that finds a reduced nonlinear state transformation that balances the system properties. The talk will illustrate the strength and scalability of the algorithm on several semi-discretized nonlinear partial differential equations, including a nonlinear heat equation, vibrating beams, Burgers' equation and the Kuramoto-Sivashinsky equation.

Posted November 14, 2023

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Micah Milinovich, University of Mississippi

TBA

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Huong Vo, Louisiana State University

TBA

Posted December 1, 2023

Last modified February 27, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233TBA

Posted January 23, 2024

Last modified January 28, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Luca Zaccarian, LAAS-CNRS and University of Trento

Lyapunov-Based Reset PID for Positioning Systems with Coulomb and Stribeck Friction

Reset control systems for continuous-time plants were introduced in the 1950s by J.C. Clegg, then extended by Horowitz twenty years later and revisited using hybrid Lyapunov theory a few decades ago, to rigorously deal with the continuous-discrete interplay stemming from the reset laws. In this talk, we provide an overview a recent research activity where suitable reset actions induce stability and performance of PID-controlled positioning systems suffering from nonlinear frictional effects. With the Coulomb-only effect, PID feedback produces a nontrivial set of equilibria whose asymptotic (but not exponential) stability can be certified by using a discontinuous Lyapunov-like function. With velocity weakening effects (the so-called Stribeck friction), the set of equilibria becomes unstable with PID feedback and the so-called ''hunting phenomenon'' (persistent oscillations) is experienced. Resetting laws can be used in both scenarios. With only Coulomb friction, the discontinuous Lyapunov-like function immediately suggests a reset action providing extreme performance improvement, preserving stability and increasing the convergence speed. With Stribeck, a more sophisticated set of logic-based reset rules recovers the global asymptotic stability of the set of equilibria, providing an effective solution to the hunting instability.

Posted February 21, 2024

Probability Seminar Questions or comments?

3:30 pm
Jessica Lin, McGill University

TBA

TBA

Posted February 19, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232
Jessica Lin, McGill University

TBA

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Nilangshu Bhattacharyya, Louisiana State University

Characteristic Classes

Posted December 6, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Joseph Breen, University of Iowa

TBA

Posted January 27, 2024

Last modified February 21, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Sergey Dashkovskiy , Julius-Maximilians-Universität Würzburg

Stability Properties of Dynamical Systems Subjected to Impulsive Actions

We consider several approaches to study stability and instability properties of infinite dimensional impulsive systems. The approaches are of Lyapunov type and provide conditions under which an impulsive system is stable. In particular we will cover the case, when discrete and continuous dynamics are not stable simultaneously. Also we will handle the case when both the flow and jumps are stable, but the overall system is not. We will illustrate these approaches by means of several examples.

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Megan Fairchild, Louisiana State University

TBA

Posted January 6, 2024

Last modified January 9, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Madalena Chaves, Centre Inria d'Université Côte d'Azur

Coupling, Synchronization Dynamics, and Emergent Behavior in a Network of Biological Oscillators

Biological oscillators often involve a complex network of interactions, such as in the case of circadian rhythms or cell cycle. Mathematical modeling and especially model reduction help to understand the main mechanisms behind oscillatory behavior. In this context, we first study a two-gene oscillator using piecewise linear approximations to improve the performance and robustness of the oscillatory dynamics. Next, motivated by the synchronization of biological rhythms in a group of cells in an organ such as the liver, we then study a network of identical oscillators under diffusive coupling, interconnected according to different topologies. The piecewise linear formalism enables us to characterize the emergent dynamics of the network and show that a number of new steady states is generated in the network of oscillators. Finally, given two distinct oscillators mimicking the circadian clock and cell cycle, we analyze their interconnection to study the capacity for mutual period regulation and control between the two reduced oscillators. We are interested in characterizing the coupling parameter range for which the two systems play the roles "controller-follower".

Posted January 28, 2024

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:20 pm Lockett 233
Greg Parker, Stanford University

TBA

Posted February 21, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232
Ben Seeger, The University of Texas at Austin

TBA

Posted September 24, 2023

Last modified January 22, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Ana Bălibanu, Louisiana State University

TBA

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Krishnendu Kar, Louisiana State University

TBA

Posted January 31, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Morgan Weiler, Cornell University

TBA

Posted January 17, 2024

Last modified February 26, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Tobias Breiten, Technical University of Berlin

On the Approximability of Koopman-Based Operator Lyapunov Equations

Computing the Lyapunov function of a system plays a crucial role in optimal feedback control, for example when the policy iteration is used. This talk will focus on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup in a specially weighted Lp-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay, which justifies finite rank approximations with numerical methods. The usefulness for numerical computations will also be demonstrated with two short examples. This is joint work with Bernhard Höveler (TU Berlin).

Posted February 1, 2024

Last modified February 11, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jean-François Lafont, The Ohio State University

TBA

Posted January 16, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Jorge Poveda, University of California, San Diego
Donald P. Eckman, NSF CAREER, and AFOSR Young Investigator Program Awardee

Multi-Time Scale Hybrid Dynamical Systems for Model-Free Control and Optimization

Hybrid dynamical systems, which combine continuous-time and discrete-time dynamics, are prevalent in various engineering applications such as robotics, manufacturing systems, power grids, and transportation networks. Effectively analyzing and controlling these systems is crucial for developing autonomous and efficient engineering systems capable of real-time adaptation and self-optimization. This talk will delve into recent advancements in controlling and optimizing hybrid dynamical systems using multi-time scale techniques. These methods facilitate the systematic incorporation and analysis of both "exploration and exploitation" behaviors within complex control systems through singular perturbation and averaging theory, resulting in a range of provably stable and robust algorithms suitable for model-free control and optimization. Practical engineering system examples will be used to illustrate these theoretical tools.