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Tomorrow, Wednesday, March 20, 2024

Posted January 18, 2024
Last modified March 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233, Zoom

Jake Murphy, LSU
Incoherent Coxeter groups

A group is coherent if every finitely generated subgroup is also finitely presented. In this talk, we will cover results of Jankiewicz and Wise showing that many Coxeter groups are incoherent using Bestvina-Brady Morse theory.

Tomorrow, Wednesday, March 20, 2024

Posted January 24, 2024
Last modified March 13, 2024

Harmonic Analysis Seminar

3:30 pm – 4:30 pm Lockett 232

Alan Chang, Washington University in St. Louis
Venetian blinds, digital sundials, and efficient coverings

Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's efficient covering theorem in which we replace lines with curves. This has a dual formulation in terms of nonlinear projections.

Tomorrow, Wednesday, March 20, 2024

Posted November 29, 2023

Geometry and Topology Seminar Seminar website

3:30 pm – 4:30 pm Lockett 233

Katherine Raoux, University of Arkansas
TBA

Thursday, March 21, 2024

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.

Monday, March 25, 2024

Posted February 12, 2024
Last modified March 4, 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. Zoom (click here to join)

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.

Monday, March 25, 2024

Posted February 17, 2024
Last modified March 18, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232

Samuel Punshon-Smith, Tulane University
Annealed mixing and spectral gap for advection by stochastic velocity fields

We consider the long-time behavior of a passive scalar advected by an incompressible velocity field. In the dynamical systems literature, if the velocity field is autonomous or time periodic, long-time behavior follows by studying the spectral properties of the transfer operator associated with the finite time flow map. When the flow is uniformly hyperbolic, it is well known that it is possible to construct certain anisotropic Sobolev spaces where the transfer operator becomes quasi-compact with a spectral gap, yielding exponential decay in these spaces. In the non-autonomous and non-uniformly hyperbolic case this approach breaks down. In this talk, I will discuss how in the stochastic velocity setting one can recover analogous results under expectation using pseudo differential operators to obtain exponential decay of solutions to the transport equation from $H^{-\delta}$ to $H^{-\delta}$ -- a property we call annealed mixing. As a result, we show that the Markov process obtained by considering the advection diffusion equation with a source term has an $H^{-\delta}$ Wasserstein spectral gap, uniform in diffusivity, and that the stationary measure has a unique limit in the zero diffusivity limit. This is a joint work with Jacob Bedrossian and Patrick Flynn.

Tuesday, March 26, 2024

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.