Posted October 15, 2023
Last modified January 10, 2024
Applied Analysis Seminar Questions or comments?
3:30 pm – 4:20 pm Lockett 232
Robert Sims, University of Arizona
Stability of the Bulk Gap for Models with Frustration-Free Ground States
We prove that uniformly small short-range perturbations do not close the bulk gap above the ground state of frustration-free quantum spin systems that satisfy a local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finite-system Hamiltonians uniform in the system size. To obtain this result, we adapt the Bravyi-Hastings-Michalakis strategy to the GNS representation of the infinite-system ground state. This is joint work with Bruno Nachtergaele (University of California, Davis) and Amanda Young (University of Illinois, Urbana-Champaign).
Posted November 29, 2023
Last modified January 26, 2024
Applied Analysis Seminar Questions or comments?
3:30 am – 4:30 pm https://lsu.zoom.us/j/92777480012
Blair Davey, Montana State University
On Landis' conjecture in the plane
In the late 1960s, E.M. Landis made the following conjecture: If $u$ and $V$ are bounded functions, and $u$ is a solution to the Schr\"odinger equation $\Delta u - V u = 0$ in $\mathbb{R}^n$ that decays like $|u(x)| \le c \exp(- C |x|^{1+})$, then $u$ must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded, complex-valued functions $u$ and $V$ that solve the Schr\"odinger equation in the plane and satisfy $|u(x)| \le c \exp(- C |x|^{4/3})$. The examples of Meshkov were accompanied by qualitative unique continuation estimates for solutions in any dimension. Meshkov's estimates were quantified in 2005 by J. Bourgain and C. Kenig. These results, and the generalizations that followed, have led to a fairly complete understanding of these unique continuation properties in the complex-valued setting. However, Landis' conjecture remains open in the real-valued setting. We will discuss a recent result of A. Logunov, E. Malinnikova, N. Nadirashvili, and F. Nazarov that resolves the real-valued version of Landis' conjecture in the plane.
Posted September 22, 2023
Last modified January 25, 2024
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett 232
Eduard-Wilhelm Kirr, University of Illinois Urbana-Champagne
Can one find all coherent structures supported by a wave equation?
I will present a new mathematical technique aimed at discovering all coherent structures supported by a given nonlinear wave equation. It relies on global bifurcation analysis which shows that, inside the Fredholm domain, the coherent structures organize themselves into manifolds which either form closed surfaces or must reach the boundary of this domain. I will show how one can find all the limit points at the Fredholm boundary for the Nonlinear Schrodinger/Gross-Pitaevskii Equation. Then I will use these limit points to uncover all coherent structures and their bifurcation points.
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.
Posted March 26, 2024
Last modified March 31, 2024
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett Hall 233
Wei Li, DePaul University
Edge States on Sharply Joined Photonic Crystals
Edge states are important in transmitting information and transporting energy. We investigate edge states in continuous models of photonic crystals with piecewise constant coefficients, which are more realistic and controllable for manufacturing optical devices. First, we show the existence of Dirac points on honeycomb structures with suitable symmetries. Then we show that when perturbed in two appropriate ways, the perturbed honeycomb structures have a common band gap, and when joined along suitable interfaces, there exist edge states which propagate along the interfaces and exponentially decay away from the interfaces. The main tools used are layer potentials, asymptotic analysis, the Gohberg-Sigal theory and Lyapunov-Schmidt reductions. This is joint work with Junshan Lin, Jiayu Qiu, Hai Zhang.
Posted February 19, 2024
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett 232
Jessica Lin, McGill University
TBA
Posted February 21, 2024
Last modified April 12, 2024
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett 232
Ben Seeger, The University of Texas at Austin
Equations on Wasserstein space and applications
The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing mean field limits of controlled multi-agent systems, which arise in applications coming from macroeconomics, social behavior, and telecommunications. Justifying this continuum description is often nontrivial and is sensitive to the type of stochastic noise influencing the population. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of the well-posedness for a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces, and how this well-posedness allows for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type.