MenuMathematical Physics and Harmonic Analysis Seminar
Spring 2024
| Date: | February 23, 2024 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Jun Kitagawa, MSU |
| Title: | The structure of sliced and disintegrated Monge-Kantorovich metrics |
| Abstract: | The sliced and max sliced Wasserstein metrics are metrics on probability measures on $\mathbb{R}^n$ with certain finite moments, exploiting the particularly simple nature of transport on the real line. These were introduced as computationally faster alternatives to the usual optimal transport distance in applications. Some basic properties are known about their geometric structure, but not much is available in the way of a systematic study. The first half of this talk will present some further properties of these sliced metrics. The second half will introduce a larger family of metric spaces into which these metrics can be embedded, which seem to have more desirable geometric properties. This talk is based on joint work with Asuka Takatsu. |
| Date: | March 22, 2024 |
| Time: | 11:00am |
| Location: | Bloc 302 |
| Speaker: | Stephen Shipman, LSU |
| Title: | Localization of Defect States in the Continuum for Bilayer Graphene |
| Abstract: | Algebraic reducibility of the Fermi surface for AB-stacked bilayer graphene provides a mechanism for creating spectrally embedded defect states. Generically, this algebraic structure is not enough to provide sufficient localization in practice. However, it turns out that AB-stacked graphene enjoys additional structure that allows extreme localization of the defect. Also, defect states in the continuum appear numerically to be much more robust to perturbations than expected. |
| Date: | March 22, 2024 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Daniel Boutros, University of Cambridge |
| Title: | On energy conservation for inviscid hydrodynamic equations: analogues of Onsager's conjecture |
| Abstract: | Onsager's conjecture states that 1/3 is the critical spatial (Hölder) regularity threshold for energy conservation by weak solutions of the incompressible Euler equations. We consider an analogue of Onsager's conjecture for the inviscid primitive equations of oceanic and atmospheric dynamics. The anisotropic nature of these equations allows us to introduce new types of weak solutions and prove a range of independent sufficient criteria for energy conservation. Therefore there probably is a 'family' of Onsager conjectures for these equations. Furthermore, we employ the method of convex integration to show the nonuniqueness of weak solutions to the inviscid and viscous primitive equations (and also the Prandtl equations), and to construct examples of solutions that do not conserve energy in the inviscid case. Finally, we present a regularity result for the pressure in the Euler equations, which is of relevance to the Onsager conjecture in the presence of physical boundaries. As an essential part of the proof, we introduce a new weaker notion of pressure boundary condition which we show to be necessary by means of an explicit example. These results are joint works with Claude Bardos, Simon Markfelder and Edriss S. Titi. |
| Date: | March 29, 2024 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Matt Powell, Georgia Institute of Technology |
| Title: | Continuity of the Lyapunov exponent for quasi-periodic Jacobi cocycles |
| Abstract: | Many spectral properties of 1D Schr\"odinger operators have been linked to the Lyapunov exponent of the corresponding Schr\"odinger cocycle. While the situation for one-frequency quasi-periodic operators with analytic potential is well-understood, the multifrequency and non-analytic situations are not. The purpose of this talk is twofold: first, discuss our recent work on multi-frequency analytic quasi-periodic cocycles, establishing continuity (both in cocycle and jointly in cocycle and frequency) of the Lyapunov exponent for non-identically singular cocycles (of which the Jacobi cocycles form a special case), and second, discuss ongoing work extending these results to suitable Gevrey classes. Analogous results for analytic one-frequency cocycles have been known for over a decade, but the multi-frequency results have been limited to either Diophantine frequencies (continuity in cocycle) or SL(2,C) cocycles (joint continuity). We will discuss the main points of our argument, which extends earlier work of Bourgain. |
| Date: | April 5, 2024 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Theo McKenzie, Stanford |
| Title: | Quantum Ergodicity for Periodic Graphs |
| Abstract: | Quantum ergodicity (QE) is a notion of eigenfunction delocalization, that large Laplacian eigenfunction entries are “well spread” throughout a manifold or graph. Such a property is true of chaotic manifolds and graphs, such as random regular graphs and Riemannian manifolds with ergodic geodesic flow. Focusing on graphs, outside of very specific examples, QE was previously only known to hold for families of graphs with a tree local limit. In this talk we show how QE is in fact satisfied for many families of operators on periodic graphs, including Schrodinger operators with periodic potential on the discrete torus and on the honeycomb lattice. In order to do this, we use new ideas coming from analyzing Bloch varieties and some methods coming from proofs in the continuous setting. Based on joint work with Mostafa Sabri. |
| Date: | April 12, 2024 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Wei Li, DePaul University |
| Title: | Edge States on Sharply Joined Photonic Crystals |
| Abstract: | 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. |
| Date: | April 26, 2024 |
| Time: | 1:50pm |
| Location: | Zoom |
| Speaker: | Gihyun Lee, Ghent University |
| Title: | A calculus for magnetic pseudodifferential super operators |
| Abstract: | The time evolution of a physical state is determined by the Liouville equation $\frac{d\rho}{dt} = -\frac{i}{\hbar}(H\rho - \rho H)$ in quantum mechanics. Here $\rho$ is the density operator describing a given physical state and $H$ is the Hamiltonian of a given system. Here we can observe that the Liouville operator $\rho\mapsto L_H \rho := -\frac{i}{\hbar}(H\rho - \rho H)$ assigns linear operators to linear operators - physicists call such an operator a super operator. On the other hand, various kinds of pseudodifferential calculi has been developed in mathematics and applied to the study of PDE, geometry and mathematical physics. The main idea behind these theories of pseudodifferential calculi is to construct systematic ways of assigning linear operators to symbol functions, which enables us to translate properties of functions to properties of linear operators. In this talk, I will introduce a novel pseudodifferential calculus of super operators in the magnetic setting and explain how the Liouville super operator $L_H$ can be incorporated into this new pseudodifferential theory. Furthermore, the $L_2$-boundedness of pseudodifferential super operators will be discussed. Based on the joint work with M. Lein. |
