MenuMathematical Physics and Harmonic Analysis Seminar
Fall 2022
| Date: | September 2, 2022 |
| Time: | 1:50pm |
| Location: | BLOC 306 |
| Speaker: | Goong Chen, Texas A&M University |
| Title: | Modes of Motion of a Coronavirus and Their Interpretations in the Invasion Process into a Healthy Cell |
| Abstract: | The motion of a coronavirus is highly dependent on its modes of vibration. If we model a coronavirus based on a elastodynamic PDEs, then the wiggling motion of the spikes and breathing motion of the capsid can be captured, for example. Here, we give a quick review of this model and the associated modal analysis through finite element computations. We then begin to study the invasion process by a coronavirus into a healthy cell, which takes place after the virus attacked the cell and its membrane began to fuse with the membrane of the cell. The fusion causes an initial opening with diameter of about 1 nm on the cell's membrane. For the invasion process to be successful, the virus must further widen the diameter to about 100nm in order for the viral genome material to enter the healthy cell to replicate. Can we model and simulate such a process by using a coupled virus-cell elastodynamic system? We will present our most recent partial findings for this question through the showing of supercomputer simulation animations. |
| Date: | September 9, 2022 |
| Time: | 1:50pm |
| Location: | Zoom |
| Speaker: | Konstantin Merz, Technische Universität Braunschweig |
| Title: | Random Schrödinger operators with complex decaying potentials |
| Abstract: | Estimating the location and accumulation rate of eigenvalues of Schrödinger operators is a classical problem in spectral theory and mathematical physics. The pioneering work of R. Frank (Bull. Lond. Math. Soc., 2011) illustrated the power of Fourier analytic methods — like the uniform Sobolev inequality by Kenig, Ruiz, and Sogge, or the Stein–Tomas restriction theorem — in this quest, when the potential is non-real and has “short range”. Recently S. Bögli and J.-C. Cuenin (arXiv:2109.06135) showed that Frank’s “short-range” condition is in fact optimal, thereby disproving a conjecture by A. Laptev and O. Safronov (Comm. Math. Phys., 2009) concerning Keller-Lieb-Thirring-type estimates for eigenvalues of Schrödinger operators with complex potentials. In this talk, we estimate complex eigenvalues of continuum random Schrödinger operators of Anderson type. Our analysis relies on methods of J. Bourgain (Discrete Contin. Dyn. Syst., 2002, Lecture Notes in Math., 2003) related to almost sure scattering for random lattice Schrödinger operators, and allows us to consider potentials which decay almost twice as slowly as in the deterministic case. The talk is based on joint work with Jean-Claude Cuenin. |
| Date: | September 23, 2022 |
| Time: | 1:50pm |
| Location: | Bloc 306 |
| Speaker: | Alain Bensoussan, University of Texas at Dallas |
| Title: | Control On Hilbert Spaces and Mean Field Control |
| Abstract: | In this work, we describe an alternative approach to the general theory of Mean Field Control as presented in the book of P. Cardaliaguet, F. Delarue, J-M Lasry, P-L Lions: The Master Equation and the Convergence Problem in Mean Field Games, Annals of Mathematical Studies, Princeton University Press, 2019. Since it uses Control Theory and not P.D.E. techniques it applies only to Mean Field Control. The general difficulty of Mean Field Control is that the state of the dynamic system is a probability. Therefore, the natural functional space for the state is the Wasserstein metric space. P.L. Lions has suggested to use the correspondence between probability measures and random variables, so that the Wasserstein metric space is replaced with the Hilbert space of square integrable random variables. This idea is called the lifting approach. Unfortunately, this brilliant idea meets some difficulties, which prevents to use it as an alternative, except in particular cases. In using a different Hilbert space, we study a Control problem with state in a Hilbert space, which solves the original Mean Field Control problem, as a particular case, and thus provides a complete alternative to the approach of Cardaliaguet, Delarue, Lasry, Lions. Based on Joint work with P. J. GRABER, P. YAM. Research supported by NSF grants DMS- 1905449 and 2204795. |
| Date: | September 30, 2022 |
| Time: | 1:50pm |
| Location: | BLOC 306 |
| Speaker: | Jie Xiao, Memorial University |
| Title: | Geometric relative capacity |
| Abstract: | Principally inspired by the very geometrically physic characteristics of the divergence theorem (regarded as one of the fundamental theorems within mathematical physics & vector calculus), this talk will address: quasilinear relative capacity; deficits via isocapacity & isoperimeter; geometric Green/Robin function; trace principle & energy evaluation; quasilinear graph mass & mean curvature. |
| Date: | October 7, 2022 |
| Time: | 1:50pm |
| Location: | BLOC 306 |
| Speaker: | Rodrigo Matos , TAMU |
| Title: | Eigenvalue statistics for the disordered Hubbard model within Hartree-Fock theory |
| Abstract: | Localization in the disordered Hubbard model within Hartree-Fock theory was previously established in joint work with J. Schenker, in the regime of large disorder in arbitrary dimension and at any disorder strength in dimension one, provided the interaction strength is sufficiently small. I will present recent progress on the spectral statistics conjecture for this model. Under weak interactions and for energies in the localization regime which are also Lebesgue points of the density of states, it is shown that a suitable local eigenvalue process converges in distribution to a Poisson process with intensity given by the density of states times Lebesgue measure. If time allows, proof ideas and further research directions will be discussed, including a Minami estimate and its applications. |
| Date: | October 28, 2022 |
| Time: | 09:00am |
| Location: | Zoom |
| Speaker: | Francesco Tudisco, Gran Sasso Science Institute, Italy |
| Title: | Nodal domain count for the generalized graph p-Laplacian |
| Abstract: | In recent years, there has been a surge in interest towards nonlinear extensions of graph operators such as the p-Laplacian and the generalized p-Laplacian (or p-Schrödinger) operators. This interest is prompted by applications connected to data clustering and semisupervised learning, where the limiting cases p=1 and p=\infty are especially noteworthy. In particular, similarly to the linear case, an important relation connects the nodal domains of the p-Laplacian and the k-th order isoperimetric constant of the graph. In this talk, we consider a set of variational eigenvalues of the generalized p-Laplacian operator and present new results that characterize several properties of these eigenvalues, with particular attention to the nodal domain count of their eigenfunctions. Just like the one-dimensional continuous p-Laplacian, we prove that the variational spectrum of the discrete generalized p-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl’s inequalities for the Laplacian operator to the nonlinear case and thus we prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized p-Laplacian on graphs, including those corresponding to variational eigenvalues. When applied to the linear case p=2, the new results imply well-known properties of the linear Schrödinger operator as well as novel ones. Based on a joint work with P.Deidda and M.Putti. |
| Date: | October 28, 2022 |
| Time: | 1:50pm |
| Location: | BLOC 306 |
| Speaker: | Matthias Hofmann, TAMU |
| Title: | Spectral minimal partitions of unbounded metric graphs |
| Abstract: | We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form $-\Delta + V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying ``landscape'' on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum $\lambda_{\text{ess}}$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other, which recalls a similar principle for the eigenvalues of the latter: for any $k\in\mathbb N$, the infimal energy among all admissible $k$-partitions is bounded from above by $\lambda_{\text{ess}}$, and if it is strictly below $\lambda_{\text{ess}}$, then a spectral minimal $k$-partition exists. We illustrate our results with several examples of existence and nonexistence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space. Joint project with James Kennedy and Andrea Serio. |
| Date: | November 4, 2022 |
| Time: | 09:00am |
| Location: | Zoom |
| Speaker: | Benjamin McKenna, Harvard |
| Title: | Random determinants, the elastic manifold, and landscape complexity beyond invariance |
| Abstract: | The Kac-Rice formula allows one to study the complexity of high-dimensional Gaussian random functions (meaning asymptotic counts of critical points) via the determinants of large random matrices. We present new results on determinant asymptotics for non-invariant random matrices, and use them to compute the (annealed) complexity for several types of landscapes. We focus especially on the elastic manifold, a classical disordered elastic system studied for example by Fisher (1986) in fixed dimension and by Mézard and Parisi (1992) in the high-dimensional limit. We confirm recent formulas of Fyodorov and Le Doussal (2020) on the model in the Mézard-Parisi setting, identifying the boundary between simple and glassy phases. Joint work with Gérard Ben Arous and Paul Bourgade. |
| Date: | November 11, 2022 |
| Time: | 1:50pm |
| Location: | BLOC 306 |
| Speaker: | Gamal Mograby, Tufts University |
| Title: | Topological quantum numbers |
| Abstract: | We present a detailed spectral analysis for a new class of fractal-type diamond graphs and provide a gap-labeling theorem in the sense of Bellissard for the corresponding probabilistic graph Laplacians using the technique of spectral decimation. Labeling the gaps in the Cantor set by the integrated density of states provides a set of topological quantum numbers that reflect the branching parameter of the graph construction and the decimation structure. The spectrum of the natural Laplacian on limit graphs is shown generically to be pure point supported on a Cantor set. However, one particular graph has a mixture of pure point and singularly continuous components. |
| Date: | November 18, 2022 |
| Time: | 09:00am |
| Location: | ZOOM |
| Speaker: | Alexey Kostenko, University of Lubljana |
| Title: | Laplacians of infinite graphs |
| Abstract: | There are two different notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to several diverse branches of mathematics and mathematical physics. The existing literature usually treats these two Laplacian operators separately. In this talk, I will focus on the relationship between them (spectral, parabolic and geometric properties). One of our main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides. Based on joint work with N. Nicolussi. |
| Date: | November 18, 2022 |
| Time: | 1:50pm |
| Location: | BLOC 306 |
| Speaker: | Jorge Villalobos, LSU |
| Title: | Embedded eigenvalues for discrete magnetic Schrodinger operators |
| Abstract: | Reducibility of the Fermi surface for a periodic operator is a key for the existence of embedded eigenvalues caused by a local defect. We consider a discrete model for a multilayer quantum system, such as stacked graphene, subject to a perpendicular magnetic field. Some techniques for constructing embedded eigenvalues extend from non-magnetic operators to magnetic ones, but the magnetic case is more complex because a typical magnetic operator on a periodic graph is merely quasi-periodic. |
| Date: | December 2, 2022 |
| Time: | 09:00am |
| Location: | Zoom |
| Speaker: | Hannah Kravitz, Portland State University |
| Title: | An application of PDEs on metric graphs to epidemiology |
| Abstract: | It has been known that a network structure enhances the spread of epidemics since the inception of the field. In the first epidemiological study John Snow (often called the father of epidemiology) determined that cholera was spreading through the 1850s London city water grid. Since then, numerous studies have supported the relationship between network structure and disease spread – parasites may be carried along fluvial networks, the colonial-era railway system in DR Congo may have contributed to the geographic spread of the first HIV outbreak, mosquitoes carrying dengue fever may be carried on trucks along highway systems, and the novel coronavirus spread from country to county first through the international air transport network. With these examples in mind, a variation of the susceptible-infected-removed (SIR) model is developed that couples travel along a network with spatially diffusive spread in a 2D region. The model begins with a metric graph structure (a network in which a distance coordinate is defined on the edges). The classic well-mixed SIR model is implemented at the vertices of the graph (may be conceived as “cities”) which is coupled to a transport model on the edges (“roads”). This structure is embedded in a 2D region, dividing it into several subregions whose boundaries are made up of the metric graph and outer borders. A diffusive SIR model is implemented in the 2D region. This talk will discuss the development of the model and present some preliminary results. |
| Date: | December 2, 2022 |
| Time: | 1:50pm |
| Location: | BLOC 306 |
| Speaker: | Tal Malinovitch, Yale University |
| Title: | Scattering for Schroedinger operators with conical decay |
| Abstract: | In this talk, I will discuss the scattering properties of Schrodinger operators with potentials that have short-range decay along a collection of rays in \mathbb{R}^d. This generalizes the classical setting of short-range scattering in which the potential is assumed to decay along all rays. For these operators, we show that any state decomposes into an asymptotically free piece and a piece that may interact with the potential for a long time. We give a microlocal characterization of the scattering states in terms of the dynamics and a corresponding description of their complement. We also show that in certain cases these characterizations can be purely spatial. In this talk, I will state our results, sketch some of the main ideas in the proof, and briefly discuss some examples of these interacting states for different systems. This is joint work with Adam Black. |
| Date: | December 8, 2022 |
| Time: | 11:00am |
| Location: | BLOC 302 |
| Speaker: | Alexander Kiselev |
| Title: | Convergence of Neumann Laplacians on thin structures: an alternative approach |
| Abstract: | Neumann Laplacians $A_\epsilon$ on thin manifolds, converging to metric graphs $G$ as $\epsilon\to0$, have been intensively studied by many authors, including Kuchment, Post, Pavlov, Exner, Zeng, among many others. The present-day state-of-the-art in this area is described in the monograph by O. Post. It was proved that the spectra of $A_\epsilon$ converge within any compact $K\in \mathbb{C}$ in the sense of Hausdorff to the spectrum of a graph Laplacian $A_G$. In the book of Post, the claimed convergence was enhanced to the norm-resolvent type, with an explicit control of the error as $O(\epsilon^\gamma),$ with $\gamma>0$ explicitly given. The matching conditions at the vertices of the limiting graph turn out to be either: (i) Kirchhoff (i.e. standard), if the vertex volumes are decaying, as $\epsilon\to0,$ faster than the edge volumes; (ii) Resonant, which can be equivalently described in terms of $\delta$-type matching conditions with coupling constants proportional to the spectral parameter $z$, if the vertex and edge volumes are of the same order; (iii) ``Dirichlet-decoupled" conditions (i.e., the graph Laplacian becomes completely decoupled), if the vertex volumes vanish slower than the edge ones. In the talk, I will be primarily interested in the most non-trivial resonant case (ii). I will provide a straightforward, alternative to that of Post, proof of the fact that the Neumann Laplacians $A_\epsilon$ in this case converge in norm-resolvent sense to an ODE acting in the Hilbert space $L^2(G)\oplus \mathbb{C}^N$, where $N$ is the number of vertices. The operator to which it converges is in fact the one first pointed out by Kuchment as the self-adjoint operator whose spectrum coincides with the Hausdorff limit of spectra for the family $A_\e$. I will show how a better error bound than that of Post is attained, namely, our estimate in the planar case is logarithmically worse than $O(\epsilon)$ and in the case of $\mathbb{R}^3$ is $O(\epsilo |
