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Mathematical Physics and Harmonic Analysis Seminar

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	Date Time	Location	Speaker	Title – click for abstract
	08/14 1:50pm	Zoom	Avy Soffer Rutgers University	*Evolution of NLS with Bounded Data* We study the nonlinear Schrodinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially bounded in time for any bounded data. In the continuum, local existence is proved for real analytic data by a Newton iteration scheme. Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm (arXiv:2003.08849 [math.AP], J.Stat.Phys, 2020). This is a joint work with Ben Dodson and Tom Spencer.
	08/21 1:50pm	Zoom	Alexander Tovbis University of Central Florida	*Soliton and breather gases for the focusing Nonlinear Schrödinger equation (fNLS): spectral theory and possible applications* In the talk we introduce the idea of an "integrable gas" as a collection of large random ensembles of special localized solutions (solitons, breathers) of a given integrable system. These special solutions can be treated as "particles". Known laws of pairwise elastic collisions allow one to write the heuristic "equation of state" for the gas of such particles. In this talk we consider soliton and breather gases for the fNLS as special thermodynamic limits of finite gap (nonlinear multi phase wave) fNLS solutions. In this limit the rate of growth of the number of bands is linked with the rate of (simultaneous) shrinkage of the size of individual bands. This approach leads to the derivation of the equation of state for the gas and its certain limiting regimes (condensate, ideal gas limits), as well as construction of various interesting examples. We also discuss the recent progress and perspectives of future work, as well as some possible applications.
	09/11 1:50pm	Zoom	Claudio Munoz CNRS/Universidad de Chile	*Sufficient condition for the asymptotic stability of kinks in general 1+1 dimensional scalar field models* In this talk, I shall discuss some new results concerning the orbital and asymptotic stability of kinks for general (1+1)-dimensional scalar field models of the form $\partial_t^2\phi -\partial_x^2\phi + W'(\phi) = 0$. The goal is to exhibit a simple and general sufficient condition on the potential $W$ for asymptotic stability, without assuming any symmetry property for the potential nor for the kink. This is joint work with M. Kowalczyk, Y. Martel, and H. Van Den Bosch, and can be found at arXiv:2008:01276
	09/18 1:50pm	Zoom	Seonghyeon Jeong MSU	*Strong MTW type condition to local Holder regularity in generated Jacobian equations* In this talk, we present a proof of local Holder regularity of solutions to generated Jacobian equations as a generalization of optimal transport case, which is proved by George Loeper. We compare structures of generated Jacobian equations with optimal transport, and point out differences with difficulties which the differences can cause. For local Holder regularity theory, we use (G3s) condition and solution in Alexandrov sense. (G3s) is a strict positiveness type condition on MTW tensor associated to the generating function G, and Alexandrov solution is a solution that satisfies pullback measure condition. (G3s) is used to show a quantitative version of (glp), which gives some room for G-subdifferentials of solutions. Then the inequality for Holder regularity is shown by comparing volumes of G-subdifferentials using the fact that our solutions is in Alexandrov sense.
	09/25 1:50pm	Zoom	Jiaqi Yang GeorgiaTech	*Persistence of Invariant Objects in Functional Differential Equations close to ODEs* We consider functional differential equations which are perturbations of ODEs in $\mathbb{R}^n$. This is a singular perturbation problem even for small perturbations. We prove that for small enough perturbations, some invariant objects of the unperturbed ODEs persist and depend on the parameters with high regularity. We formulate a-posteriori type of results in the case when the unperturbed equations admit periodic orbits. The results apply to state-dependent delay equations and equations which arise in the study of electrodynamics. The proof is constructive and leads to an algorithm. This is a joint work with Joan Gimeno and Rafael de la Llave.
	10/01 10:00am	Zoom	Jonathan Breuer Hebrew University of Jerusalem	*Periodic Jacobi Matrices on Trees* The theory of periodic Jacobi matrices on the line is extremely rich and very well studied. Viewing the line as a regular tree of degree 2 leads to a natural generalization to periodic Jacobi matrices on general trees. This family of operators, which is at least as rich (by definition), but considerably less well understood, is at the center of this talk. We review some of the few known results, present some examples, and discuss open problems and directions for future research. The talk is based on joint work with Nir Avni and Barry Simon.
	10/02 1:50pm	Zoom	Rodrigo Matos TAMU	*Dynamical Contrast on Highly Correlated Anderson-type models* We present examples of random Schödinger operators obtained in a similar fashion but exhibiting distinct transport properties. The models are constructed by connecting, in different ways, infinitely many copies of the one dimensional Anderson model. Spectral aspects of the models will also be presented. In particular, we obtain a physically motivated example of a random operator with purely absolutely continuous spectrum where the transient and recurrent components coexist. This can be interpreted as a sharp phase transition within the absolutely continuous spectrum. Time allowing, I will discuss some tools related to harmonic analysis, including a version of Boole's equality which, to the best of our knowledge, is new. Based on joint work with Rajinder Mavi and Jeffrey Schenker.
	10/08 10:00am	Zoom	Delio Mugnolo University of Hagen	*Bi-Laplacians on graphs: self-adjoint extensions and parabolic theory* Elastic beams have been studied by means of hyperbolic equations driven by bi-Laplacian operators since the early 18th century: several properties of the corresponding parabolic equation on the whole Euclidean space have been discovered since the 1960s by Krylov, Hochberg, and Davies, among others. On a bounded domain or a metric graph, the bi-Laplacian is generally not anymore acting as a squared operator, though: this strongly affects the features of associated PDEs. I am going to present a full characterization of self-adjoint extensions of the bi-Laplacian, focusing on a class of realizations that encode dynamic boundary conditions. Maximum principles of parabolic equations will also be discussed: after a transient time, I am going to show that solutions often display Markovian features. This is joint work with Federica Gregorio.
	10/23 1:50pm	Zoom	Jun Kitagawa Michigan State University	*On free discontinuities in optimal transport* It is well known that regularity results for the optimal transport (Monge-Kantorovich) problem require rigid geometric restrictions. In this talk, we consider the structure of the set of ``free discontinuities'' which arise when transporting mass from a connected domain to a disconnected one, and show regularity of this set, along with a stability result under suitable perturbations of the target measure. These are based on a non-smooth implicit function theorem for convex functions, which is of independent interest. This talk is based on joint work with Robert McCann (Univ. of Toronto).
	10/29 10:00am	Zoom	Sebastian Egger Technion	*Well-defined spectral position for Neumann domains* A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition generated by specific gradient flow lines of the eigenfunction. The restricted eigenfunction onto the partition's components satisfies Neumann boundary conditions and the components are therefore coined 'Neumann domains'. Neumann domains represent a complementary path to the famous nodal-domain partition to study elliptic eigenfunctions where the latter is associated with the Dirichlet Laplacian. A very basic but fundamental property of nodal domains is that the restricted eigenfunction onto a nodal domain always gives the ground-state of the Dirichlet Laplacian. That feature becomes significantly more complex for Neumann domains due to the presence of possible cusps and cracks. In this talk, we focus on this problem and show that the spectral position for Neumann domains is well-defined. Moreover, we provide explicit examples of Neumann domains displaying a fundamentally different behavior in their spectral position than their nodal-domain counterparts.
	10/30 1:50pm	Zoom	Farhan Abedin MSU	*Hele-Shaw Flow and Parabolic Integro-Differential Equations* I will present a regularization result for a special case of the two-phase Hele-Shaw free boundary problem (a.k.a. interfacial Darcy flow), which models the evolution of two immiscible fluids flowing in the narrow gap between two parallel plates and subject to an external pressure source. Assuming that the fluid interface is given by the graph of a function, recent work of Chang-Lara, Guillen, and Schwab establishes the equivalence between the Hele-Shaw free boundary problem and a first-order parabolic integro-differential equation. By exploiting this equivalence and using available regularity theory for nonlocal parabolic equations, we show that if the gradient of the graph of the fluid interface has a Dini modulus of continuity for all times, then the gradient must be Holder continuous. This is joint work with Russell Schwab (MSU).
	11/06 1:50pm	Zoom	Mengxuan Yang Northwestern	*Wave Diffraction on Cones and The Aharonov-Bohm Effect* Hormander's classic theorem shows the singularities of solutions to the wave equation propagate along the geodesic flow. This is generalized by Cheeger-Taylor and Melrose-Wunsch to manifolds with conic singularities, where the singularities propagate along the so-called diffractive geodesics. In this talk, we will discuss the finer structure of the diffractive wave on cones as well as propagation of singularities for the wave equation with a singular magnetic Laplacian. This latter equation, closely analogous to wave equations with conic singularities, is the setting of the celebrated Aharonov-Bohm effect.
	11/19 10:00am	Zoom	James Kennedy University of Lisbon	*Spectral partitions of metric graphs* We introduce a theory of partitions of metric graphs via spectral-type functionals, inspired by the theory of spectral minimal partitions of domains but also with a view to understanding how to detect "clusters" in metric graphs. The goal is to associate with any given partition a spectral energy built around eigenvalues of differential operators like the Laplacian, and then minimize (or maximize) this energy over all admissible partitions. Since metric graphs are essentially one-dimensional manifolds with singularities (the vertices), the range of well-posed problems is much greater than on domains. We first sketch a general existence theory for optimizers of such partition functionals, and discuss a number of natural functionals and optimization problems. We also illustrate how changing the functionals and the classes of partitions under consideration -- for example, imposing Dirichlet versus standard conditions at the cut vertices or considering min-max versus max-min type functionals -- may lead to qualitatively different optimal partitions which seek out different features of the graph. Finally, we show how for many problems the optimal energies behave very similarly to the eigenvalues of the Laplacian (with Dirichlet or standard vertex conditions), in terms of Weyl asymptotics, upper and lower bounds, and interlacing inequalities. This is based on joint works with Matthias Hofmann, Pavel Kurasov, Corentin Léna, Delio Mugnolo and Marvin Plümer.
	11/20 1:50pm	Zoom	Eduardo Teixeira University of Central Florida	*Shaping diffusion in non-variational models* Mathematical models yielding diffusion adjustments are relevant in several fields of research and by now a comprehensive variational theory for treating such problems is well established. The corresponding non-variational theory has been completely open, despite of its potential applicabilities and in this talk I will describe some recent efforts towards launching such a theory. We will introduce a new class of non-divergence form elliptic operators whose degree of degeneracy/singularity varies accordantly to a prescribed power law. Under rather general conditions, we prove viscosity solutions are differentiable, with appropriate (universal) estimates. This result opens a number of new lines of investigation and I'll describe some of these endeavors towards the end of the talk.

The organizer for this seminar is Stephen Fulling.