
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, quasiperiodic 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 Gsubdifferentials of solutions. Then the inequality for Holder regularity is shown by comparing volumes of Gsubdifferentials 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 aposteriori type of results in the case when the unperturbed equations admit periodic orbits. The results apply to statedependent 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 Andersontype 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 
Parabolic theory of biLaplacians
Properties of the parabolic equation associated with the fourth derivative operator (the biLaplacian) on $\mathbb R^d$, including a semiexplicit formula for the heat kernel, have been known since early investigations by Krylov, Hochberg, and Davies. On a domain or a metric graph conditions, the biLaplacian is generally not anymore acting as a squared operator, though: this strongly affects its analytic features.
We are going to describe selfadjoint and eventually markovian extensions of the biLaplacian, focusing on how the properties of the semigroup generated by biLaplacians on metric graphs strongly depend on the boundary conditions. I will also discuss how classical extension theory can be applied to characterize realizations of higher order parabolic equations with dynamic boundary conditions.
This is joint work with Federica Gregorio.


10/09 1:50pm 
Zoom 
Frank Sottile TAMU 
Critical points of discrete periodic operators
It is believed that the dispersion relation of a Schroedinger operator with a periodic potential has nondegenerate critical points. In work with Kuchment and Do, we considered this for discrete operators on a periodic graph G, for then the dispersion relation is an algebraic hypersurface. A consequence is a dichotomy; either almost all parameters have all critical points nondegenerate or almost all parameters give degenerate critical points, and we showed how tools from computational algebraic geometry may be used to study the dispersion relation.
With Matthew Faust, we use ideas from combinatorial algebraic geometry to give an upper bound for the number of critical points at generic parameters, and also a criterion for when that bound is obtained. The dispersion relation has a natural compactification in a toric variety, and the criterion concerns the smoothness of the dispersion relation at toric infinity. This toric variety has a twisted real structure with the dispersion relation a real algebraic subvariety whose real points compactify the real dispersion relation.


10/23 1:50pm 
Zoom 
Jun Kitagawa Michigan State University 
TBA 

10/30 1:50pm 
Zoom 
Farhan Abedin MSU 
TBA 

11/06 1:50pm 
Zoom 
Mengxuan Yang Northwestern 
TBA 

11/13 1:50pm 
Zoom 
Chris Marx Oberlin College 
TBA 

11/20 1:50pm 
Zoom 
Eduardo Teixeira University of Central Florida 
TBA 