
Date Time 
Location  Speaker 
Title – click for abstract 

09/01 1:50pm 
BLOC 302 
Gregory Berkolaiko Texas A&M University 
Duistermaat index and eigenvalue interlacing for selfadjoint extensions of a symmetric operator
Eigenvalue interlacing is a tremendously useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rankone positive perturbation shifts the eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as the "Weyl interlacing" (additive perturbations), "Cauchy interlacing" (for principal submatrices of Hermitian matrices), "DirichletNeumann bracketing" and so on.
We discuss the extension of this idea to general "perturbations in boundary conditions", encoded as interlacing between eigenvalues of two selfadjoint extensions of a fixed symmetric operator with finite (and equal) defect numbers. In this context, even the terms such as "signature of the perturbation" are not immediately clear, since one cannot take the difference of two operators with different domains. However, it turns out that definitive answers can be obtained, and they are expressed most concisely in terms of the Duistermaat index, an integervalued topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the Lagrangian planes describe the two selfadjoint extensions being compared, while the third one corresponds to the distinguished Friedrichs extension.
We will illustrate our general results with simple examples, avoiding technicalities as much as possible and giving intuitive explanations of the Duistermaat index, the rank and signature of the perturbation in the selfadjoint extension, and the curious role of the third extension (Friedrichs) appearing in the answers.
Based on a work in progress with Graham Cox, Yuri Latushkin and Selim Sukhtaiev. 

09/08 1:50pm 
BLOC 302 
Yi Sheng Lim University of Bath 
An operatorasymptotic approach to periodic homogenization applied to equations of linearized elasticity
We explain an operatorasymptotic approach to homogenization for periodic composite media. This approach was developed by Cherednichenko and Velčić in the context of thin elastic plates, and here we demonstrate the approach under the simpler setting of equations of linearized elasticity.
As a consequence, we obtain L2 → L2, L2 → H1, and higherorder L2 → L2 normresolvent estimates. The correctors for the L2 → H1, and higherorder L2 → L2 results are constructed from boundary value problems that arise during the asymptotic procedure.
This is joint work with Josip Žubrinić (University of Zagreb). 

09/15 1:50pm 
BLOC 302 
Emmanuel Trelat Sorbonne University 
From microscopic to macroscopic scale equations: mean field, hydrodynamic and graph limits
Considering finite particle systems, we elaborate on various ways to pass to the limit as the number of agents tends to infinity, either by mean field limit, deriving the Vlasov equation, or by hydrodynamic or graph limit, obtaining the Euler equation. We provide convergence estimates. We also show how to pass from Liouville to Vlasov or to Euler by taking adequate moments. Our results encompass and generalize a number of known results of the literature.
As a surprising consequence of our analysis, we show that sufficiently regular solutions of any quasilinear PDE can be approximated by solutions of systems of N particles, to within 1/log(log(N)).
This is a work with Thierry Paul. 

09/22 1:50pm 
BLOC 302 
Matthias Maier Texas A&M University 
Lorentz Resonance in the Homogenization of Plasmonic Crystals
We explain the Lorentz resonances in plasmonic crystals that consist
of 2D nano dielectric inclusions as the interaction between resonant
material properties and subwavelength geometric resonances of
electrostatic nature. One example of such plasmonic crystals is
graphene nanosheets that are periodically arranged within a
nonmagnetic bulk dielectric.
We identify local geometric resonances on the length scale of the
small scale period. From a materials perspective, the graphene surface
exhibits a dispersive surface conductance captured by the Drude
model. Together these phenomena conspire to generate Lorentz
resonances at frequencies controlled by the surface geometry and the
surface conductance.
The Lorentz resonances found in the frequency response of the
effective dielectric tensor of the bulk metamaterial is shown to be
given by an explicit formula, in which material properties and
geometric resonances are decoupled.
Joint work with Wei Li (DePaul University) and Robert Lipton
(Louisiana State University)


09/29 1:50pm 
BLOC 302 
Yannick Sire Johns Hopkins 
Harmonic maps between singular spaces
After reviewing briefly the classical theory of harmonic maps between smooth manifolds, I will describe some recent results related to harmonic maps with free boundary, emphasizing on two different approaches based on recent developments by Da Lio and Riviere. This latter approach allows in particular to give another formulation which is wellsuited for such maps between singular spaces. After the works of Gromov, Korevaar and Schoen, harmonic maps between singular spaces have been instrumental to investigate superrigidity in geometry. I will report on recent results where we introduce a new energy between singular spaces and prove a version of Takahashi’s theorem (related to minimal immersions by eigenfunctions) on RCD spaces. 

10/06 1:50pm 
BLOC 302 
Hung Tran UW Madison 
Periodic homogenization of HamiltonJacobi equations: some recent progress
I first give a quick introduction to front propagations, HamiltonJacobi equations, levelset forced mean curvature flows, and homogenization theory. I will then show that the optimal rate of convergence is O(epsilon) in the convex setting and some nonconvex cases. I will also mention finer results on the effective fronts in two dimensions. Connections to stable norms in Riemannian geometry will also be made. Based on various joints work with W. Jing and Y. Yu. 

10/20 1:50pm 
BLOC 302 
Xiaoxu Wu Texas A&M University 
TBA 

10/27 1:50pm 
BLOC 302 
Gheorghe Craciun UW Madison 
TBA 

11/10 1:50pm 
Zoom 
Angeliki Menegaki IHES 
TBA 

11/17 1:50pm 
BLOC 302 
Enrique Zuazua FriedrichAlexanderUniversität ErlangenNürnberg 
TBA 