Mathematical Physics and Harmonic Analysis Seminar
Date Time 
Location  Speaker  Title – click for abstract  

01/28 11:00am 
Zoom  Ralph Kaufmann Purdue University 
Local Models and Global Constraints
Motivated by current research in condensed matter, we consider multiparameter families of Hamiltonians which may have multiple Eigenvalues at socalled critical points. In this setting, we address several questions. In particular, for 3d families with isolated critical points, we can give classification theorems for the local structures.
We also show more generally that one can associate topological invariants to such systems, by various methods such as Chern classes and slicing.
Finally, we investigate global constraints on the local structures and give theorems for the case in which the parameter space is a threetorus, which is relevant to real world periodic systems. We illustrate these constraints on a concrete system, the Gyroid, and its deformations.  
01/29 2:00pm 
Zoom  Chris Marx Oberlin College 
Potential dependence of the density of states: deterministic, ergodic, and random potentials
In this talk we will address the potential dependence of the density of states and related spectral functions for discrete Schr\"odinger operators on infinite graphs. Following ideas by J. Bourgain and A. Klein, we will consider the density of states {\em{outer}} measure (DOSoM), a {\em{deterministic}} quantity, which is well defined for {\em{all}} Schr\"odinger operators.
We will explicitly quantify the potential dependence of the DOSoM in weak topology by proving a modulus of continuity with respect to the potential in $\ell{l}^\infinity$norm. The resulting modulus of continuity reflects the geometry of the graph at infinity. For the special case of operators on $\mathbb{Z}^d$ our result implies Lipschitz continuity of the DOSoM, in the case of the Bethe lattice, we obtain that the DOSoM is $\frac{1}{2}$logH\"older continuous. Applications of this result to ergodic, and in further consequence, for random Schr\"odinger operators will be presented.
This talk is based on joint work with Peter Hislop (University of Kentucky).  
02/11 11:00am 
Zoom  Marvin Plümer FernUniversität in Hagen 
Eigenvalue bounds for the Laplacian on embedded metric graphs
In this talk, we present some new developments in the theory of Laplacians on metric graphs. We discuss the role played by planarity or  more generally  the graph's genus and derive bounds for the eigenvalues of the Laplacian. For the derivation of these bounds, we make use of a recently developed transference principle by Amini and CohenSteiner that compares eigenvalues of continuous and discrete objects in a very convenient way. As a byproduct of our methods we also obtain eigenvalue bounds for Laplacian on discrete weighted graphs.  
02/12 2:00pm 
Zoom  Benito JuarezAubry IIMASUNAM 
Semiclassical gravity in static spacetimes as a constrained initial value problem
Semiclassical gravity is the theory in which gravity is treated classically and matter is treated in the framework of quantum field theory. The spacetime metric tensor, which encodes gravitational effects in its curvature, interacts with matter through the semiclassical Einstein equations: matter sources the dynamics of the spacetime metric via the expectation value of the stressenergy tensor of the quantum fields, while the quantum fields propagate in curved spacetime. It is currently unknown whether semiclassical gravity has a wellposed initial value formulation even for free fields. In this talk, I will discuss the situation in static spacetimes, where timetranslation symmetry greatly reduces the difficulty of the problem, and where one can show wellposedness. I will also discuss the main ideas on how to generalise these results to nonstatic spacetimes under some special circumstances. Based on arXiv:2011.05947 and on some unpublished work in progress with S. Modak.  
02/25 11:00am 
Zoom  Salma Lahbabi ENSEM, UHII/MSDA, UM6P 
A meanfield model for disordered crystals
In this talk, we consider disordered quantum crystals in the reduced Hartree
Fock (rHF) framework. The nuclei are supposed to be classical particles ar
ranged around a reference periodic configuration. In particular, we consider a family of nuclear distributions μ(ω, ·), where ω spans a probability space Ω.
Under some assumptions on the nuclear distribution μ, the average energy per
unit volume admits a minimizer, which is a solution of the rHF equations with
shortrange Yukawa interaction [2, 1]. We obtain partial results for Coulomb
interacting systems. We also study localization properties of the meanfield
Hamiltonian numerically [3].
References
[1] Éric Cancès, Salma Lahbabi, and Mathieu Lewin. Meanfield electronic
structure models for disordered materials. In Proceeding of the International
Congress on Mathematical Physics, Aalborg (Denmark), August 2012.
[2] Éric Cancès, Salma Lahbabi, and Mathieu Lewin. Meanfield models for
disordered crystals. J. math. pures appl., 100(2):241274, 2013.
[3] Salma Lahbabi. Étude mathématique de modèles quantiques et classiques
pour les matériaux aléatoires à l'échelle atomique. PhD thesis, Université de
CergyPontoise, 2013.  
03/04 11:00am 
Zoom  Pavel Kurasov Stockholm 
Crystalline measures: from quantum graphs to stable polynomials
Quantum graphs surprised researchers by their extraordinary spectral properties and possibility to carry out explicit analysis. One such example is given by the trace formula connecting spectra of quantum graphs to the set of periodic orbits on the underlying metric graph (without any additional correction terms). It appears that the corresponding spectral measure provides an explicit example of socalled crystalline measures generalising classical Dirac comb and Poisson summation formula. Crystalline measures are tempered distributions given by locally finite purely atomic measures whose Fourier transform is also a purely atomic measure. Such measures were studied by J.P. Kahane, A.P. Guinand, and S. Mandelbrojt in the fifties. It remained unclear whether Dirac combs provide the only type of examples of crystalline measures with uniformly discrete support. We are going to show how to construct a wide family of crystalline measures using quantum graphs (differential operators on metric graphs) and more generally via stable polynomials. The measure we obtain are:
 
03/05 2:00pm 
Zoom  Rostislav Grigorchuk Texas A&M University 
Spectra of groups and graphs: a short survey.
In my talk I will touch on such topics as the shape of the spectrum of Cayley and Schreier graphs of finitely generated groups, type of spectral measures, the question of A.Valette "Can one hear the shape of a group", and the relation to the random Schodinger operator. Based on numerous results with coauthors: L.Bartholdi, A.Zuk, Z.Sunic, D.Lenz, T.Nagnibeda, A.Perez, B.Simanek, A.Dudko and others.  
03/11 11:00am 
Zoom  Raphael Ducatez UNIGE 
Spectrum of critical Erdos Renyi graph
We analyze the spectrum of the (scaled) adjacency matrix A of the Erdős
Rényi graph G(N, d/N ) in the critical regime d = b log N. We establish a onetoone
correspondence between vertices of degree at least 2d and nontrivial eigenvalues outside the asymptotic bulk [−2, 2]. This correspondence implies a transition at an explicit b*. For d>b* log N the spectrum is just the bulk [−2, 2] and the eigenvectors are completely delocalized. For d< b* log N another phase appears. The spectrum outside [−2, 2] is not empty and there the eigenvectors concentrate around the large degree vertices.(joint work with Antti Knowles and Johannes Alt)
 
03/12 2:00pm 
Zoom  Jon Harrison Baylor University 
Periodic orbit evaluation of a spectral statistic of quantum graphs without the semiclassical limit
Spectral statistics of classically chaotic quantum systems are often analyzed using their periodic orbit structure via Gutzwiller's trace formula, which holds in the semiclassical limit. We show that for chaotic 4regular quantum graphs the variance of coefficients of the characteristic polynomial of the quantum evolution operator can be evaluated via periodic orbits without taking the semiclassical limit. The variance of the nth coefficient is precisely determined by the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) with n edges that fall in the following classes: those with no selfintersections, and those where all the selfintersections consist of two sections of the pseudo orbit crossing at a single vertex (2encounters of length zero).  
03/18 11:00am 
Zoom  Silvius Klein PucRio 
Mixed randomquasiperiodic Schrödinger operators
The purpose of this talk is to describe a discrete, one dimensional Schrödinger operator with mixed randomquasiperiodic potential and to discuss the positivity and the continuity of the corresponding Lyapunov exponent.
This is part of a larger joint project with Ao Cai and Pedro Duarte (both from the University of Lisbon) in which we study the spectral properties of such operators as well as the stability under random perturbations of the Lyapunov exponent of quasiperiodic Schrödinger operators.  
03/25 11:00am 
Zoom  Dmitry Jakobson Univ. Montreal 
Nodal set and negative eigenvalues in conformal geometry
We first describe conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators (such as Yamabe or Paneitz operator). We discuss applications to curvature prescription problems. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension greater than 2. We show that 0 is generically not an eigenvalue of the conformal Laplacian. If time permits, we shall discuss related results on manifolds with boundary, as well as for weighted graphs. This is joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M. Levitin, M. Karpukhin, G. Cox and Y. Sire.  
03/26 2:00pm 
Zoom  Sergei Tabachnikov Penn State 
Flavors of bicycle mathematics
This talk concerns a naive model of bicycle motion: a bicycle
is a segment of fixed length that can move so that the velocity of the
rear end is always aligned with the segment. Surprisingly, this simple
model is quite rich and has connections with several areas of research,
including completely integrable systems. Here is a sampler of problems
that I hope to touch upon:
1) The trajectory of the front wheel and the initial position of the
bicycle uniquely determine its motion and its terminal position; the
monodromy map sending the initial position to the terminal one arises.
This mapping is a Moebius transformation, a remarkable fact that has
various geometrical and dynamical consequences.
2) The rear wheel track and a choice of the direction of motion uniquely
determine the front wheel track; changing the direction to the opposite,
yields another front track. These two front tracks are related by the
bicycle (Backlund, Darboux) correspondence, which defines a discrete
time dynamical system on the space of curves. This system is completely
integrable and it is closely related with another, well studied,
completely integrable dynamical system, the filament (a.k.a binormal,
smoke ring, local induction) equation.
3) Given the rear and front tracks of a bicycle, can one tell which way
the bicycle went? Usually, one can, but sometimes one cannot. The
description of these ambiguous tire tracks is an open problem,
intimately related with Ulam's problem in flotation theory (in dimension
two): is the round ball the only body that floats in equilibrium in all
positions? This problem is also related to the motion of a charge in a
magnetic field of a special kind. It turns out that the known solutions
are solitons of the planar version of the filament equation.  
04/02 2:00pm 
Zoom  Peter Kuchment Texas A&M University 
The nodal sets mysteries
Nodal patterns of oscillating membranes have been known for hundreds of years. Leonardo da Vinci, Galileo Galilei, and Robert Hooke have observed them. By the nineteenth century they acquired the name of Chladni figures. Mathematically, they represent zero sets of eigenfunctions of the Laplace (or a more general) operator. In spite of such long history, many mysteries about these patterns (even in domains of Euclidean spaces, and even more on manifolds) still abound and attract recent attention of leading researchers working in physics, mathematics (including PDEs, math physics, and number theory) and even medical imaging. The talk will survey these issues, with concentration on the most recent results, where A&M mathematicians have been playing a significant role. No prior knowledge is assumed.  
04/09 2:00pm 
Zoom  Jared Wunsch Northwestern University 
Semiclassical analysis and the convergence of the finite element method
An important problem in numerical analysis is the solution of the Helmholtz equation in exterior domains, in variable media; this models
the scattering of timeharmonic waves. The Finite Element Method (FEM) is a flexible and powerful tool for obtaining numerical solutions, but difficulties are known to arise in obtaining convergence estimates for FEM that are uniform as the frequency of waves tends to infinity. I will describe some recent joint work with David Lafontaine and Euan Spence that yields new convergence results for the FEM which are uniform in the frequency parameter. The essential new tools come from semiclassical microlocal analysis. No knowledge of either FEM or semiclassical analysis will be assumed in the talk, however.  
04/16 2:00pm 
Zoom  Cosmas Kravaris Texas A&M University 
On the density of eigenvalues on discrete periodic graphs
Using the FloquetBloch transform, we show that Zdperiodic graphs have finitely many finite support eigenfunctions up to translations and linear combinations and show that this can be used to calculate the density of eigenvalues. We study the Kagome lattice to illustrate these techniques and generalize the claims to amenable quasihomogeneous graphs whose acting group has Noetherian group algebra (this includes all virtually polycyclic groups). Finally, we provide a formula for the von Neumann dimension (i.e. density) of eigenvalues on Zdperiodic graphs using syzygy modules.  
04/22 11:00am 
Zoom  Mikael Sundqvist Lund University 
Spectral flow for pair compatible equipartitions
In this talk we discuss how a recent spectral flow approach, proposed by Berkolaiko–Cox–Marzuola for analyzing the nodal deficiency of the nodal partition associated to an eigenfunction, can be extended to more general partitions. To be more precise, we discuss how the method can be extended to spectral equipartitions that satisfy a pair compatible condition. Nodal partitions and spectral minimal partitions are examples of such partitions. Based on joint work with Bernard Helffer.  
04/23 2:00pm 
Zoom  Jason Metcalfe University of North Carolina at Chapel Hill 
Local energy in the presence of degenerate trapping
Trapping is a known obstruction to local energy estimates for the wave equation and local smoothing estimates for the Schrödinger equation. When this trapping is sufficiently unstable, it is known that estimates with a logarithmic loss can be obtained. On the other hand, for very stable trapping, it is known that all but a logarithmic amount of local energy decay is lost. Until somewhat recently, explicit examples of scenarios where an algebraic loss (of regularity) was both necessary and sufficient for local energy decay had not be constructed. We will review what is known in these specific examples. We will also examine the relationship between the trapping and the existence of a boundary. In this highly symmetric case, a relatively simple proof showing a bifurcation in the behavior of local energy as the boundary passes through the trapping is available. This is related, e.g., to the instability of ultracompact neutrino stars.  
04/29 11:00am 
Zoom  Constanza RojasMolina CY Cergy Paris Université 
Fractional random Schrödinger operators, integrated density of states and localization
In this talk we will review some recent results on the fractional Anderson model, a random Schrödinger operator driven by a fractional laplacian. The interest on the latter lies in their association to stable Levy processes, random walks with long jumps and anomalous diffusion. We discuss in this talk the interplay between the nonlocality of the fractional laplacian and the localization properties of the random potential in the fractional Anderson model, in both the continuous and discrete settings. In the discrete setting we study the integrated density of states and show a fractional version of Lifshitz tails. This coincides with results obtained in the continuous setting by the probability community. This is based on joint work with M. Gebert (LMU Munich).

The organizers for this seminar are Bob Booth and Rodrigo Matos. Email them with talk suggestions and to request the Zoom link.