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Schedule 

Venue: New Classroom Building 320

Monday, May 20

8:30 - 9:00, Registration

9:00 - 9:15, Opening

9:15 - 10:00, Jan Derezinski

10:00 - 10:30, Coffee Break

10:30 - 11:15, Fritz Gesztesy

 11:15  -12:00, Caroline Lasser

12:00 - 14:00, Lunch Break

14:00 - 14:45, Michael I. Weinstein

15:00 - 16:00, Short Talks

Tuesday, May 21

9:00 - 9:45, Yosi Avron

9:45 - 10:15, Coffee Break

10:15 - 11:00, Alain Joye

11:05 - 11:50, Lana Jitomirskaya

12:00 - 14:00, Lunch Break

14:00 - 15:00, Short Talks

15:00 - 15:30, Coffee Break

15:30 - 16:30, Math Colloquium, Barry Simon

19:00                 Banquet


Wednesday, May 22

9:00 - 9:45, Marius Lemm

9:45 - 10:15, Coffee Break

10:15 - 11:00, Anatoli Polkovnikov

11:05 - 11:50, Percy Deift

12:00 - 14:00, Lunch Break

14:00 - 14:45, Israel Klich

15:00 - 16:00 Short Talks


Thursday, May 23

9:00 - 9:45, Wojciech De Roeck

9:45 - 10:15, Coffee Break

10:15 - 11:00, Stefan Teufel

11:05 - 11:50, Clotilde Fermanian

12:00 - 14:00, Lunch Break

14:00 - 14:45, Ovidiu Costin

15:00 - 16:00 Short Talks

Friday, May 24

9:00 - 9:45, Clement Tauber

9:45 - 10:15, Coffee Break

10:15 - 11:00, Abel Klein

11:05 - 11:50, Ian Jauslin

12:00 - 14:00, Lunch Break

14:00 - 14:45, John Imbrie

15:00 - 16:00 Short Talks

Monday

9:15 - 10:00

Jan Derezinski, On some families of exactly solvable Schrödinger operators

Abstract: I will discuss various realizations of one dimensional Schrödinger operators with 1/x^2 and 1/x potentials on L^2[0, infty). It is natural to organize them into holomorphic families, allowing for complex coupling constants. Their properties are sometimes quite surprising.

10:30 - 11:15

Fritz Gesztesy,
On Birman-Hardy-Rellich-type inequalities

Abstract: We will illustrate how factorizations of singular, even-order partial (and ordinary) differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. In particular, we will derive a general inequality and demonstrate how particular choices of the parameters contained in it yield well-known inequalities, such as the classical Hardy and Rellich inequalities, as special cases.

Going beyond the factorization method we will also describe a variety of generalized situations involving higher-order operators yielding Birman-type inequalities. In addition, we will describe refinements of these inequalities involving radial derivatives as well additional logarithmically weaker singularities. The emphasis throughout will be to obtain sharp constants in these inequalities.

This is based on joint work with Lance Littlejohn, Isaac Michael, Michael Pang, and Richard Wellman.

11:15 - 12:00

Caroline Lasser,
Semiclassical wave packets in various flavours

Abtsract: Semiclassical wave packets have been introduced by George Hagedorn as a powerful anisotropic wave packet generalization of the Hermite functions. We revisit their construction, highlighting structural properties of their polynomial part. Then we discuss their approximation properties for time dependent Schrödinger equations, both in the Hermitian and the anti-Hermitian setting. The results are  joint-work with Roman Schubert and Stephanie Troppmann.

14:00 - 14:45

Michael I. Weinstein,
Dynamics of waves in systems with spectral band degeneracies

Abstract: Periodic media with spectral band degeneracies are a source of novel phenomena with implications for the physical properties of naturally occurring materials as well as for the design of metamaterials. We shall review recent work on a) the propagation in 1D of semi-classical wavepackets through band degeneracies (w/ A.B. Watson and J. Lu), and b) the dynamics of waves, spectrally localized near Dirac points, in honeycomb media and their perturbations (w/ A. Drouot, C. L. Fefferman, J.P. Lee Thorp, and Y. Zhu).

15:00 - 16:00

Short Talks

Rui Han, TBA
Tom VandenBoom, Finite-gap CMV matrices: Periodic coordinates and a Magic Formula
Benjamin Eichinger, KDV Hierarchy via Abelian Coverings and Operator Identities
Blake Allan, On Critical Dipoles in Dimensions n > 2

Tuesday

9:00 - 9:45

Yosi Avron, 
Sagnac interferometers: a tribute to George Hagedorn

Abstract: I shall describe the Sagnac interferometer, and give an elementary, special relativistic, derivation of Sagnac formula for deformable fiber optics interferometers. I shall then describe a fully relativistic eikonal equation for optical fibers in arbitrary motion. In the case that the eikonal changes adiabatically I shall describe the first adiabatic correction to the phase shift. The talk is based on joint works with A. Ori and O. Kenneth.

10:15 - 11:00

Alain Joye, George’s Travels in Molecular Dynamics

Abstract: A personal view on some of George’s contributions in the adiabatic, semiclassical and Born-Oppenheimer theories, in the service of molecular dynamics.

11:05 - 11:50

Lana Jitomirskaya, Fractal properties of the Hofstadter's butterfly and singular continuous spectrum of the critical almost Mathieu operator.

Abstract: Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly. It reduces to the critical almost Mathieu family, indexed by phase. We discuss the proof of singular continuous spectrum for this family, for all phases, finishing a program with a long history. We also present a result (with I. Krasovsky) that proves one half of the Thouless' one half conjecture from the early 80s: that Hausdorff dimension of the spectrum of Harper's operator is bounded by 1/2 for all irrational fluxes.

14:00 - 15:00

Short Talks

Charles Hagedorn, Precision measurements of Newton's gravitational inverse-square law and the Equivalence Principle
Maria Perel, Non-adiabatic transitions for the Schroedinger type equation: avoided and unavoidable crossing
Alexander Watson, Wave-packet dynamics in locally periodic media
Ksenia Kyzyurova, Bayesian statistical framework for analysis of agreement between experimental data and theoretical developments

15:30 - 16:30

Barry Simon,
Poncelet’s Theorem, Paraorthogonal Polynomials and the Numerical Range of Truncated GGT matrices 

Abstract: During the last 20 years there has been a considerable literature on a collection of related mathematical topics: higher degree versions of Poncelet’s Theorem, certain measures associated to some finite Blaschke products and the numerical range of finite dimensional completely non-unitary contractions with defect index 1. I will explain that without realizing it, the authors of these works were discussing Orthogonal Polynomials on the Unit Circle (OPUC). This will allow us to use OPUC methods to provide illuminating proofs of some of their results and in turn to allow the insights from this literature to tell us something about OPUC. This is joint work with Andrei Martínez-Finkelshtein and Brian Simanek. Background will be provided on the topics discussed.

Wednesday

9:00 - 9:45

Marius Lemm,
Finite-size criteria for spectral gaps in quantum spin systems

Abstract: A central question concerning a quantum spin system is whether the spectral gap above its ground state persists in the thermodynamic limit. We survey some recent applications of finite-size criteria for deriving spectral gaps in frustration-free spin models. This talk is based on joint works with H. Abdul-Rahman, A. Lucia, E. Mozgunov, B. Nachtergaele, A. Sandvik, S. Yang, and A. Young.

10:15 - 11:00

Anatoli Polkovnikov,
Variational approach to adiabatic transformations in Hamiltonian systems

Abstract: In this talk I will outline a general approach to construct the adiabatic gauge potential (adiabatic connection), which allows for finding generators of adiabatic transformations (parallel transport) in quantum and classical Hamiltonian systems. In integrable systems this approach leads to construction of the exact gauge potential; in chaotic systems it allows for finding an accurate local approximate gauge potentials. I will review various applications of the gauge potentials from establishing connections between the mass tensor and the quantum geometric tensor to establishing fundamental quantum speed limits.

11:05 - 11:50

Percy Deift, 
Universality in numerical computation with random data

Abstract: We consider various standard algorithms in numerical analysis and computation and show that
fluctuations in the time to achieve a desired accuracy in a given algorithm are universal, independent of the statistics of the random data. Some of the results are numerical/experimental, and some are analytical. This is joint work with many authors over the years, C.Pfrang, G.Menon, S.Olver, S.Miller and mostly T.Trogdon.

14:00 - 14:45

Israel Klich, TBA

Abstract: TBA

15:00 - 16:00

Short Talks

Alex Bols, TBA
Fan Yang, TBA
Rodrigo Matos, Recent Progress on Localization theory for correlated random systems
Carlos Villegas-Blas, On semiclassical eigenvalue distribution theorems for perturbations of the Landau problem

Thursday

9:00 - 9:45

Wojciech De Roeck, 
Topological affects in quantum many-body theory

Abstract: TBA

10:15 - 11:00

Stefan Teufel,
Non-equilibrium almost-stationary states and linear response for gapped quantum systems

Abstract: I will first review the problem of justifying linear response theory for gapped extended quantum systems and briefly discuss existing mathematical results. Then I will present a recent approach to the rigorous justification of linear (and higher order) response based on non-equilibrium almost-stationary states and an adiabatic theorem for time-dependent perturbations that close the spectral gap.
My talk is based on [arXiv:1708.03581, CMP Online First] and an ongoing joint project with Giovanna Marcelli.

11:05 - 11:50

Clotilde Fermanian,
Propagation of Wave Packets and Application to Hermann-Kluk Propagators.

Abstract: In this talk, we are interested in the description of the solution of semi-classical Schrödinger’s type equations for initial data which are wave packets. Such data, also called coherent states, are families of wave functions which concentrate in the phase space and the main examples are Gaussian wave packets and their generalizations, the Hagedorn’s wave packets. We shall discuss how one can use these wave packets to deduce Hermann-Kluk's type representations of the propagators associated with semi-classical PDEs by combining the Hermann-Kluk approach with ideas issued from surface hopping semi-groups.  

14:00 - 14:45

Ovidiu Costin,
Reconstruction of the global behavior of special functions from 
divergent expansions.

Abstract: I will describe a method of reconstructing special functions (more generally, resurgent functions) from a divergent series conventionally placed at infinity-- in a rigorous and numerically particularly efficient way. The information obtained includes very accurate values at the origin (central connection) and throughout the domain of analyticity, the position and local expansions at other possible singularities, and Stokes transitions. The approach has intricate connections with recent deep results in Szego asymptotics. I will take the Painleve P1 tritronquee solutions as an example, important in mathematics and mathematical physics.

15:00 - 16:00

Short Talks

Nishant Rangamani, Spectral and Dynamical Localization for Random Polymer Models
Paul Bergold, Fourier series windowed by a bump function
Victor Bernardo Chabu, Asymptotic limits of pure quantum states as classical mixtures

Friday

9:00 - 9:45

Clement Tauber, 
Topology of periodically driven systems

Abstract: Floquet topological insulators describe independent electrons on a lattice driven out of equilibrium by a time-periodic Hamiltonian, beyond the adiabatic approximation. In dimension two, such systems are characterized by integer-valued topological indices associated with the unitary propagator, alternatively in the bulk or at the edge of a sample. In this talk I will discuss a definition of the two indices relying neither on translation invariance nor on averaging, and show that they are equal. This feature is called the bulk-edge correspondence. In this approach disorder is intrinsically taken into account, even in the strong (mobility gap) regime. This is based on joint works with Gian Michele Graf and Jacob Shapiro.

10:15 - 11:00

Abel Klein,
Localization for the random XXZ quantum spin chain via an eigensystem multiscale analysis

Abstract: We will discuss work in progress with Alex Elgart towards proving localization for the random XXZ quantum spin chain via an eigensystem multiscale analysis.

11:05 - 11:50

Ian Jauslin,
Time-evolution of electron emission from a metal surface

Abstract:  The electrons in a metal can be extracted from the metal by a variety of processes: for instance by applying a strong electric field (field emission), or shining an intense laser pulse (photoemission). In the case of field emission, the celebrated and widely used Fowler-Nordheim equation predicts a value for the current of electrons outside the metal. In this talk, I will show that the Fowler-Nordheim equation emerges as the long-time asymptotic solution of a Schrodinger equation with a realistic initial condition, thereby justifying the use of the Fowler Nordheim equation in real setups. As for photoemission, the expression for the current has been predicted by Faisal et al, and I will discuss ongoing work to show that this prediction emerges as the long-time asymptotic solution of a Schrodinger equation. In both cases, I will discuss the time scale of the convergence.

14:00 - 14:45

John Imbrie,
Smoothing of eigenvalue distributions for quantum systems with discrete disorder

Abstract: For disordered quantum systems such as the Anderson model, degeneracies provide avenues for long-range tunneling, and hence are a barrier to localization. To handle these effects in the case of discrete disorder distributions, one needs to build up smoothness of the density of states and separate eigenvalues in a multi-scale procedure. This can be accomplished with a careful analysis of the way eigenvalues depend on the disorder.

15:00 - 16:00

Short Talks

Stephanie Gamble, Semiclassical solutions for a conical intersection
Rajinder Mavi, Dynamical and spectral properties of random Schrodinger operators with strongly correlated potentials
Christoph Fischbacher, Entanglement entropy for the higher spin XXZ model

Short Talk Abstracts

  • Blake Allan, On Critical Dipoles in Dimensions n > 2, Abstract: We consider generalizations of Hardy's inequality corresponding to the case of dipole potentials V(x) = gamma (e, x) |x|^{-3} x in R^n\{0}, gamma in R, n in N, n > 2, of the type int_{R^n} d^n x |(nabla f)(x)|^2 geq d_n int_{R^n} d^n x  (e, x) |x|^{-3} |f(x)|^2, f in C_0^{infty}(R^n), where e in R^n, |e| = 1 and (a,b) denotes the Euclidean scalar product of a, b in R^n. In particular, we (implicitly) determine the largest value of d_n > 0 (i.e., the critical dipole moment) for which this inequality holds. Thus, for gamma leq d_n, \overline{\big[- \Delta + \gamma (e, x) |x|^{-3}\big]\big|_{C_0^{\infty}(\bbR^n)}} \geq 0, where overline represents the operator closure in L^2(R^n).  We also study the behavior of d_n as a function of the space dimension n.  In addition, we extend these inequalities to the case of finitely many and countably infinitely many (but suitably screened w.r.t. their support) dipole potentials whose singularities are uniformly spaced apart. This is based on joint work with F. Gesztesy.
  • Paul Bergold, Fourier series windowed by a bump function
  • Alex Bols, TBA
  • Victor Bernardo Chabu, Asymptotic limits of pure quantum states as classical mixtures. Abstract: If a family of quantum pure states submitted to a smooth potential concentrates as h --> 0 to a pure classical one at a given instant, then necessarily the concentrated state keeps being pure for any time. However, in this poster we exhibit a family of solutions to the Schrödinger equation whose associated Wigner (semiclassical) measures do correspond to pure classical states for t > 0, but split into a mixture of two of them for t > 0. Such a phenomenon may only exist due to the presence of singularities in the potential; indeed, this example is a spin-off of a work in which we study the asymptotic behaviour of systems evolving under conical potentials.
  • Benjamin Eichinger, We prove global existence and uniform almost-periodicity in the time and space coordinates of solutions to the Cauchy problem for the KdV hierarchy with reflectionless initial conditions having spectrum satisfying a certain moment condition. Our methods involve the development of generalized Abelian integrals on noncompact Riemann surfaces and a spectrally-dependent Fourier transform, with respect to which the Schrödinger and Lax operators associated to the KdV hierarchy become greatly simplified. This talk is based on joint work with T. VandenBoom and P. Yuditskii. 
  • Christoph Fischbacher Entanglement entropy for the higher spin XXZ model, Abstract: Recently, Beaud and Warzel considered so called droplet states from the lowest energy band of the spin-1/2 XXZ chain and showed that the entanglement entropy for such states satisfies a log-corrected area law in the deterministic case and a true area in the disordered case. In this talk, we consider the XXZ chain for higher spins. We will discuss how it is still possible to write the Hamiltonian as a direct sum of Schroedinger-type operators on rather complicated graphs, where - under suitable assumptions on the anisotropy parameter - the potential controls the kinetic term. This fact can be used to prove suitable Combes-Thomas estimates and bounds on spectral projections similar to those previously shown for the spin-1/2 case. We then finish by presenting first results on entanglement entropy for states from the lowest energy regime. This is joint work with G. Stolz (UAB).
  • Stephanie Gamble Semiclassical solutions for a conical intersection, Abstract: We investigate some preliminary solutions using semiclassical wave packets for a conical intersection.
  • Charles Hagedorn Precision measurements of Newton's gravitational inverse-square law and the Equivalence Principle, Abstract:  I'll mention a small open problem for mathematical physicists -- the notion of helicity as a charge in gravitational experiments is, so far as experimentalists are aware, undefined. Searches for gravitational couplings to helicity have been performed, but cannot be interpreted without a mathematical framework.
  • Rui Han TBA
  • Ksenia Kyzyurova Bayesian statistical framework for analysis of agreement between experimental data and theoretical developments, Abstract: In my PhD dissertation I have developed and analyzed a fully Bayesian statistical framework for testing theoretical scientific models, often realized as scientific computer models, with respect to limited and noisy experimental data. This framework allows to obtain probabilistic answers to analysis of agreement between experimental data and the theory. I have previously considered this framework for analysis of computer models of volcano pyroclastic flows and volcano ash transport and dispersal and engineering computer models. I'm interested to learn from the participants of the conference if this framework is fruitful for application in other fundamental science and engineering problems; and if so, then to propose interesting collaborations.
  • Rodrigo Matos, Recent Progress on Localization theory for correlated random systems. Abstract: We intend discuss results in the theory of Anderson localization (and related questions) for two types of random systems. The first type is exemplified by the Hartree Aproximation for the Hubbard model, where dynamical localization in the form of decay of eigenfunction correlators is shown. The second one involves random operators on graphs which look similar at first glance but where we are able to verify contrasting quantum dynamical properties . This is joint work with Jeff Schenker.
  • Rajinder Mavi, Dynamical and spectral properties of random Schrodinger operators with strongly correlated potentials Abstract: We consider a random Schrodinger operator with strongly correlated potentials at arbitrarily large distances. We will study the dynamical and spectral properties of the Hamiltonian. We apply our results to an Anderson localized polaron, which is the motivation of our study. Joint w/ Rodrigo Matos and Jeffrey Schenker.
  • Maria Perel Non-adiabatic transitions for the Schroedinger type equation: avoided and unavoidable crossing, Abstract: The adiabatic approximation for monochromatic modes in slowly irregular waveguides may fail near the sections where modes degenerate or almost degenerate, i.e., where phase velocities of two modes coincide or are close to each other. We find that the transformation of modes near such sections can be studied in a general form for waveguides of different nature - acoustic, elastic or electromagnetic - by means of the techniques elaborated in Quantum Mechanics for the Schroedinger equation. The crucial step in the solution of the problem is the reduction of equationsof motion to a Schroedinger type equation. It turns out that the Hamiltonianin this equation is non self-adjoint even for waveguides without dissipation.We study the transformation of modes of the Schroedinger type equation with a non self-adjoint Hamiltonian ofspecial form. We assume that the Hamiltonian is a small perturbation of one with a crossing point of eigenvaluesas functions of time and the parameter characterizing an order of the perturbation is a square root of  the adiabatic parameter. The non perturbed Hamiltonian is assumed to be dialonalizable.  After the perturbation, two regimes of eigenvalue behavior are possible: avoided crossingand two close points of degeneration. We have obtained asymptotic solutions and transfer matrices  in both cases. The Landau-Zener formulas are obtained in the case of avoided crossing of eigenvalues.The transfer matrix for the case of unavoidable crossing is different. Two distinct physical processes correspond to both cases.
  • Nishant Rangamani, Spectral and Dynamical Localization for Random Polymer Models, Abstract: We will give a new proof of spectral localization for random polymer models. We will then show that once one has the existence of a complete orthonormal basis of eigenfunctions, the same estimates used to prove existence naturally lead to stronger dynamical results. In particular, we will show that on any compact interval not containing a finite set of critical energies we have dynamical localization in expectation (EDL). The dynamical localization results are new.
  • Tom VandenBoom Finite-gap CMV matrices: Periodic coordinates and a Magic Formula, Abstract: (Joint work with Jacob Christiansen and Benjamin Eichinger) We prove a bijective unitary correspondence between 1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class of operators is related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices.
  • Carlos Villegas-Blas On semiclassical eigenvalue distribution theorems for perturbations of the Landau problem, Abstract: We study two different ways to consider the semiclassical limit of the eigenvalue distribution in clusters around the Landau levels associated to suitable perturbations of the Landau Hamiltonian. In the first one, we consider perturbations of the Landau problem keeping both the magnetic field strength and the value of the Planck's parameter fixed and then study the high energy asymptotics (joint work with a. Pushnitsky and G . Raikov). We obtain that such a distribution is determined by averages of the perturbation along straight lines on plane. In the second one, we take both the magnetic field strength and the Planck's parameter depending on a parameter in such a way that the eigenvalue distribution is now determined by averages of the perturbation along classical orbits with fixed classical energy (joint work with G. Hernandez Dueñas, S. Perez-Esteva and A.Uribe).
  • Alexander Watson, Wave-packet dynamics in locally periodic media, Abstract:  We study the dynamics of wave-packet solutions of Schrödinger’s equation and Maxwell’s equations in media with a local periodic structure which varies adiabatically (over many periods of the periodic lattice) across the medium. We focus in particular on the case where symmetries of the periodic structure lead to degeneracies in the Bloch band dispersion surface. We derive systematically and rigorously the `anomalous velocity’ of wave-packets due to the Bloch band’s Berry curvature, and the dynamics of a wave-packet incident on a Bloch band degeneracy in one spatial dimension. Joint work with Michael Weinstein and Jianfeng Lu.
  • Fan Yang, TBA