Title: Asymptotics of the radiation field on cones

Abstract:

Radiation fields are rescaled limits of solutions of wave equations near “null infinity” and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. We consider the wave equation on a product cone and show that the associated radiation field has an asymptotic expansion; the exponents seen in this expansion are the resonances of the hyperbolic cone with the same link. This talk is based on joint work with Jeremy Marzuola (building on prior work with Andras Vasy and Jared Wunsch).

Title: A tale of two resolvent estimates

Abstract:

I will discuss two new results concerning the best of resolvent estimates and the worst of resolvent estimates. In the former, case, that of nontrapping obstacles or metrics, we have obtained (in joint work with Galkowski and Spence) optimal, dynamically determined, constants in the standard non-trapping estimate for the (chopped off) resolvent. In the latter case, that of obstacles or metrics that may have very strong trapping, I will discuss joint work with Lafontaine and Spence that shows the estimates to be a far, far better thing than you might have expected, provided you omit a small set of frequencies from consideration.

Title: Adapting analysis/synthesis pairs to pseudodifferential operators

Abstract:

Many problems in harmonic analysis are resolved by producing

an analysis/synthesis of function spaces. For example the Fourier or

wavelet decompositions. In this talk I will discuss how to use Fourier

integral operators to adapt analysis/synthesis pairs (developed for the

constant coefficient PDE case) to the pseudodifferential setting. I will

demonstrate how adapting a wavelet decomposition can be used to prove

$L^{p}$ bounds for joint eigenfunctions.

Title: Pollicott-Ruelle resonances and Betti numbers

Abstract:

In joint work with Tobias Weich, we study the multiplicity of

the Pollicott-Ruelle resonance 0 of the Lie derivative along the

geodesic vector field on the cosphere bundle of a closed negatively

curved Riemannian manifold, acting on flow-transversal one-forms. We

prove that if the manifold admits a metric of constant negative

curvature and the Riemannian metric is close to such a constant

curvature metric, then the considered resonance multiplicity agrees with

the first Betti number of the manifold, provided the latter does not

have dimension 3. In dimension 3 and for constant curvature, it turns

out that the resonance multiplicity is twice the first Betti number.

Title – The stochastic nonlinear Schrödinger equations: defocusing mass and energy critical cases

Abstract – In this talk we will present our recent results on stochastic nonlinear Schrödinger equations with linear multiplicative noise, particularly, in the defocusing mass-critical and energy-critical cases. More precisely, for general initial data, we obtain the global existence and uniqueness of solutions in both mass-critical and energy-critical case. When the quadratic variation of noise is globally bounded, we also prove the rescaled scattering behavior of stochastic solutions in the spaces L2, H1 as well as the pseudo-conformal space. Furthermore, the Stroock-Varadhan type theorem is derived for the topological support of solutions to stochastic nonlinear Schrödinger equations in the Strichartz and local smoothing spaces.

]]>Title: On the Cauchy problem for the Hall-magnetohydrodynamics equations

Absract:

In this talk, I will describe a recent series of work with I.-J. Jeong on the Cauchy problem for the Hall-MHD equation without resistivity. This PDE, first investigated by the applied mathematician M. J. Lighthill, is a one-fluid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field). Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character.

Title: Diabatic Surface Hopping, Marcus Rate and Ehrenfest dynamics

Abstract: Surface hopping algorithms are popular tools to study dynamics of the quantum-classical mixed systems. In this talk, we will present a surface hopping algorithm in diabatic representations, in the view point of time dependent perturbation theory and semiclassical analysis. The algorithm is validated numerically in both weak coupling and avoided crossing regimes. We then discuss some recent progress on the asymptotics of the algorithm in weak and large coupling regimes.

Title:

Skyrmions and stability of degree ±1 harmonic maps from the plane to the two-dimensional sphere.

Abstract: Skyrmions are topologically nontrivial patterns in the magnetization of extremely thin ferromagnets. Typically thought of as stabilized by the so-called Dzyaloshinskii-Moriya interaction (DMI), or antisymmetric exchange interaction, arising in such materials, they are of great interest in the physics community due to possible applications in memory devices.

In this talk, I will characterize skyrmions as local minimizers of a

two-dimensional limit of the full micromagnetic energy, augmented by DMI and retaining the nonlocal character of the stray field energy. In the regime of dominating Dirichlet energy, I will provide rigorous predictions for their size and “wall angles”. The main tool is a quantitative stability result for harmonic maps of degree ± 1 from the plane to the two-dimensional sphere, relating the energy excess of any competitor to the homogeneous H¹-distance to the closest harmonic map. This is joint work with Anne Bernand-Mantel and Cyrill B. Muratov.

Title:

Concentration and Growth of Laplace Eigenfunctions.

Abstract: In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration including Weyl laws; in each case obtaining quantitative improvements over the known bounds.

]]>Title:

Resonances on asymptotically flat black holes

Abstract:

A fundamental problem in the context of Einstein’s equations of general relativity is to understand the dynamical evolution of small perturbations of stationary black hole solutions. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late times and carry information about the nature of the black hole, much like how the normal frequencies of a vibrating guitar string play an important role in the resulting sound wave. These frequencies are called quasinormal frequencies or resonant frequencies and they are closely related to scattering resonances in the study of Schrödinger-type equations. I will consider the linear wave equation on black hole backgrounds as a toy model for Einstein’s equations and give an introduction to resonances in this setting. Then I will discuss a new method of defining and studying resonances on asymptotically flat spacetimes, developed from joint work with Claude Warnick, which puts resonances on the same footing as normal modes by showing that they are eigenfunctions of a natural operator acting on a Hilbert space.