The APDE seminar on Monday, 09/09 will be given by Di Fang in Evans 939 from 4:10 to 5pm.
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.
The APDE seminar on Monday, 08/26 will be given by Thilo Simon in Evans 939 from 4:10 to 5pm.
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.
The next APDE seminar will be given on Monday, 03/04 by Mihaela Ifrim in Evans 740 from 4:10 to 5pm.
Title: Dispersive decay of small data solutions for the KdV equation
Authors: Mihaela Ifrim, Herbert Koch, Daniel Tataru
We consider the Korteweg-de Vries (KdV) equation, and prove that small
localized data yields solutions which have dispersive decay on a quartic
time-scale. This result is optimal, in
view of the emergence of solitons at quartic time, as predicted by
inverse scattering theory.
The APDE seminar on Monday, 04/08 will be given by Maciej Zworski in Evans 740 from 4:10 to 5pm.
Rough control for Schr\”odinger operators on 2-tori.
Abstract: I will explain how the results of Bourgain, Burq and the speaker ’13 can be used to obtain control and observability by rough functions and sets on 2-tori. We show that for the time dependent Schr\”odinger equation, any set of positive measure can be used for observability and controllability. For non-empty open sets this follows from the results of Haraux ’89 and Jaffard ’90, while for sufficiently long times and rational tori this can be deduced from the results of Jakobson ’97. Other than tori (of any dimension; cf. Komornik ’91, Anantharaman–Macia ’14) the only compact manifolds for which observability holds for any non-empty open sets are hyperbolic surfaces. That follows from results of Bourgain–Dyatlov ’16 and Dyatlov–Jin ’17 and I will discuss the difficulty of passing to rougher rougher sets in that case. Joint work with N Burq.
The next APDE seminar will be given on Monday, 02/11 by Jean-Michel Coron in Evans 740 from 4:10 to 5pm.
Some methods to use the nonlinearities in order to control a system
A control system is a dynamical system on which one can act thanks to what is called the control. For example, in a car, one can turn the steering wheel, press the accelerator pedal etc. These are the control(s). One of the main problems in control theory is the controllability problem. One starts from a given situation and there is a given target. The controllability problem is to see if, by using some suitable controls depending on time, one can move from the given situation to the desired target. We study this problem with a special emphasis on the case where the nonlinearities play a crucial role. We first recall some classical results on this problem for finite dimensional control systems. We explain why the main tool used for this problem in finite dimension, namely the iterated Lie brackets, is difficult to use for many important control systems modeled by partial differential equations. We present methods to avoid the use of these iterated Lie brackets. We give applications of these methods to various physical control systems (Euler and Navier-Stokes equations of incompressible fluids, shallow water equations, Korteweg-de Vries equations).
The next APDE seminar will be given on Monday, 12/11 by Benjamin Harrop-Griffiths in Evans 740 from 4:10 to 5pm.
Title: Vortex filament solutions of the Navier-Stokes equations
Abstract: From Helmholtz to vaping hipsters, the dynamics of vortex filaments, i.e. fluids with vorticity concentrated along a smooth curve, has been a topic of significant interest in fluid dynamics. The global well-posedness of vortex filaments with small circulation follows from the theory of mild solutions of the 3d Navier-Stokes equations at critical regularity. However, for filaments with large circulation these results no longer apply. In this talk we discuss a proof of well-posedness (in a suitable sense) for vortex filaments of arbitrary circulation. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of the 3d Navier-Stokes without additional symmetry assumptions. This is joint work with Jacob Bedrossian and Pierre Germain.
The next APDE seminar will take place Wednesday, Nov 28, in 740 Evans from 3-4pm.
Title: Illusions: curves of zeros of Selberg zeta functions
Abstract: It is well known (since 1956) that the Selberg Zeta function
for compact surfaces satisfies the “Riemann Hypothesis”: any zero in the
critical strip 0<R(s)<1 is either real or Im(s)=1/2. The question of
location and distribution of the zeros of the Selberg Zeta function
associated to a noncompact hyperbolic surface attracted attention of the
mathematical community in 2014 when numerical experiments by
D. Borthwick showed that for certain surfaces zeros seem to lie on
smooth curves. Moreover, the individual zeros are so close to each other
that they give a visual impression that the entire curve is a zero set.
We will give an overview of the computational methods used, present
recent results, justifying these observations as well as state open
The next APDE seminar will take place Monday, Nov 19, in 740 Evans from 4-5pm.
Title: Rough control for Schr\”odinger operators on 2-tori.
Abstract: I will explain how the results of Bourgain, Burq and the
speaker ’13 can be used to obtain control and observability by rough
functions and sets on 2-tori. We show that for the time dependent
Schrödinger equation, any set of positive measure can be used for
observability and controllability.
For non-empty open sets this follows from the results of Haraux ’89
and Jaffard ’90, while for sufficiently long times and rational tori
this can be deduced from the results of Jakobson ’97.
The next APDE seminar will take place Monday, Nov 5, in 740 Evans from 4-5pm.
title: The Marked Length Spectrum of Anosov manifolds
Abstract: We discuss new results on the geometric problem of determining a Riemannian metric with negative curvature on a closed manifold from the lengths of its periodic geodesics. We obtain local rigidity results in all dimensions using combination of dynamical system results with microlocal analysis. Joint work with Thibault Lefeuvre.
The Analysis and PDE seminar will take place Monday Oct 1st in 740 Evans from 4-5pm.
Title: Construction of unstable quasi-periodic solutions for a system of coupled NLS equations.
Abstract: The systems of coupled NLS equations occur in some physical problems, in particular in nonlinear optics (coupling between two optical waveguides, pulses or polarized components…). From the mathematical point of view, the coupling effects can lead to truly nonlinear behaviors, such as the beating effect (solutions with Fourier modes exchanging energy) of Grébert, Paturel and Thomann (2013).
In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi-periodic solutions. The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see of this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions.
This is a work in collaboration with Benoît Grébert.