The APDE seminar on
Monday, 01/13 Monday, 01/27 will be given by Alexander Volberg in Evans 939 from 4:10 to 5pm.
Title: Box condition versus Chang–Fefferman condition for weighted multi-parameter paraproducts.
Abstract: Paraproducts are building blocks of many singular integral
operators and the main instrument in proving “Leibniz rule” for
fractional derivatives (Kato–Ponce). Also multi-parameter paraproducts
appear naturally in questions of embedding of spaces
of analytic functions in polydisc into Lebesgues spaces with respect to
a measure in the polydisc. The latter problem (without loss of
information) can be often reduced to boundedness of weighted dyadic
We find the necessary and sufficient condition for this boundedness in
n-parameter case, when n is 1, 2, or 3. The answer is quite unexpected
and seemingly goes against the well known difference between box and
Chang–Fefferman condition that was given by
Carleson quilts example of 1974.
The APDE seminar on Monday, 11/25 will be given by Charles Hadfield in Evans 939 from 4:10 to 5pm.
Title: Dynamical zeta functions at zero on surfaces with boundary
Abstract: The Ruelle zeta function counts closed geodesics on a Riemannian manifold of negative curvature. Its zeroes are related to Pollicott-Ruelle resonances which have been heavily studied in the setting of Anosov dynamical systems. In 2016, Dyatlov-Zworski proved an unexpected result relating the structure of the zeta function near the origin to the topology of the manifold. This extended a formula previously only known to hold in the constant curvature setting.
This talk will consider the situation where the manifold has boundary. A similar story can be told and the ultimate result extends the constant curvature setting (understood in 2001) to the variable curvature setting.
The microlocal tools required to consider this problem had been well developed in earlier papers (principally Dyatlov-Guillarmou 2016) and it remained to manipulate correctly relative cohomology (in this case à la Bott-Tu) in order to understand the space of 1-form Pollicott-Ruelle resonances.
The APDE seminar on Monday, 10/07 will be given by Deng Zhang in Evans 939 from 4:10 to 5pm.
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.
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