MIT PDE/Analysis Seminar

Abstract: Consider the triadic Cantor set equipped with the "uniform" Cantor law. It happens that its cumulative distribution function, the devil’s staircase, is Hölder regular, and its best exponent of regularity is ln(2)/ ln(3), which is exactly the Hausdorff dimension of the Cantor set. Moreover, one can show that the Fourier transform of the Cantor law decay like |ξ|^{− ln 2/ ln 3} "on average". This is no coincidence, and hint for a deeper link between Fractal Geometry and Fourier Analysis. In this talk we will detail and explore this link through the notion of Fourier Dimension. We will introduce the Fourier dimension, compute it on some easy examples, quote some natural questions that arise, and then discuss a (partial) state of the art on the topic.

Abstract: The speaker aims to report some advances in the study of irrotational oscillation of a water droplet under zero gravity. The governing physical laws, resembling the well-studied capillary water waves equation, are converted to a quasilinear dispersive para-differential system defined on the 2-sphere. Regarding this conversion, a coordinate-independent, global para-differential calculus defined on compact Lie groups and homogeneous spaces is developed as a toolbox. After discussing Cauchy theory under this novel formalism, the speaker will propose several unsolved problems concerning existence of periodic solutions, normal form reduction and generic lifespan estimates. It is pointed out that all of these problems are closely related to certain Diophantine equations.

Abstract: Although the laws of physics are quantum mechanical, the world we live in and observe is well described by classical physics. One mathematical justification for this is Egorov’s theorem which shows that the quantum and classical descriptions match as the semiclassical parameter ℏ tends to 0. However, the Ehrenfest timescale for which Egorov’s theorem holds may only be seconds or minutes for realistic systems. In this talk we consider quantum systems that are weakly coupled to a noisy external environment, as modelled by the Lindblad equation. The semiclassical limit of this Lindblad equation is a Fokker-Planck equation which includes the effects of friction and diffusion. In this talk I will show that even for very weak diffusion, the agreement between the Lindblad and Fokker-Planck equations persists for times much longer than the Ehrenfest time. This is joint work with Daniel Ranard and Jess Riedel.

Abstract: In this talk, we prove global well-posedness and scattering for the conformally invariant, radially symmetric nonlinear wave equation in the defocusing case. This result is sharp in the radial case. We spend the first half of the talk discussing the broader context of the nonlinear wave equation. We also explain why the result is sharp.