Abstract:In this talk we will discuss some recent progresses on the study of dynamics of energy critical wave equations, specifically on the soliton resolution conjecture (SRC). SRC predicts that for many dispersive equations, generic solutions should asymptotically de-couple into solitary waves and radiation as time goes to infinity. The conjecture is open for most equations except integrable ones, but is better understood in the case of energy critical wave equations. We will give a sketch of the proof of this conjecture for a sequence of times, in the case of semilinear wave equations. The proof uses many ideas, including optimal perturbation theory, monotonicity formula, unique continuation property for elliptic equations, and most interestingly a channel of energy argument for outgoing waves.

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Abstract:A remarkable feature for dispersive equations is the simplification" of solutions at large times. For linear dispersive equations, this is well understood. But for nonlinear equations where there are complicated solitary waves, the mechanism by which the solution de-couple" into the solitary waves plus radiation is still mysterious, except for integrable systems. We will review some history on this fascinating topic, and explain some recent progress in the energy critical wave equations, such as defocusing wave with potential, focusing wave equations, and wave maps.

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Abstract:The global regularity problem for 3D Euler equations is an important open problem in PDEs. The main issue is to control vorticity, which could grow due to a stretching term in the equation. The main difficulty is to understand the interplay between the vorticity transportation and vorticity stretching. De Gregorio proposed a one dimensional model, based on a modification of the famous Constantin-Lax-Majda model, to gain insight on this effect. It turns out that this one dimensional model is very interesting. Numerical simulations show global existence, but we do not have a proof. In this talk, we will give a proof of global existence in the perturbative regime near the ground state. The proof reveals some interesting features which are relevant in the large data case as well. It also reveals the distinction between several notions of criticality" for some quasilinear equations: critical space for well-posedness, persistence of regularity, and the critical space for global existence and long time behavior.

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2018.06.12（星期二），10:00-12:00

Abstract :I will first review the (classical) Taylor-Wiles-Kisin patching construction and its application for attacking some cases of the Fontaine-Mazur conjecture. Then I will introduce Emerton's completed homology for GL_2/Q and explain how to modify the patching argument in this setting. One key ingredient is Paskunas' work on the p-adic local Langlands correspondence for GL_2/Q_p.

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Abstract :The famous conjecture of Fontaine and Mazur predicts that certain l-adic Galois representations come from geometry. I will talk about some recent progress on this conjecture.