2019-11-19
Speaker: Xianchao Wu (McGill University)
Title: Reverse Agmon Estimate and some applications
Abstract: We consider L^2-normalized eigenfunctions of the semiclassical Schrodinger operator on a compact manifold. The well-known Agmon-Lithner estimates are exponential decay estimates (ie. upper bounds) for eigenfunctions in the forbidden region. The decay rate is given in terms of the Agmon distance function which is associated with the degenerate Agmon metric with support in the forbidden region.
The point of this talk is to prove a partial converse to the Agmon estimates (ie. exponential lower bounds for the eigenfunctions) in terms of Agmon distance in the forbidden region under a control assumption on eigenfunction mass in the allowable region arbitrarily close to its boundary. And some improvement estimates in the analytic setting will also be considered.
We then give some applications to hypersurface restriction bounds for eigenfunctions in the forbidden region along with corresponding nodal intersection estimates.
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2019-10-22
Speaker: 王超 Wang Chao (北京大学数学学院)
Title: The local well-posedness of water wave equations
Abstract: In this talk, I will present our recent results on the water wave equations. First, I give the proof of the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to $C^{\f32+\varepsilon}$. Second part, I will talk about the water-waves problem in a two-dimensional bounded corner domain $\Om_t$ with an upper free surface $\Gamma_t$ and a fixed bottom $\Gamma_b$.
We prove the local well-posedness of the solution to the water-waves system when the contact angles are less than $\f{\pi}{16}$.
2019-10-15
Speaker: Cyril Imbert (ENS de Paris)
Title: Global regularity estimates for the Boltzmann equation without cut-off
Abstract: In this talk, we will review a series of results related to the Boltzmann equation, an important nonlinear PDE from statistical physics. The global well-posedness of such an equation is a major open problem in mathematical physics and probably still very far from reach. We will see that as long as some quantities, observable at the macroscopic scale, stay under control, then the solution remains smooth.
This is a work in collaboration with L. Silvestre.
2019-3-12
Speaker: Hongkun Zhang (Department of Math & Stat. University of Massachusetts Amherst)
Title: Optimal decay rates of correlation for nonuniformly hyperbolic systems
Abstract: We obtain the optimal bound on polynomial decay rates of correlations for rather general non uniformly hyperbolic maps, including semi-dispersing Sinai billiards, Bunimovich stadium, etc. The main tools we use are the combination of renewal theory and coupling method, as well as the concept of standard pairs, which was firstly brought by by Dolgopyat and Chernov. This is a join work with Sandro Vaienti at CPT, Marseille.
2019-3-5
Speaker: Lovy Singhal (PKU)
Title: Cylinder absolute games on solenoids
Abstract: In 1988, S. G. Dani showed that the set of points on the torus $\mathbb{T}^n$ with non-dense orbits under any semisimple automorphism is large in the sense of Hausdorff dimension even though it has Haar measure zero. This was achieved using the technology of Schmidt games. Winning behaviour of subsets moreover tells us about their incompressible nature. Using a refinement of Schmidt games, we have shown that a statement similar to Dani's theorem holds for (affine) surjective endomorphisms of finite solenoids as well as for the full solenoid over the unit circle. Time permitting, we will also like to discuss the issues faced when dealing with general infinite solenoids over $S^1$.
