Analysis Mini-course Series
Student No.:20
Instructor:兰洋,张瑞祥,邓煜,金龙,陈大卫,王健,Dietrich Hafner,朱晖,孙晨旻,范晨捷  
Place:Jin Chun Yuan West Bldg.
Starting Date:2018-6-26
Ending Date:2018-8-30

Course 1: On asymptotic dynamics for mass critical gKdV equations

Speaker: 兰洋 Lan Yang

Time: Tue & Thu, 09:50-11:50, Jun.26-Jul.19

Place: Lecture Hall (Tue) & Conference Room 1(Thu), Jin Chun Yuan West Bldg.

Course 2: 抛物面解耦定理的证明

Speaker: 张瑞祥 Zhang Ruixiang

Time: Mon & Wed & Fri, 13:30-15:05, Jun.29/Jul.2/4/6

Place: Conference Room 1, Jin Chun Yuan West Bldg.

Abstract: 最近Bourgain和Demeter给出的抛物面解耦(decoupling)定理在和限制性定理相关的一系列问题中有非常重要的应用。我们将讨论该定理的证明,并简述其影响。希望你知道Fourier变换的定义和简单性质。不需要关于限制性定理和Bennett-Carbery-Tao的多线性Kakeya定理的基础知识,但若知道会有帮助。

参考资料:Larry Guth的课堂记录: 课程介绍里有对解耦定理的描述。

Course 3: Stability and instability of the Couette flow

Speaker: 邓煜 Deng Yu

Time: Mon & Wed & Fri, 09:50-11:25, Jun.29/Jul.2/4/6

Place: Conference Room 2(Fri, Jun.29/Jul.6) & Conference Room 3(Mon & Wed, Jul.2/4), Jin Chun Yuan West Bldg.

Abstract: I will talk about recent results in the study of small perturbations to the 2D inviscid Couette flow. Here the major interplay is between the effects of inviscid damping, which leads to decay, and "fluid echoes", which causes growth. In the first lecture I will briefly go over the background and the heuristic picture, in the second lecture I will discuss the 2015 paper of Bedrossian and Masmoudi which establishes stability in Gevrey (1/2+), and in the third lecture I wil explain my recent work with Masmoudi which establishes instability in Gevrey (1/2-).

Course 4: Quantum Chaos and Fractal Uncertainty Principle

Speaker: 金龙 Jin Long

Time: Tue & Thu 09:50-11:50, Jul.24-Aug.30

Place: Lecture Hall (Tue) & Conference Room 1(Thu), Jin Chun Yuan West Bldg.

Abstract: In this course we discuss recent development in quantum chaos via a new tool, called fractal uncertainty principle.
1. Introduction to quantum chaos
2. A simple model: quantum baker's map
3. Fractal uncertainty principle in one dimension
4. Control and stabilization on compact hyperbolic surfaces
5. Resonances for convex compact hyperbolic surfaces
Prerequisite: Basic knowledge about differential geometry of surfaces, dyamical system and Fourier analysis.

1. Semyon Dyatlov, Lecture notes on fractal uncertainty principle, online lecture notes, (working in progress),
2. Recent papers on quantum chaos and fractal uncertainty principle:
-Semyon Dyatlov and Joshua Zahl, Spectral gaps, additive energy, and a fractal uncertainty principle, Geometric and Functional Analysis 26(2016), 1011–1094, arXiv:1504.06589
-Semyon Dyatlov, Improved fractal Weyl bounds for hyperbolic manifolds, with an appendix with David Borthwick and Tobias Weich, to appear in Journal of the European Mathematical Society, arXiv:1512.00836
-Semyon Dyatlov and Long Jin, Resonances for open quantum maps and a fractal uncertainty principle, Communications in Mathematical Physics 354(2017), 269–316, arXiv:1608.02238
-Jean Bourgain and Semyon Dyatlov, Spectral gaps without the pressure condition, with Jean Bourgain, to appear in Annals of Mathematics, arXiv:1612.09040
-Semyon Dyatlov and Long Jin, Dolgopyat's method and the fractal uncertainty principle, to appear in Analysis & PDE, arXiv:1702.03619
-Jean Bourgain and Semyon Dyatlov, Fourier dimension and spectral gaps for hyperbolic surfaces, Geometric and Functional Analysis 27(2017), 744–771, arXiv:1704.02909
-Semyon Dyatlov and Long Jin, Semiclassical measures on hyperbolic surfaces have full support, arXiv:1705.05019
-Jian Wang, Strichartz estimates for convex co-compact hyperbolic surfaces, arXiv:1707.06310
-Long Jin, Control for Schr?dinger equation on hyperbolic surfaces, arXiv:1707.04990
-Long Jin and Ruixiang Zhang, Fractal uncertainty principle with explicit exponent, arXiv:1710.00250
-Long Jin, Damped wave equations on compact hyperbolic surfaces, arXiv:1712.02692
-Semyon Dyatlov and Maciej Zworski, Fractal uncertainty for transfer operators, to appear in International Mathematics Research Notices, arXiv:1710.05430
-Semyon Dyatlov, Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle, Proceedings of Journées EDP, 2017, arXiv:1710.08762

Course 5: Moduli of differentials and Teichmueller dynamics

Speaker: 陈大卫 Chen Dawei

Time: Wed & Fri 09:50-11:50, Jul.25-Aug.3

Place: Conference Room 1, Jin Chun Yuan West Bldg.

Abstract: An abelian differential defines a Euclidean metric with conical singularities such that the underlying Riemann surface can be realized as a polygon with edges pairwise identified via translation. Varying the shape of the polygon induces a GL(2,R)-action on the moduli space of abelian differentials, called Teichmueller dynamics, whose study has provided fruitful results and fascinating connections to many fields in mathematics, including (but not limited to) the works of a number of Fields Medalists (Avila, Kontsevich, McMullen, Mirzakhani, Okounkov, Yoccoz, etc). In this lecture series I will give an accessible introduction to this beautiful subject, with a focus on a combination of analytic, algebraic, dynamical, and combinatorial viewpoints.

Course 6: Contractible 3-manifolds and positive scalar curvature

Speaker: 王健 Wang Jian

Time: Tue & Thu, 13:30-15:00, Jul.31-Aug.9

Place: Conference Room 1, Jin Chun Yuan West Bldg.

Course 7: Scattering theory for the Dirac equation on Kerr space-time

Speaker: Dietrich Hafner

Time: Mon & Wed & Fri, 09:50-11:50, Aug.1/3/6/8

Place: Conference Room 2, Jin Chun Yuan West Bldg.

Course 8: Control of Dispersive Equations

Speaker: 朱晖 ZhuHui, 孙晨旻 Sun Chenmin

Time: Tue & Wed & Thu & Fri, 13:30-16:30, Jul.31-Aug.3

Place: Lecture Hall, Jin Chun Yuan West Bldg.

Abstract: The main purpose of this course is to review some classical results about the control theory of linear and non-linear wave equations and Schr?dinger equations. The proofs will use a micro-local approach developed in the works of Bardos-Lebeau-Rauch, Lebeau, Burq-Gérard, Burq-Zworski, etc. When time permits, some recent progresses in other models, e.g. KdV, KPI and water waves will be discussed.

Course 9: A introduction to concentration compactness and energy induction

Speaker: 范晨捷 Fan Chenjie

Time: Wed & Fri, 13:30-16:30, Aug.8/10

Place: Conference Room 3(Wed, Aug.8), Conference Room 1(Fri, Aug.10), Jin Chun Yuan West Bldg.

Abstract: In this mini course, we review the concentration compactness and energy induction strategy in the study of nonlinear dispersive equation. The model we will focus on will be nonlinear Schrodinger equation. We will focus on scattering type problems. We hope to also mention the application of this strategy in soliton resolution type problems.