Program
Microlocal Analysis and Applications
Student No.:50
Time:Mon/Wed 10:10-12:00, 2016-07-04 ~ 2016-07-27
Instructor:Long Jin  [CMSA, Harvard University]
Place:Conference room 3, floor 2, Jin Chun Yuan West Building
Starting Date:2016-7-4
Ending Date:2016-7-27

 

 

Description:

 

 

The goal of the course is to give a brief introduction to the method of microlocal analysis, mostly in the semiclassical setting. We shall also discuss the applications to Anosov flows. Details are as follows:

 

 

1.    Oscillatory integrals. Method of stationary phase.

 

2.    Pseudodifferential operators, symbol calculus.

 

3.    Wavefront set, microlocality.

 

4.    Microlocal estimates: elliptic estimates, propagation estimates, etc.

 

5.    Anosov flows, zeta functions, resonances and trace formula.

 

6.    Anisotropic Sobolev spaces and resolvent estimates.

 

7.    Smale’s conjecture on meromorphic continuation of zeta functions.

 

8.    (Optional) Counting resonances.

 

 

 

Prerequisite:

 

 

1.    Theory of distributions and Fourier transforms.

 

2.    Basic theory of differential manifolds, vector bundles and flows.

 

 

 

 

Reference:

 

 

1. Alain Grigis, Johannes Sj ̈ostrand, Microlocal analysis for differential operators, an introduction, Cambridge University Press, 1994.

 

2. Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138 AMS, 2012.

 

3. Fr ́ed ́eric Faure and Johannes Sj ̈ostrand, Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys. 308(2011), 325–364.

 

4. Semyon Dyatlov and Maciej Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis, arXiv:1306.4203.

 

5. Long Jin and Maciej Zworski, A local trace formula for Anosov flows, with an appendix by Fr ́ed ́eric Naud, preprint, arXiv:1411.6177.