October 20, 2010

Working Seminar in Semiclassical Analysis

Mondays, 3-5 PM, 2866 East Hall

Semiclassical analysis is the study of the relationships between the mathematics of quantum mechanics on one hand, and classical mechanics on the other, which arise as one lets Planck's constant tend to zero. Taken broadly, as in this seminar, the field makes connections between a number of branches of mathematics (analysis, differential geometry, complex geometry, number theory...).

In the seminar we'll work through significant papers in the field, with a view towards possiblitities for new developments. Participants will be encouraged to present papers they are interested in. New results by local and outside researchers will also be presented.

The two-hour slot allows time for interruptions from the audience, questions, discussions, etc. All of these are encouraged!

Theme for Fall 2010:

Semiclassical behavior of the eigenfunctions of the Schrödinger (including Laplace-Beltrami) operator.

It turns out that the eigenfunctions of a quantum system that is classically chaotic behave very differently from those whose classical system is quasi-periodic, as in completely integrable. There are many aspects to this issue though, and exploring them is the theme of this fall's seminar.

Here are some interesting papers that could be on the agenda, depending on participants' interest.

Schedule:

Monday, 9/20/2010. 

Speaker: A. Uribe

Title: Introduction to the seminar's themes and background material.

Monday, 9/27/2010. 

Speaker: Z. Wang

Title: The quantum ergodicity theorem of Shnirelman, Zelditch and Colin de Verdière.




Monday, 10/04/2010.

Speaker: Z. Wang

Title: On Hassell's stadia examples of non QUE.



Monday, 10/25/2010.

Speaker: Z. Wang

Title: On Hassell's stadia examples of non QUE, conclusion.



Monday, 11/1/2010.

Speaker: A. Uribe

Title: Pointwise behavior of the Husimi function of eigenfunctions (work with T. Paul).




 Monday, 3/28/2011.

 Speaker: Kin Kwan Leung 
			     
 Title:  Introduction to Zoll Manifolds and a Study in Zoll Surfaces, I



 Monday,  4/4/2011. 

 Speaker: Kin Kwan Leung 

 Title:  Introduction to Zoll Manifolds and a Study in Zoll Surfaces, II. 


 Moday, April 11, 2011. Special Double Presntation 
 First Speaker: Kin Kwan Leung

 Title: Zoll Manifolds and Complex Surfaces (Abstract below.)

 Second Speaker: Becky Hoai 

 Title: Compact Rank One Symmetric Spaces, or CROSSes (Abstract below)




 Zoll Manifolds and Complex Surfaces

 Abstract: In the talk, I will summarize the results of LeBrun and Mason `Zoll Manifolds and Complex Surfaces' by  stating the results when M=S^2 thatany Zoll metrics which is sufficiently close to the standard one is going to be represented by a totally real RP^2 in CP^2 which satisfies certain properties.



 
 Compact Rank One Symmetric Spaces 

 Abstract:  The compact rank one symmetric spaces (CROSSes) are precisely the Euclidean spheres, the projective spaces $\mathbb KP^n$ (where $\mathbb K$ is the field of real numbers, the field of complex numbers, or the skew field of quaternions), and the Cayley plane. They have a rich geometric structure and are the basic examples of Zoll manifolds, i.e. manifolds all of whose geodesics are closed and of equal length. In this talk and next week's, I will discuss some of the geometric and topological properties of the CROSSes, as well as their relation to the Blaschke conjecture.

If you have any questions please contact the organizers, Alejandro Uribe and/or Zuoqin Wang.