Zygmund-Calderón Lectures in Analysis

What is Microlocal Analysis?

Monday April 20th 4:00pm-5:00pm in E 202

One possible answer is that microlocal analysis explores the classical/quantum correspondence in the theory of partial differential equations. Its historical origins in mathematics were rather different, and the Calderon-Zygmund theory of singular integrals played a central role in the development of the subject. But over the years the parallels with the classical/quantum correspondence have become a unified theme of the subject in its many guises.

I will attempt to present this unifying theme by discussing the following questions (with the answers provided by the listed authors):
- Can random matrices enhance classical/quantum correspondence? (Bordeaux, Hager, Sjostrand)
- Do smooth (non-analytic) functions appear in nature? (Melrose, Taylor, Lebeau)
- Does considering primes as semiclassical parameters, p = hbar, make sense? (Belov-Kanel, Kontsevitch)


What are Quantum Resonances?

Tuesday April 21st 4:30pm-5:30pm in E 206

Quantum resonances describe metastable states and replace eigenvalues in more realistic situations in which decay or escape are possible. Under many names (scattering poles, quasinormal modes, zeros of zeta functions) they appear in many branches of mathematics and physics: typically, the real part is interpreted as rest energy or frequency and the imaginary part as the rate of decay. I will give clear mathematical definitions in some simple settings illustrated by "live" numerical experiments. I will also present some applications of results on resonances to linear and non-linear PDEs.


Microlocal Methods in the Study of Resonances.

Wednesday April 22nd 4:15pm-5:15pm in E 203

Metastable states are both intuitively and mathematically associated with trapping: that is already seen in calling eigenstates "bound states". Trapping is a geometric or classical condition and the study of the relation between trapping and the distribution of resonances has been successfully explored by microlocal methods. I will discuss recent results on chaotic scattering: the fractal Weyl laws for the density of resonances and estimating quantum decay rates using topological pressure of the dynamical system. These results have been obtained in joint work with Sjostrand, and with Nonnenmacher. They will also be illustrated using numerical and experimental results.