Victor Ivrii

Wed, Oct 22:
5.30pm, room 310
Fri, Oct 24:
7.10pm, room 308
Wed, Oct 29:
5.30pm, room 310

Description

What is Microlocal Analysis?

Microlocal Analysis is a theory of Pseudo-Differential Operators and Fourier Integral Operators (PDOs and FIOs). It draws on many rich traditions in mathematics including the links between analysis and geometry and has a powerful application to partial differential equations (PDEs). For example, all the significant progress in linear PDEs for the last 35 years has been based on microlocal analysis and many important advances in non-linear PDEs over this period were also based on microlocal analysis.

PDO are generalization of (partial) differential operators (DO) and PDO appear as parametrices (almost inverse) of elliptic DO (and parabolic DO unless we stick with classical'' PDOs). 0-order PDOs are singular integral operators. There is a very nice calculus of PDOs which make them extremely useful tool.

FIO appear as propagators of hyperbolic and similar equations.

Microlocal analysis comes in few flavors. One of the possible classifications is: semi-classical with explicit asymptotics with respect to a small parameter $h$ (Plank constant or wavelength) and classical (standard or more general) where there is no explicit parameter and asymptotics is with respect to the smoothness (or something like this). However, there is still implicit small parameter and microlocal analysis is asymptotical by its nature.

We will consider semi-classical PDOs and FIOs: one can always get from there to standard PDOs and FIOs but semi-classical theory has more applications.

Less formal description of (semi-classical) microlocal analysis is a semi-classical limit of quantum mechanics'' or high-frequency limit of electrodynamics'' (etc). This is really difficult and exciting topic to understand how quantum mechanics yields classical mechanics as Plank constant $h$ goes to $0$ (in contrast to trivial Special relativity yields Newton mechanics as light speed $c\to \infty$; the fundamental difference is that quantum and corresponding classical objects are mathematically completely different while special relativity and Newton mechanics objects are exactly the same). However course will not be physical.

So, Microlocal means local both in coordinates and momenta. There are direct and deep connections with Global Analysis.

Knowledge of PDE would be an asset but is not necessary.

Lecture 1: Pseudodifferential operators

• Quantization, symbols and operators. Calculus. Boundedness.
• Wave Front sets. Microlocalization. Logarithmic uncertainty principle.

Lecture 2: Propagation and Fourier Integral Operators

• Propagation of Singalarities: the problem. Heisenberg approach (evolution of PDOs). Classical and Quantum evolution.
• Propagation of Singularities: the problem. Heisenberg approach (evolution of PDOs). Classical and Quantum evolution.
• Energy estimates method.

Lection 3: Spectral Asymptotics

• Problem. Sharp spectral asymptotics. Tauberian Method.
• Poisson relation. Sharp spectral asymptotics and closed geodesics. Weyl conjecture.