HUT / Department of Mathematics / teaching

Mat-1.3355 Microlocal Analysis (4-6 credits), autumn 2005

In microlocal analysis we analyse functions and operators in the phase space. For example R³'s phase space consists of pairs (x,p), where x is a point in the space and p is a direction. When we study, for instance, the solutions of the wave equation we can ask how the discontinuities and other singularities of the wave propagate.

The tools of microlocal analysis can be used in the research of the solutions of various differential operators. Some of these tools are the concept of the wave front set of a distribution, pseudodifferential operators and Fourier integral Operators. The goal of the course is to give the fundamentals of these tools. Examples of the applications of microlocal analysis can be found in the inverse problems (especially in medical and geophysical imaging), the spectral theory of operators and differential geometry.

Microlocal analysis has formed an important bridge between analysis, geometry and mathematical physics. On 2004 Sir Michael Atiyah and Isadore Singer got an Abel prize for their research on pseudodifferential operators.

Prerequirements: Mat-1.015 Fundamentals of Modern Analysis (ModA) and Mat-1.140 Fundamentals of Functional Analysis.

Lectures: Wed. 12--14 and Fri. 12--14, U345 (starts 14.9.2004.), professor Matti Lassas, e-mail: Matti.Lassashut.fi, Y319, p. 451 3069.

Exams: The exam date is agreed on during the course

Exercises: Wed. 14--16 U345 Assistant Pekka Tietäväinen, e-mail: Pekka.Tietavainenmath.hut.fi, Y336.

Exercises are available also on a folder next to the assistants door. Students solve the exercises independently at home and the solutions are discussed during the exercise sessions. For the done exercises one gets points, which are taken into account when grades are given. Model solutions for the exercises can be found in the course folder in the math departments library. There will be 11 sets of exercises.

Exercises: Can be found in the finnish site .

Course Book: Grigis and Sjöstrand: Microlocal analysis for differential operators, Cambridge University press, 2000.

Background Information:
Walter Rudin, Functional Analysis.
Lars Hörmander, The Analysis of Linear Partial Differential Operators I

Progress of the Lectures: Lecture notes are in the library. For independent study there is a list of  relevant sections of the course book underneath.

Welcome to the course!

Matti Lassas

Pekka Tietäväinen