Inverse problems on Riemann surfaces, fall 2009

Lecturer

Leo Tzou

Type

Advanced studies.

Lectures

Weeks 39-43 and 45-48, Tuesday 12-14 in room B120, Thursday 10-12 in room D123.

Note that the first lecture is Tuesday 22.9. and the course will end in late November.

Lecture notes

Lecture notes (in progress)

Exercices

The course may be taken for credit (8 credit units) by doing exercises which are given out in class and by writing an essay.

Exercise sheet (due on 04.12.2009)

Course description

In this course we will provide a complete solution for the inverse boundary value problem for the elliptic operator $\Delta + V$ on a Riemann surface. That is, on a Riemann surface $M$ we will recover the coefficient $V$ from boundary data of the solutions of the equation $(\Delta + V)u = 0$.

This problem turns out to have rich topological and geometrical structure. In this course we will see where these geometrical considerations arise and what are the tools needed to understand these problems. The course also gives an introduction to Riemann surfaces (no prior knowledge is assumed).

Prerequisites

Basic knowledge of real, complex, and functional analysis. Some familiarity with partial differential equations and differential geometry may be helpful.

Bibliography

We will follow the articles

C. Guillarmou and L. Tzou, Calderón inverse problem for the Schrödinger operator on Riemann surfaces, arXiv:0904.3804.
 C. Guillarmou and L. Tzou, Calderón inverse problem with partial data on Riemann surfaces, arXiv:0908.1417.

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