## Advanced topics in analysis

Lecturers: Andreas Deuchert and Alessandro Olgiati

Lecture: Tuesday and Thursday from 3:00 - 4:45 pm

Exercises class: Monday from 3:00 - 4:45 pm

ECTS: 5 points

General description: The lecture covers advanced topics in analysis that are essential for the understanding of several areas of modern analysis as e.g. the study of partial differential equations, mathematical physics, and geometric analysis. Topics to be covered are: Fourier transform, distributions (generalized functions), weak derivatives, Sobolev spaces, weak and strong convergence, and Sobolev inequalities. Afterwards, these tools will be applied to study the Schrödinger equation (existence and properties of solutions) via the direct method in the calculus of variations.

The lecture can be booked by Bachelor students and runs for eight weeks, that is, it ends after the first half of the semester. Afterwards, it continues as a Bachelor/Master course in the second half of the semester under the title “Variational Methods in Analysis”. The second part of the lecture is based on the first half. It aims at introducing the audience to techniques from the calculus of variations. Although these techniques are very general, we will, for the sake of concreteness, introduce them in the framework of certain models originating from atomic physics. More details can be found in the description of the lecture “Variational Methods in Analysis”.

Prior Knowledge: Analysis 1-3, Linear Algebra 1

Course Material: Handwritten lecture notes

Learning outcome: Understanding of advanced techniques in analysis that are necessary for the understanding of modern pde theory, mathematical physics and geometric analysis.

Lecture notes:

`Chapter 0 and 1`

`Chapter 2`

`Chapter 3`

`Chapter 4`

`Chapter 5`