# Regular Lectures - Winter semester 2011/2012

## Mathematical structures of economic theory: Jürgen Jost

- Date: Friday, 13:30 - 15:00 h
- Location: MPI MIS, room A 02
- First lecture: October 14th

In this course, I shall discuss the conceptual structure of contemporary economic theory and introduce the relevant mathematical methods, like game theory, dynamical systems, stochastic analysis.

Thus, the course will also give an overview of and an introduction to a wide range of mathematical topics.

## Topics in geometric analysis and the calculus of variations: Jürgen Jost

- Date: Monday, 13:30 - 15:00 h
- Location: MPI MIS, room A 02

Topics in geometric analysis and the calculus of variations

I shall discuss a variety of interesting topics, results, and problems in nonlinear analysis.

This course will meet somewhat irregularly.

## Techniques of variational analysis: Miroslav Bačák

- Date: Thursday, 10:15 - 11:45 h
- Location: MPI MIS, room A 02
- First lecture: October 13th

Description of the lecture: Variational arguments can be traced back to the early development of the calculus of variations. The discovery of modern variational principles and nonsmooth analysis further expand the range of applications of these techniques.

The lecture is an introduction to (mostly infinite-dimensional) first-order variational analysis, which includes several variational principles (Ekeland, Borwein-Preiss, DGZ, Stegall), subdifferential calculus, rudiments of convex analysis (convex subgradient, Fenchel duality theory, maximal monotone operators), and, as the peak of the lecture, nonsmooth analysis on smooth (finitely dimensional) Riemannian manifolds and its applications to spectral functions.

Our intention is to accommodate a broader audience, long and technical proofs will therefore be omitted. Various examples, applications, and open problems make the lecture more lively.

An elementary knowledge of functional analysis is the only prerequisite we anticipate.

The main reference for this lecture is the monograph "Techniques of variational analysis" by Borwein and Zhu.

## Selected topics in applied analysis: Felix Otto

- Date: Tuesday, 9:00 - 11:00 h
- Location: MPI MiS, room A01
- First lecture: October 11th

## Introduction to sets of finite perimeter: Emanuele Spadaro

- Date: Friday, 9:15 - 11:00 h
- Location: MPI MiS, room A01
- First lecture: October, 14th

Sets of finite perimeter are the most general class of sets

for which the notions of boundary measure and external normal can be

defined. They play an important role in many geometric and analytical

problems, such as minimal surface theory, capillarity problems, phase

separation models etc.

The aim of the course is to present the basic theory up to covering the

structure theorem. Depending on the time and the interests of the

audience, further regularity issues or applications to phase separation

problems can also be covered.

## IMPRS Ringvorlesung "Higher Dimensions": Felix Otto, Nihat Ay, Wolfgang Hackbusch

- Date: Wednesday, 13:30 - 15:00 h
- Location: MPI MiS, room G 10
- First lecture: October 12th
- see more info at IMPRS

## Seminar: The mathematical theory of Homogenization: Stephan Luckhaus

- Date: Thursday, 13:30 - 15:00 h
- Location: MPI MiS, room A 01
- First seminar: October 20th

The theory of Homogenization is about trying to understand relations between macroscopic and microscopic scales. More precisely, one is interested in recovering equations satisfied by macroscopic quantities from those satisfied by the physical quantities.

Mainly we will deal with the homogenization or upscaling of elliptic or parabolic partial differential equations with rapidly varying coefficients. In all problems considered existence of solutions for the microscopic problems is known. The difficulty is the derivation of equations for the averaged quantities. Here special notions of convergences like the two-scale convergence introduced by Nguetseng and Allaire, H-convergence and Gamma-convergence come into play, as well as the method of asymptotic expansion. There will be weekly presentations on selected papers. The seminar is addressed to graduate students planning a deep in the area of pure or applied analysis, Ph.D. students and everybody who is just interested in. Feel very welcome to participate.