Prospectus

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Topics in analysis

Course 2010-2011

The topic is “Gradient flows in metric spaces”. A gradient flow is a function y satisfying y’(t)=F(y(t)), where F=-grad H for some (convex) potential function H. Thus y describes a flow in the direction of the steepest descent of H. During the last decade a theory for gradient flows in metric spaces has been developed with many connections with other active fields of research. The course will focus on three aspects: (I) the formulation of a gradient flow in a metric space as an Evolution Variational Inequality, an existence and uniqueness result by Ambrosio, Gigli and Savare (2005), and examples in Hilbert spaces; (II) spaces consisting of probability measures equipped with a Wasserstein metric and the relation with famous optimal transportation problems of Monge and Kantorovich; (III) the Fokker-Planck partial differential equation for densities of the laws of stochastic processes given by certain stochastic differential equations can be interpreted as gradient flows in Wasserstein spaces.

By taking this course you will learn some advanced measure theory, a functional analytic approach to differential equations, and you will get an introduction to various topics such as the Wasserstein metric, optimal transportation problems, and stochastic differential equations.

Hours per week
2×45 min lecture.
Exam
Homework assignments and (possibly) an oral exam
Compulsory literature
Lecture notes by the lecturer, which will become available on the webpage of the course
Prerequisites
Basic knowledge of metric spaces (Topologie), measure theory, differential equations (Analyse 3), and some introductory course to functional analysis (Linear Analysis is more than enough). Probability theory is not a prerequisite.
Recommended for
Those with an interest in metric spaces, functional analysis, measure theory and/or connections with differential equations and stochastic processes
Eligible for
Master track Applied Mathematics or Bachelor study
Remarks
If all participants have access tot the Leiden Blackboard environment, then we can use that. Otherwise a link tot the course’s webpage will be provided on the lecturer’s homepage, see below.

Links
Lecturer’s homepage