Webrelaunch 2020

RTG Lecture “Asymptotic Invariants and Limits of Groups and Spaces” (Wintersemester 2017/18)

Vorlesung: Dienstag 9:45-13:00 SR 2.58
Dienstag 9:45-13:00
Dienstag 15:00-16:30 SR 1.067
Dienstag 15:00-16:30
Dozent Prof. Dr. Roman Sauer
Sprechstunde: nach Vereinbarung
Zimmer 1.001 Kollegiengebäude Mathematik (20.30)
Email: roman.sauer@kit.edu

This semester’s RTG lecture will be split between courses by Tobias Lamm on “The Willmore functional” and by Daniele Alessandrini on “Thurston's theory of surfaces”.

The Willmore functional (Tobias Lamm)

Abstract: After reviewing some useful tools and definitions in Riemannian geometry we will introduce the Willmore functional

\mathcal{W}(f)=\tfrac{1}{4}\int_{\Sigma} |\vec{H}|^{2}d\mu_g ,
where M is an n-dimensional Riemannian manifold, f:\Sigma\rightarrow M an immersed surface and \vec{H} the mean curvature vector.
We will discuss the most important properties of \mathcal{W} such as its scale invariance and invariance under Möbius transformations. This property is crucial as it prevents us from using standard theory from the Calculus of Variations to prove existence of minimizers, the so called Willmore surfaces.
Further we will discuss some aspects of the associated gradient flow.
Later on in the lecture we will address more recent developments such as the proof of the Willmore conjecture by Marques and Neves and the cost of the minmax sphere eversion by Rivière.

Thurston's theory of surfaces (Daniele Alessandrini)

Abstract: The course is about the theory of surfaces, as developed by Thurston at the end of the 70s. The main aim is to prove Thurston's theorem of classification of homeomorphisms of surfaces up to isotopy, in principle a purely topological statement. To prove this theorem Thurston used geometric structures on surfaces, mainly hyperbolic metrics and singular measured foliations. The study of the parameter spaces of these geometric objects will give, as a corollary, the classification of homeomorphisms.