|Analysis and geometry in metric spaces - Fall 2017 and Spring 2018|
|Lecturer||Prof. S. Wenger|
Fridays 10:15-12:00, Science de la terre 2.312 in the Fall, Physics building 2.52 in the Spring|
22 September 2017
Course description: This course gives an introduction to basic topics from metric geometry and analysis on metric spaces relevant in many areas of mathematics and also of current research interest. Metric geometry is concerned with the study geometric properties of (singular) metric spaces and with notions such as curvature bounds via triangle comparison. Analysis on metric spaces is the study of first order differential calculus in spaces with a priori no differentiable structure.
Singular metric spaces often appear naturally as limits of smooth objects. For example, the surface of a cube can be obtained by blowing up a balloon inside the cube until it fills the whole cube. Similarly, when observing a Riemannian manifold from further and further away one typically sees a singular metric space rather than a smooth Riemannian manifold "in the limit". Indeed, when seen from infinitely far away the hyperbolic plane looks like a tree. This course will introduce basic notions and tools from metric geometry and analysis on metric spaces, two active fields of research.
Topics of part I (Fall 2017) include: length structures, Lipschitz and biLipschitz mappings between metric spaces, quasi-isometries, Gromov-Hausdorff convergence of metric spaces. Topics of the second part II (Spring 2018) include: Rademacher type theorems, differentiable structures, Sobolev mappings between metric spaces, Poincare inequalities, quasisymmetric homeomorphisms, Bonk-Kleiner uniformization theorem.
Important information: The first part of this course (Fall 2017) can either be combined with the second part (Spring 2018) of the course or with the course "Metric Geometry" given by C. Ciobotaru in Spring 2018. The second part of this course (Spring 2018) can also be combined with the course "Quasiconformal mappings" given by K. Faessler in Fall 2017.
Outline of Part 1 - Fall 2017:
A) Metric spaces:
Tentative outline of Part 2 - Spring 2018 (might change):
A) Measure theoretic background:
D) Sobolev functions in metric spaces
F) Quasisymmetric embeddings