Analysis and geometry in metric spaces - Fall 2017 and Spring 2018
Lecturer Prof. S. Wenger

Start date
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:
  1. Definition and basic examples
  2. Lengths of curves, length structures
  3. Geodesics, Hopf-Rinow theorem
  4. Constructions (products, gluing, quotients)
B) Lipschitz extensions:
  1. McShane's lemma
  2. Kirszbraun's theorem
  3. Whitney type extensions
C) Embeddings of metric spaces:
  1. Kuratowski embedding
  2. Doubling property and Assouad embedding
  3. BiLipschitz non-embeddings
D) Quasi-isometries:
  1. Definition and examples
  2. Svarc-Milnor lemma
  3. Growth of groups
E) Limits of metric spaces:
  1. Hausdorff distance
  2. Gromov-Hausdorff convergence
  3. Gromov's compactness theorem

Tentative outline of Part 2 - Spring 2018 (might change):

A) Measure theoretic background:
  1. Outer measures
  2. Covering theorems
  3. Maximal functions
B) Differentiability of Lipschitz maps:
  1. Rademacher's theorem
  2. Metric derivatives
  3. Cheeger differentiable structures
C) Modulus of curves

D) Sobolev functions in metric spaces
  1. Classical Sobolev functions
  2. Upper gradients
  3. Comparison with Hajlasz's definition
E) Poincare inequalities

F) Quasisymmetric embeddings
  1. Basic Theory
  2. The Bonk-Kleiner uniformization theorem

  • D. Burago, Yu. Burago, S. Ivanov: A course in metric geometry; Graduate Studies in Mathematics, Vol 33, AMS.
  • M. Bridson, A. Haefliger: Metric spaces of non-positive curvature, Springer Verlag.
  • J. Heinonen: Lectures on Analysis on Metric Spaces, Springer Verlag.
  • J. Heinonen: Lectures on Lipschitz Analysis, University of Jyvaskyla.
  • J. Heinonen: Geometric embeddings of metric spaces, University of Jyvaskyla.
  • B. Kleiner, J. Mackay: Differentiable structures on metric measure spaces: A primer.
  • J. Heinonen, P. Koskela, N. Shanmugalingam, J. Tyson: Sobolev Spaces on Metric Measure Spaces, Cambridge University Press.