Title:
  "Rademacher's theorem for metric measure spaces"
Lecturer:
  David Bate (Helsinki)



The abstract:
Rademacher's theorem states that any Lipschitz function on Euclidean space is differentiable Lebesgue almost everywhere. It is a fundamental result in geometric measure theory. For example, it shows 
that any rectifiable set possesses a weak tangent plane at almost every point. Moreover the statement itself is very interesting in its own right; the fact that a seemingly rather simple condition 
can impose such strong regularity is quite remarkable. Recently there has been a wealth of interest in generalising results of classical analysis to the setting of metric spaces. Naturally, 
Rademacher's theorem is a candidate for such a generalisation.

This course will focus on a new proof of Cheeger's generalisation which replaces the domain with a doubling metric measure space that satisfies a Poincare inequality.  We will naturally cover many 
concepts used throughout the field of analysis on metric spaces.

Specific titles:
Analysis on metric spaces and the Poincare inequality.
Rademacher's theorem for metric measure spaces: Cheeger's theorem and Alberti representations.
Alberti representations from the Poincare inequality.
Gromov--Hausdorff convergence, weak tangent spaces, and concluding the proof of Cheeger's theorem.