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.