The Ohio State University


announces the
1998 Radó Lectures
to be given by

Jeff Cheeger

Jeff Cheeger
Courant Institute of Mathematical Sciences
New York University

on the topic

Differentiability of Lipschitz Functions
on Metric Measure Spaces

Tue. October 20 4:30 p.m. EA160
Wed. October 21 4:30 p.m. SM1153
Thurs. October 22 4:30 p.m. EA160

Room EA160 is in the Math Annex, 209 West 18th Avenue,
and SM1153 is in Smith Laboratory, 174 W. 18th Avenue.


A fundamental theorem of Rademacher asserts that Lipschitz functions f: Rn -->R, are differentiable almost everywhere. In our lectures, we will describe an extension of Rademacher's theorem to a certain class of metric measure spaces, (Z,\mu). As a consequence of this extension, much of that part of calculus which concerns first derivatives (although not, of neccessity, the implicit function theorem) generalizes to spaces in this class. In various natural examples, for instance, the boundaries at infinity of 2-dimensional hyperbolic buildings, the measure \mu, is Hausdorff measure and the Hausdorff dimension exceeds the topological dimension. In particular, the differentiability of Lipschitz functions (suitably interpreted) does not imply the existence of points at which the underlying space has a linear structure at the infinitesimal level.

Lecture 1. Overview and basic concepts
Lecture 2. Doubling measures, Poincaré inequalities and differentials
Lecture 3. Length spaces and a generalization of Rademacher's theorem

About Jeff Cheeger

Jeff Cheeger received his Ph.D. in 1967 from Princeton, writing his dissertation under the direction of Salomon Bochner. He has held faculty positions at Berkeley and SUNY at Stony Brook. Since 1990 he has been affiliated with the Courant Institute of Mathematical Sciences. He has won numerous scientific awards and distinctions, including a Sloan, Guggenheim, the Max Planck Research Award of the Alexander von Humboldt Society, 1992 Marston Morse Lecturer at the Institute for Advanced Study, 1997 Blyth Lecturer at the University of Toronto, 1997 Andrejewski Lecturer at the University of Göttingen, and invited speaker at the International Congress of Mathematicians in 1974 and 1986. He was elected a member of the National Academy of Sciences in 1997. His major research interests are differential geometry and its connections to topology and analysis.