Tue. October 20 | 4:30 p.m. | EA160 |
Wed. October 21 | 4:30 p.m. | SM1153 |
Thurs. October 22 | 4:30 p.m. | EA160 |
A fundamental theorem of Rademacher asserts that Lipschitz functions
f: R^{n} -->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,). 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 , 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
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.