UCSB Distinguished Lectures in the Mathematical Sciences

Jeff Cheeger, April 26-28, 2011

 Tuesday, April 26, 2011, 3:30 p.m. South Hall 6635 (Refreshments at 3:00 p.m.) Jeff Cheeger Courant Institute of Mathematical Sciences, New York University Title: Quantitative Differentiation Abstract: The lectures will deal with the notion of quantitative differentiation and its applications. The simplest instance concerns functions $f:[0,1]\to \mathbb{R}$ with $|f'|\leq 1$. The basic assertion, which appears in work of Peter Jones from 1988, can be paraphrased as stating that in a precise quantitative sense, "$f$ is as close as one likes to being linear at most locations and scales". In the first lecture, it will be explained how the above is actually a particular case of something considerably more general. An "axiomatic" formulation is given in an appendix to a joint paper with B. Kleiner and A. Naor. This paper is discussed in the second lecture. The specific quantitative differentiation result concerns Lipschitz maps from the Heisenberg group to $L_1$. It turns out that there is an application to theoretical computer science. In the third lecture we will explain a quantitative differentiation result in riemannian geometry, which leads to curvature estimates for Kähler-Einstein manifolds off sets of small volume.
 Wednesday, April 27, 2011, 3:30 p.m. South Hall Room 6635 Jeff Cheeger Courant Institute of Mathematical Sciences, New York University Title: Lipschitz maps to L1 Abstract: The lectures will deal with the notion of quantitative differentiation and its applications. The simplest instance concerns functions $f:[0,1]\to \mathbb{R}$ with $|f'|\leq 1$. The basic assertion, which appears in work of Peter Jones from 1988, can be paraphrased as stating that in a precise quantitative sense, "$f$ is as close as one likes to being linear at most locations and scales". In the first lecture, it will be explained how the above is actually a particular case of something considerably more general. An "axiomatic" formulation is given in an appendix to a joint paper with B. Kleiner and A. Naor. This paper is discussed in the second lecture. The specific quantitative differentiation result concerns Lipschitz maps from the Heisenberg group to $L_1$. It turns out that there is an application to theoretical computer science. In the third lecture we will explain a quantitative differentiation result in riemannian geometry, which leads to curvature estimates for Kähler-Einstein manifolds off sets of small volume.
 Thursday, April 28, 2011, 3:30 p.m. South Hall Room 6635 Jeff Cheeger Courant Institute of Mathematical Sciences, New York University Title: Curvature estimates for Kähler-Einstein manifolds Abstract: The lectures will deal with the notion of quantitative differentiation and its applications. The simplest instance concerns functions $f:[0,1]\to \mathbb{R}$ with $|f'|\leq 1$. The basic assertion, which appears in work of Peter Jones from 1988, can be paraphrased as stating that in a precise quantitative sense, "$f$ is as close as one likes to being linear at most locations and scales". In the first lecture, it will be explained how the above is actually a particular case of something considerably more general. An "axiomatic" formulation is given in an appendix to a joint paper with B. Kleiner and A. Naor. This paper is discussed in the second lecture. The specific quantitative differentiation result concerns Lipschitz maps from the Heisenberg group to $L_1$. It turns out that there is an application to theoretical computer science. In the third lecture we will explain a quantitative differentiation result in riemannian geometry, which leads to curvature estimates for Kähler-Einstein manifolds off sets of small volume.