美国科学院院士Jeff Cheeger教授公众报告

Title: Quantitative Differentiation and Generalized Differentiation.

Speaker: Professor Jeff Cheeger

Time: July 2, Monday, 16:00-17:00

Classroom: 565net必赢客户端教二楼927报告厅

Abstract:

       The simplest case of “quantitative differentiation” deals with functions , with derivative satisfying , but with no control of the second derivative, which might not even exist. There is a natural measure on the (countable) collection of all diadic intervals , where  and , which weights every interval by its length. The sum of the lengths is infinite. However, given , the sum of the lengths on which  fails to be -close to being linear (in a suitable scaled sense) is finite and bounded by . What makes the proof work can be abstracted and axiomatized. When combined with certain other ideas, this general version has many strong applications in geometric analysis. Until recently, were missed. Quite possibly, this happened because the above simplest case was not widely known to mathematicians.

       One instance of “generalized differentiation” refers to the possibility of doing first order calculus on certain spaces  with a metric  and a measure . Namely, there is a (nonobvious) notion of a “differentiable structure” on such spaces, generalizing the usual notion for say Lipschitz manifolds. If a differentiable structure exists, then a version of Rademacher’s Theorem asserting the almost everywhere differentiability of Lipschitz functions holds. There are two conditions of compatibility between  and  which taken together, guarantee that a differentiable structure exists: 1) a doubling condition, and 2) a Poincaré inequality. There are many examples including fractal spaces for which these conditions are verified. Using this generalized differentiation theory, the majority of these spaces can be shown not to bi-Lipschitz embed in  for any  or even in any  space for . Thus, they are truly exotic. 

Jeff Cheeger教授简介

Jeff Cheeger教授，美国科学院院士。主要研究领域为微分几何学，以及几何与拓扑、几何与分析之间的联系。本科毕业于哈佛大学（1964），在普林斯顿大学获得博士学位（1967）。先后于普林斯顿大学、密西根大学做博士后，在纽约州立大学石溪分校任副教授（1969），正教授（1971），leading正教授（1985），杰出正教授（1990）。1993年至今，为纽约大学柯朗数学科学研究所希尔弗教授。

Jeff Cheeger教授发现了很多当今黎曼几何学中最深刻的结果。诸如Laplace-Beltrami算子谱估计，对于torsion的几何定义和分析定义的等价性等方面的工作，直接导致了拓扑、图论、数论、马尔可夫过程等不同学科中一系列问题的解决。他与M. Gromov共同建立了度量几何学。这方面的开创性工作，在包括黎曼几何、几何分析等现代微分几何研究领域中，有着广泛、深刻的应用。

Jeff Cheeger教授因其杰出的学术贡献，先后于1974年和1986年在国际数学家大会上做特邀报告，1984年获得Guggenheim奖，1992年获得Max Planck Research奖。1997年当选美国科学院院士，1998年当选芬兰科学院外籍院士。2001年美国数学会授予Cheeger教授第十四届Oswald Veblen几何学奖。2012年当选美国数学会会士。