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学术报告：Series of Lectures by Prof. Christ from UCB

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报 告 人：Prof. Michael Christ, Department of Mathematics, University of California, Berkeley

简 介： Professor Michael Christ is a Professor of Mathematics at UC Berkeley. He earned his B. S. from Harvey Mudd College and his Ph.D. from the University of Chicago. He had received numerous honors and awards, including an NSF Presidential Young Investigator Award and a Sloan Fellowship in 1986, the 1997 Bergman Prize from the AMS, and a Miller Research Professorship for 2000-2001. In 2002, he received the Mathematics Distinguished Teaching Award from the Mathematics Undergraduate Student Association at UC Berkeley. He also received a 2004 Distinguished Teaching Award by the Office of Educational Development at UC Berkeley. He has been an invited lecturer twice at the International Congress of Mathematicians, first in Kyoto in 1990 and then in Berlin in 1998. In 2007, he was elected a Fellow of the American Academy of Arts and Sciences. His areas of research are harmonic analysis, partial differential equations, and complex analysis in several variables.

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Lecture 1:

报告题目：Near-extremizers for Young's inequality in Euclidean space
报告地点：数学中心201多媒体教室
报告时间：6月21日下午2点
摘 要:

Inequalities for operators between Lebesgue spaces are a fundamental aspect of modern analysis. While most substantive inequalities involve unspecified finite constant factors, there are a few instances in which optimal constants and extremizing functions can be determined. Among these is Young's convolution inequality in Euclidean spaces, for which the optimal constant is strictly less than for general groups. Extremizers are Gaussians, as was first shown by Beckner and by Brascamp-Lieb in the mid-1970s. Subsequently other illuminating perspectives on the problem have been discovered, in particular by Lieb, Ball, Barthe, and Carlen-Lieb-Loss.

In this talk we analyze those functions which nearly, but not exactly, extremize the inequality, and show that such functions are necessarily nearly equal to Gaussians. The main ingredients in the analysis come from additive combinatorics.

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Lecture 2:

报告题目：On near equality in the Brunn-Minkowski inequality
报告地点：数学中心201多媒体教室
报告时间：6月21日下午3:30点
摘 要:

The Brunn-Minkowski inequality is a basic analytic inequality which relates Lebesgue measure, geometry of Euclidean space, and the algebraic structure of the Euclidean groups. It is related to the isoperimetric inequality, and to arithmetic progressions. It provides a lower bound for the measure of a sum set in terms of the measures of the summands.

It has long been known which sets are extremal, in the sense that the inequality becomes an exact equality for them. This talk concerns a characterization of those sets which nearly realize equality. An analogue of such a characterization, for the case of finite sets of integers, with cardinality replacing Lebesgue measure, is a basic result of additive combinatorics due to Freiman. This characterization involves arithmetic progressions. In this talk, we discuss the corresponding result for Euclidean space, as well as connections with other inequalities.