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發布時間:2019-05-06 10:19    瀏覽次數:    來源:

   本活動計劃連續地用三到五年甚至更長的時間,瞄準Minkowskiw問題相關主流問題,以"學研"為出發點, 從凸幾何、偏微分方程、質量輸運三個專題基礎開始, 在打好凸幾何分析、Monge-Ampere方程理論、最優輸運基礎的同時向團隊青年教師和研究生介紹Minkowski問題相關熱門問題,從而推動湖南大學“偏微分方程與幾何分析”團隊的發展,加深國際同行特別是青年科研人員在該領域的合作研究。

資助: 國家自然基金天元專項(11826014)

課程一:Interior regularity for Monge-Ampere equations


時間: 2019520, 21日 19:30-21:30.   5月 23日15:00-17:00


摘要: In this short course we discuss the interior a priori estimates for the Monge-Ampere equation, such as the strict convexity, interior $C^{1,\alpha}$, $C^{2,\alpha}$ and $W^{2,p}$ estimates for convex solutions to the Monge-Ampere type equation. We also give a brief discussion on the regularity for more general Monge-Ampere type equations arising in optimal transportation.


課程二:An Introduction to Minkowski-type problems in convex geometry

主講人:趙翌銘(博士后, Massachusetts Institute of Technology

時間: 2019年6月16日晚上19:00一21:30;17-19日上午9:00一11:30.


摘要:The target audience for this mini-lecture series are graduate students or junior/senior undergrad students with an interest in convex geometry, differential geometry, geometric analysis, or nonlinear PDE. Throughout the lecture series, we will be working with convex bodies in \mathbb{R}^n. The boundaries of these convex bodies are in general not smooth, which makes them natural in the setting of Euclidean geometry. We will see how the missing smoothness assumption impacts our study of boundary shapes; for example, what is the natural replacement of Gauss curvature.

These eventually lead to a family of Minkowski-type problem which characterize geometric measures related to convex bodies. Two problems that we are going to talk about are the classical Minkowski problem and the recently posed dual Minkowski problem (Huang-Lutwak-Yang-Zhang, Acta 2016). Minkowski problems link many fields of mathematics. In particular, in differential geometry, it is known as the problem of prescribing Gauss curvature; in PDE, it reduces to Monge-Ampere type equations. But, we shall discuss how these problems can be solved without any smoothness assumptions on the given data using calculus of variation.

This mini-lecture series is a gentle introduction to get the audience up-to-speed with the most recent research on Minkowski type problems, in particular the dual Minkowski problem which has received much attention in recent years.


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