本活动计划连续地用三到五年甚至更长的时间，瞄准Minkowskiw问题相关主流问题，以"学研"为出发点, 从凸几何、偏微分方程、质量输运三个专题基础开始, 在打好凸几何分析、Monge-Ampere方程理论、最优输运基础的同时向团队青年教师和研究生介绍Minkowski问题相关热门问题，从而推动湖南大学“偏微分方程与几何分析”团队的发展，加深国际同行特别是青年科研人员在该领域的合作研究。
资助: 国家自然基金天元专项(11826014)
课程一：Interior regularity for Monge-Ampere equations
主讲人：刘佳堃（澳大利亚Wollongong大学）
时间： 2019年5月20, 21日 19:30-21:30. 5月 23日15:00-17:00
地点：数学院432
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.
课程二：An Introduction to Minkowski-type problems in convex geometry
主讲人：赵翌铭（博士后, Massachusetts Institute of Technology）
时间： 2019年6月16日晚上19：00一21：30；17-19日上午9：00一11：30.
地点：数学院425
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.
