孙奥博士短期课程
2020年12月29日 21:30-23:00 腾讯会议 https://meeting.tencent.com/s/4m4F1sa0v4Vs 会议 ID：280 317 626
2021年1月2日 21:30-23:00 腾讯会议 https://meeting.tencent.com/s/8TqU0kmy4991 会议 ID：715 817 583
2021年1月5日 21:30-23:00 腾讯会议 https://meeting.tencent.com/s/iFA4VK24fZGC 会议 ID：508 617 973
2021年1月7日 21:30-23:00 腾讯会议 https://meeting.tencent.com/s/oPqVKot2HnZQ 会议 ID：679 197 119
标题：Complexity of elliptic and parabolic systems
摘要：This series of talks will be focused on Colding-Minicozzi's recent work “Complexity of parabolic systems”
Publications mathématiques de l'IHéS volume 132, pages83–135(2020).
We will discuss the first part of this paper. One of the main goals is to understand the dimension of the space
of linear parabolic equations. This problem is originated from Colding-Minicozzi's earlier work on the space of
harmonic functions, see "Harmonic functions on manifolds" Ann. of Math. (2) 146 (1997), no. 3, 725–747. The talks
are organized as follows:
1. We will first discuss the growth of harmonic functions on a manifold. We sill start from the classical Liouville's theorem.
Then we will discuss Yau's of Liouville type theorem on a Riemannian manifold. Then we will move forward to Colding-Minicozzi's
result on the dimension of the space of harmonic functions.
2. Next, we will discuss the space of ancient solutions to parabolic equations. This includes the related work by Lin-Zhang
"On ancient solutions of the heat equation", Commun. Pure Appl. Math., LXXII (2019), 2006–2028, and Colding-Minicozzi
"Optimal bounds for ancient caloric functions", preprint.
3. Then, we will introduce some basic backgrounds on mean curvature flow. Then we will discuss how previous work on ancient
parabolic equations can be applied to the study of mean curvature flow.
4. If time permits, we will discuss some possible generalizations, including the speaker's work "Codimension Bounds
and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$" (joint with Douglas Stryker)
