一、 課程背景與目的：

Geometric flow is an active field in geometric analysis. It has played a prominent role in several recent breakthrough in geometry, topology and mathematical physics: Perelman used Ricci flow to prove the Poincare conjecture in 2003 (based on Hamilton's work from 1982 to 1997); Huisken-Ilmanen used inverse mean curvature flow to prove the Riemannian Penrose inequality in general relativity. Geometric flow also appears ubiquitously in applications. Applied mathematicians use mean curvature flow to smooth out rough data and study surface tension in fluids and materials.

This minicourse introduces students the basic ideas geometric flow through curve shortening flow. We will follow professor R. Haslhofer's lecture notes and learn Huisken's proof of the classical theorem of Gage-Hamilton and Grayson using maximum principle, Harnack inequality and blowup analysis.

幾何流是幾何分析裡非常活躍的領域，在近期幾何、拓樸、數學物理的重大突破中扮演重要的角色：Perelman 2003 年用 Ricci flow 證明 Poincare 猜想(奠基於 Hamilton 1982-1997 的工作)；Huisken-Ilmanen 2001 年用 inverse mean curvature flow 證明廣義相對論的 Riemannian Penrose 猜想。幾何流也有廣泛的實際應用，應用數學家用 mean curvature flow 光滑化粗糙的圖片、資料以及研究流體與材料中的表面張力。

這個短期課程藉由 curve shortening flow 介紹幾何流的基本想法與技巧。課程教材參考 Haslhofer 教授的講義。我們會學到 Huisken 如何用極大值原理, Harnack 不等式以及 blowup analysis證明 Gage-Hamilton 和 Grayson 的經典定理。

 

二、課程之大綱與講者：

王業凱 Curve Shortening Flow

李國瑋 Review of plane curves and heat equations

陳志偉 Introduction to geometric flow

 

三、課程詳細時間地點以及方式：

1.13 9:30-10 am Registration, 10-12 am Lecture 1, 1-4 pm Problem session 1

1.14 10-12 am Lecture 2, 1-4 pm Problem session 2

1.15 10-12 am Lecture 3, 1-4 pm Problem session 3

1.16 10-12 am Lecture 4, 1-4 pm Problem session 4

1.17 10-12 am Lecture 5, 1-2 pm Free discussion

 

四、課程及報名網頁：

http://myweb.ncku.edu.tw/~ykwang/2020CSF