Atiyah-Singer Index Theorem

這學期會決定修幾何除了排課上的技術問題外
主要的動機應該源於三年前丘成桐的演講
而最後一根稻草則是去年讀到的這篇nature review

如果說我學classical diff geo有什麼實用目的
大概就是弄清楚Gauss-Bonnet Thm的來龍去脈
近日蔡宜洵開始談G.-B.T.時說
G.-B.T.只是Atiyah-Singer Index Thm的special case
才驚覺Gauss-Bonnet Thm的優雅深邃
並不止於其最為人熟知的表象上(幾何與拓樸的橋樑)
其根源於Gauss的測地實務而至管窺宇宙最深層的實在
......
證明了一個定理真的不代表了解該定理背後可能的蘊涵
A.-S.I.T.某些版本的證明手法甚至是受到物理的啟發而來
隱藏在世界繽紛樣貌背後那一以貫之的道啊！

下節錄自Atiyah和Singer獲頒Abel Prize時的prize announcement──

Scientists describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas, called differential equations, involving their rates of change. Such formulas may have an "index," the number of solutions of the formulas minus the number of restrictions that they impose on the values of the quantities being computed. The Atiyah–Singer index theorem calculated this number in terms of the geometry of the surrounding space.

A simple case is illustrated by a famous paradoxical etching of M. C. Escher, "Ascending and Descending," where the people, going uphill all the time, still manage to circle the castle courtyard. The index theorem would have told them this was impossible.

不得不提一下
title連結的那篇文章寫得十分清晰易懂
沒什麼接觸數學的人看了應該也會大受感動才是XD

BTW,
因為那篇文章我才知道原來theorem跟theater同字源──
theorem: worthy of closer examination and appreciation
僅僅一個字就道盡了數學本質上的真與美！

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posts in this series:
Hairy Ball Theorem
Brouwer Fixed Point Theorem,
Nash Equilibrium