也許開始看一點Lecture 問題是 看得懂嗎

以下是AMS的公告(Bulletin)

[The isospectral problem for flat tori from three perspectives]

 Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed.

In 1964, Milnor used a construction of Witt to find an example of isospectral nonisometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science of the USA.

Milnor's example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric.

A natural question is, What is the lowest dimension in which such isospectral nonisometric pairs exist?

This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language.

We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s.

Moreover, we share a number of open problems.

還有[Sunflows,from soil to oil]  [Stable black hole : in vaccum and beyond]

這兩天一直在讀偏微分方程 孤粒子(soliton)---KdV方程的故事 預作Ricci soliton(Ricci flow)的準備

當然是為了讀Poincare猜想的證明

Ricci flow為何重要 看看Fields Medal得主 有四個人都與之有關 就可見一般

1962 John Milnor  1966 Stephen Smale  1986 Michael Freedman  2006 Grigori Perelman

然後問自己 可以開始讀一點論文了嗎 例如 [Isospectral Problem]

12/30 何謂可積系統

可積系統的現代理論 因為Martin D. Kruskal(1925~2006)與Norman Zabusky(1929~2018)在1965年發現一些孤立子而復活

(Korteweg-de Vries 方程中的孤立子)

Frobenius定理是局部的 而 可積性 在動力系統的意義上是大域性質

以現代意義而言 是相空間(phase space)系統解的幾何或拓撲性質

以上是今天早上 看了[可積的動力系統] [孤立子] 的一點結論 對孤立波有多一點認識

[A brief introduction to Solitons]

12/22到境主 修緣禪寺參拜 早上M說 前天 "似乎"濟公活佛有來看她一下

俊哥說 或許因為她那裡有一些寶物吧