投稿

科普：复数与几何之间让人惊讶的联系

Name: %E7%A7%91%E6%99%AE%EF%BC%9A%E5%A4%8D%E6%95%B0%E4%B8%8E%E5%87%A0%E4%BD%95%E4%B9%8B%E9%97%B4%E8%AE%A9%E4%BA%BA%E6%83%8A%E8%AE%B6%E7%9A%84%E8%81%94%E7%B3%BB
Uploaded: 2024-06-03T14:50:36.422Z

1.7万

2024-04-29 16:44:36

https://www.youtube.com/watch?v=5RHSS-zMaAQ 这个视频简单介绍了黎曼面，展示了复分析与代数几何之间的深刻联系。作者以黎曼球面/二次曲线，复环面/椭圆曲线作为具体实例，通俗易懂地描述了如何将黎曼面对应到代数曲线。黎曼面虽然是相对简单的一类数学对象，但其中却蕴含了极其丰富的数学结构与数学现象，与几乎所有数学分支交汇。紧黎曼面与复射影曲线的对应，建立起一维紧复流形和一维复射影代数簇之间的桥梁。Serre在GAGA中将其进一步推至了高维，为解析几何与代数几何带来了丰富的比较定理。这不仅是上世纪复代数几何的高光，其思想也启发着非阿基米德解析几何的发展。原视频作者是 Aleph0，标题为 The shocking connection between complex numbers and geometry. 以下是原视频描述 A peek into the world of Riemann surfaces, and how complex analysis is algebra in disguise. SOURCES and REFERENCES for Further Reading: This video is a quick-and-dirty introduction to Riemann Surfaces. But as with any quick introduction, there are many details that I gloss over. To learn these details rigorously, I've listed a few resources down below. (a) Complex Analysis To learn complex analysis, I really like the book "Visual Complex Functions: An Introduction with Phase Portraits" by Elias Wegert. It explains the whole subject using domain coloring front and center. Another one of my favorite books is "A Friendly Approach To Complex Analysis" by Amol Sasane and Sara Maad Sasane. I think it motivates all the concepts really well and is very thoroughly explained. (b) Riemann Surfaces and Algebraic Curves A beginner-friendly resource to learn this is "A Guide to Plane Algebraic Curves" by Keith Kendig. It starts off elementary with lots of pictures and visual intuition. Later on in the book, it talks about Riemann surfaces. A more advanced graduate book is "Algebraic Curves and Riemann Surfaces" by Rick Miranda.

知识