This post records the roadmap and resources for my mathematics learning.

My Math Learning Roadmap

Before listing the books, here are a few suggestions for myself and for anyone following a similar path:

• Ask AI.
• Do exercises.

For readers from Mainland China (中国大陆), I have an additional suggestion section in the end of this post.

Stage 1 (Current)

• Sets, Models and Proofs
• Linear Algebra Done Right
• Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 5th edition
• The Princeton Companion to Mathematics
• The Princeton Companion to Applied Mathematics

Stage 2

• Topology — James R. Munkres
• Topology Without Tears
• Visual Group Theory
• Measure, Integration & Real Analysis, written by Sheldon Axler, the author of Linear Algebra Done Right.

Interesing but not necesssary for me:

• Visual Complex Analysis: 25th Anniversary Edition
• Proofs from THE BOOK

Stage 3

• Category Theory in Context
  • It's always good to learn the language of category theory as it's very high-level, and it's especially helpful for communcation with pedantic researchers :)
• Elementary Functional Analysis - Barbara D. MacCluer
• Fourier Analysis: An Introduction - Elias Stein and Rami Shakarchi
• Ordinary Differential Equations - Vladimir Arnold

I'm not into Fourier Analysis and ODE (and, of course, PDE), but they're prerequisite for some Physics and enginnering topics so I put them here. If you're neither interested in Physics nor have passion to these Math, you generally don't have to learn them.

Stage 4 (Hope to learn before my death)

• The Rising Sea, Foundations of Algebraic Geometry by Ravi Vakil

Interesting Blogs

• Who Can Understand the Proof? A Window on Formalized Mathematics

  by Stephen Wolfram, creator of Mathematica.

• Terrence Tao’s Blog

  Some books and lecture notes written by Tao are also useful.

Book Recommendations

I often use ChatGPT and Reddit for book recommendations and find them quite helpful. However, some curated recommendations written by humans are also excellent:

• A Fast Track to Learn Math and Physics — book recommendations.

给中国读者的建议

以下建议是我给中国的朋友的, 出自我和我朋友的亲身经历, 希望对你们有帮助:

1. 看英文原版书, 可以用让AI来给你推荐, 用Google, Zlibrary和Anna‘s Archive来找电子书资源. 读英文书会让很多中国学生感到畏惧, 但是现在的英文教材都是用非常简单的英语写的, 因此不需要花很大时间就能看懂, 况且你还可以用AI翻译. 英文的理工科书籍比中文的学起来容易很多.

2. 如果你觉得某门课, 某本书很困难, 那可能不是你的问题, 而是教材和老师的问题. 特别地, 如果你在国内大学学习, 那么肯定是教材和老师的问题, 特别是教材. 相信我, 找英文原版书, 你会发现原来知识学起来这么容易.

3. 如果你还没高考, 我建议你把数学仅仅当成数学, 而不是当成“人生的依靠”之类的东西, 因为世俗的成功(高考)比满足一时的求知欲重要得多, 而且这个建议并不残酷, 因为你在高考后有充足的时间学任何东西.

   我见过很多聪明且勤奋的人因为偏科或者固执之类的原因而在应试教育中表现不佳, 于是埋头于学习高阶的数学和物理知识以获得满足感, 这很好, 非常非常好, 但是如果你能多花点时间在考试上, 进入一个更好的大学, 你获得的会更多. 有两个理由:

   1. 好的大学能让你学得更快, 因为中国大学的资源分配是严格按照招生分数线排名的, 更好的大学拥有更多的科研资源 (不过你也不必为没考上最好的大学而气馁, 因为说到底这些只是外在的辅助).
   2. 更重要的是这一点: 一个更好的文凭会让你在中国找工作容易很多. 我知道很多痴迷学习的人懒得去找实习和社交, 而且其大学专业往往是数学/物理这些很难就业等的专业, 在这种情况下, 如果你来自名校, 那么你会更容易得到工作. 反过来说, 这也意味着读更好的大学的人可以在本科阶段更肆无忌惮地学习, 不用过多担心就业之类的事情.

   我甚至见过15岁就学完QFT的人, 最后考的学校却很普通, 当然, 我知道这样的人注定拥有一个成功的人生, 但是没有人会拒绝一个更好的起点 (好吧, 我想这个例子不太好, 因为15岁学完QFT的人的确不需要在乎学校了). 对于高中生来说, 不要和分数过不去.

   最后, 如果你来自很有钱的家庭, 或者可以出国读书, 或者你已经被保送了, 那么这条建议可以忽略.