My Math Learning Roadmap

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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
  • armstrong的basic topology
  • Topology Without Tears. It's a thick book.
  • Visual Group Theory. Not very professional but it's interesting.
  • 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

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:

UTM，UTX, springer的GTM系列, SBM(Springer Briefs in Mathematics)

给中国读者的建议

  1. 看英文原版书，可以用 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，你会获得更大的物质回报——最直观的例子是，对于选择数学 / 物理这些不容易找工作的专业，并且大学阶段缺乏企业实习经历的人来说，一个好学校的文凭对找第一份工作非常有用。不过，学历对于知识学习和科学研究都不重要，所以你最终还是能成为很好的工程师 / 科学家。

    如果你来自很有钱的家庭 / 可以出国读书 / 已经被保送了，那么这条建议可以忽略。