Algebraic Topology

What Hacker News says

1. orlandpmFeb 2022
   _Algebraic Topology_ by Allen Hatcher. This is quite an advanced book, but it was the first topology book I picked up and it blew my mind. Without a serious math background you may only be able to read the first parts (some of chapters 0 and chapter 1), but even so it may amaze you. It was insane to see how much mathematical machinery can be built up to understand concepts as simple as “space” and “continuity.” Then these tools can be used to quickly prove facts about mind-boggling higher dimensional objects.

   _Introduction to Quantum Mechanics_ by David Griffiths. Not CS or Math, but an amazing book because if you sit down and work through it it gives you a manageable intro to a completely non-intuitive and mysterious scientific field. Prerequisites are “only” multivariable calc and linear algebra.

   _Algebra_ by Michael Artin. This was the book I decided to grind through at the right time to learn abstract algebra and get better at rigorous proofs, and it was worth it. Part of why I loved this book and subject is because it feels dry and mechanical at first, but if you work on it long enough you can see the beautiful bigger picture come together.

   _Introduction to Topological Manifolds_ by John M. Lee. This was another “right book at the right time” for me. The title is a bit jargony, but it is a rigorous introduction to the foundational objects of modern geometry (namely topological spaces and manifolds in particular). Great warm-up for the first book on my list.

   _Algebra: Chapter 0_ by Paulo Aluffi. Weird title, but the first few chapters are the best perspective on abstract algebra I’ve seen. He focuses early on categories in a useful and philosophically interesting way, which is unique.

   _Gravitation_ by Misner, Thorne, and Wheeler. Didn’t read the whole thing, but this book taught me a tensor is just a higher-order linear function. Who knew? Most physicists give really crappy explanations of tensors. Also I love that this book, affectionately known as the “phone book”, is so heavy that one imagines spacetime curving around it.

   Show morethread
2. MrBraJan 2015
   "An overview of Inter-universal Teichmüller Theory and [...] how we can begin to understand this theory":

   http://www.quora.com/Joseph-Heavner/Posts/An-overview-of-Int...

   Also, from an anonymous post on 4chan @ http://boards.4chan.org/sci/thread/6931488/are-you-ready-for... :

   "How to understand IUTeich from the bottom up:

   - Algebra (Israel Gelfand)

   - The Method of Coordinates (Israel Gelfand)

   - How to Prove It: A Structured Approach (Daniel Velleman)

   - Kiselev's Geometry - Planimetry and Stereometry

   - Trigonometry (Israel Gelfand)

   - What Is Mathematics? An Elementary Approach to Ideas and Methods (Richard Courant)

   - A Course of Pure Mathematics (G.H. Hardy)

   - Linear Algebra (Kenneth Hoffman and Ray Kunze)

   - Elementary Differential Equations (William Boyce and Richard DiPrima)

   - Topology (James Munkres)

   - Calculus On Manifolds (Michael Spivak)

   - Principles of Mathematical Analysis (Walter Rudin)

   - Real and Complex Analysis (Walter Rudin)

   - Functional Analysis (Walter Rudin)

   - Partial Differential Equations (Lawrence Evans)

   - Analysis On Manifolds (James Munkres)

   - Abstract Algebra (David Dummit and Richard Foote)

   - Algebraic Topology (Allen Hatcher)

   - Introduction to Smooth Manifolds (John Lee)

   - Foundations of Differentiable Manifolds and Lie Groups - (Frank Warner)

   - Galois Theory (Harold Edwards)

   - Linear Representations of Finite Groups (Jean-Pierre Serre and Leonhard Scott)

   - A Classical Introduction to Modern Number Theory (Kenneth Ireland and Michael Rosen)

   - Introduction to Analytic Number Theory (Tom Apostol)

   - Modular Functions and Dirichlet Series in Number Theory (Tom Apostol)

   - Riemannian geometry (Peter Petersen)

   - The Theory of the Riemann Zeta-Function (Edward Charles Titchmarsh)

   - An Introduction to Teichmüller Spaces (Yoichi Imayoshi and Masahiko Taniguchi)

   - A Course in p-adic Analysis (Alain Robert)

   - Foundations of p-Adic Teichmuller Theory (Shinichi Mochizuki)

   - Mochizuki's papers at http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.htm... "

   Show morethread