I am a knot theorist (PhD student) and always tickled when the subject pops up on HN. Curious what the author had in mind including this in his algebraic topology notes. Even the Jones polynomial which is both algebra and topology is not usually called "algebraic topology", but rather "quantum topology".
If anyone is interested in a conversational intro to the subject with lots of pictures, I suggest these semi-famous "Knots Knotes" (amazing title)
_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.
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I'm Paul Orland and I'm co-founder of Tachyus. For the last ~7 years at Tachyus, we've built predictive analytics software for mature oil and gas fields, and successfully deployed it to increase operational efficiency in some of the biggest oilfields in the world. This summer, I'm kicking-off an exciting new chapter at Tachyus by building and commercializing a new product helping energy companies (and other kinds of companies) minimize the carbon intensity of their operations.
Even though we will be leveraging existing technology, this will be a fun return to start-up square one. We will be designing a brand new product, meeting and learning from new users, building out a prototype, and helping new customers run their businesses more sustainably. Coding experience (full-stack web) and UX design are nice-to-have, but the main requirement is a track record of taking on ambitious projects and driving them to completion. If you want to come embark on this (re-)founding journey with me, shoot me an email at [email protected]
If anyone is interested in a conversational intro to the subject with lots of pictures, I suggest these semi-famous "Knots Knotes" (amazing title)
https://mathweb.ucsd.edu/~justin/Roberts-Knotes-Jan2015.pdf