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)
Well, all these closed surfaces you mention are (topologically) spheres. The theorem doesn’t apply to some other closed surfaces, like the torus, which does admit a continuous non-vanishing vector field.
Heard a mathematician friend call this the “hairy sphere theorem” once. At first I thought he was being a prude, but now I appreciate that the theorem is about spheres, as opposed to balls.
Yes! I’m working on this in my PhD research. Here’s a relevant Caltech lecture aimed at a general scientific audience, discussing use of ML to detect equivalent knots.
_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.
After I understood monads, I didn't write a short blog post about how simple they are. In fact, I took the opposite approach and am writing a ~500 page book about them :).
In seriousness, there's a lot that goes into appreciating a concept like a monad, especially if you want to combine the FP understanding with a rigorous mathematical understanding (monoid in a category of endofunctors approach). My new book is coming out next year with No Starch Press, and my goal is to give programmers a serious but doable introduction to abstract algebra, in the case that they care about this kind of stuff. Ping me if you want to read some early chapters and give feedback! [email protected]
Tachyus | Product Development Intern | INTERNS, REMOTE or ONSITE in Houston TX or Los Angeles CA. | https://tachyus.com
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]
Tachyus | San Mateo, CA or Houston, TX | ONSITE | Full-Time | Scientific Software Engineer | http://www.tachyus.com/
Tachyus creates technology to optimize energy production for the oil and gas industry. With Tachyus software, petroleum engineers integrate all relevant data sources in real-time to explore millions of scenarios and identify the optimal operational and development plans resulting in 10%+ increases in net present value of their assets. Our Scientific Software Engineers use the F# programming language to bring our latest scientific research to production.
For anyone curious, a Lie group is a continuous collection of symmetries. While a square, for example, has 8 symmetries (identity, three rotations, and 4 reflections), a circle has infinitely many: you can reflect it about any axis and rotate it by any angle and get the same circle you started with.
Since Lie groups are indexed by continuous parameters, they can be studied as spaces in their own right. For example, the Lie group of rotations of the circle (called SO(2)) is topologically the same as the circle: you can put the rotations of the circle in 1-1 correspondence with the points on the circle. For the (2-)sphere, the Lie group of rotations actually looks more like a higher dimensional (3-)sphere.
Lie algebras consist of vectors (rather than symmetries), and encode most of the structure of corresponding lie groups. For example, the Lie algebra for the circle rotation group is the vector space of the real line: each rotation is indexed by a real number (the angle, with redundancy) and composition of rotations is obtained by adding angles. In higher dimensions, especially where symmetries don't commute, the Lie algebra is indispensable and often the starting point for study.
At Tachyus (http://tachyus.com) we have made a huge bet on F#, and it has paid off in spades. The back end of our SaaS product is 99%+ F#, and so far we have had a great experience using Azure for hosting. Our iOS apps are also built using 99%+ F# (using Xamarin, on Mono rather than .Net). The language allows us to write highly expressive code, and fewer lines of code overall.
Using a Microsoft language has given us huge productivity benefits (most notably Visual Studio and Azure support). There is a fantastic F# community (http://fsharp.org) which is untethered to Microsoft, and it's exciting to see the language grow as an open-source and cross-platform tool.
At Tachyus (tachyus.com), we are using 100% F# to build the back end of our platform. Getting from zero to our first product deployment in three months, we've proven that F# is a great language for real world web projects -- especially when iteration speed and application correctness are both indispensable. Visual Studio and Windows Azure have actually been a huge part of the leverage we've gotten from choosing F#, as we can build, debug, deploy, and manage deployments directly from the IDE. With an Azure trial account created, you can literally get your first F# site up in minutes.
Unfortunately, I can't answer your question about Mono just yet. We are exploring the practicality of Mono (for mobile development in F#), and while I am optimistic, I can't speak to the reliability of the platform. So my answer (a complete cop-out) is to try out Visual Studio unless you are completely tethered to Linux/Mac.
How many of these are you paying for out of pocket??