Ask HN: What kind of math do you study in your free time? | Hacker News Reader

Ask HN: What kind of math do you study in your free time?

81 pointsdunefox96 comments

Currently, I'm working through Gilber Strangs new book "Linear Algebra and Learning from Data"

Bartosz Milewski's lectures on category theory are great. Also, the accompanying book is well written (and free)!

lectures -> https://www.youtube.com/user/DrBartosz

book -> https://github.com/hmemcpy/milewski-ctfp-pdf

Another good category theory read is Pierce's Category Theory for Computer Scientists

What practical applications does it present for programmers? I've looked at this a bit and it just seems like a regular category theory book.

Sometimes it is helpful to really understand what the hell you are actually doing when you write code (design patterns, data types and all). Otherwise, not much (unless you write Haskell, that is.)

You don't get scared when you read stuff about Haskell because you know what most the fancy words mean

On the other hand it would be easier to just avoid Haskell.

Some math topics I like to read about / play with:

- Measure theory / lebesque/daniell integration / stochastic calculus -- super useful but very beautiful. I have a background in mathematical finance.

- Combinatorial topology -- Simplicial complexes, polytopes. A more finite/computational flavor of algebraic topology.

- Dynamical systems: Highly interdisciplinary. Brings together physics, fractals, calculus, and computer simulations.

- Multilinear Algebra -- tensors, grassman algebras.

- History of Mathematics -- love reading about the development of mathematics throughout the centuries.

“History of Mathematics"

Anyone have good book recommendations?

The World of Mathematics by James R. Newman is a classic, filled with great biographies and short essays about various parts of math.

I just finished The Nothing That Is - it was alright. The illustrations were cool, but the writing and content were only okay imo.

I really enjoyed Fermat's Last Theorem.

I've also heard good things about

- Zero: The Biography of a Dangerous Idea

- Journey Through Genius

"Huygens and Barrow, Newton and Hooke" by Vladimir Arnold (would literally recommend any of his books).

Stillwell's Mathematics and Its History.

Seconding this. His other books are great as well (Reverse Mathematics, Naive Lie Theory). Great writer.

I only took 1 semester of a grad-level Linear algebra course. Is there any newbie friendly book for Multilinear Algebra with emphasis on Application that you know of? Thank you!

There is a wonderful book "Linear Algebra via Exterior Products" by Sergei Winitzki. You can download it for free from his site - https://sites.google.com/site/winitzki/linalg

Thanks. But this book seems to be mostly theoretical. I was looking for one with an emphasis on practical/real life problems.

Hm, this book applies exterior algebra to linear algebra. I don't think there are a lot of applications of multilinear algebra outside of math, it is mostly needed to study differential geometry (forms and so on).

It's "Lebesgue," not "Lebesque." See here for more on the history of this unfortunately common typo: https://math.stackexchange.com/questions/1433899/why-is-lebe....

Any stochastic calculus resource recommendations? Tangentially, I've been interested in getting into SDEs, specifically parameter estimation, but don't know enough about the topic to know good resources from bad.

In grad school, I used "Stochastic Calculus for Mathematical Finance II" by Shreve. Its pretty rigorous/technical (don't bother with volume I, imo).

I don't know of any decent books on estimating parameters for SDE's. Though I suppose one way of going about it is as follows: convert the SDE to a discrete time series using Euler method and use maximum likelihood estimation.

Not sure if it counts as "free time" as I study math for a living :) but optimization theory is always on my list.

If you like Strang's new book, I think you'll be quite partial to Boyd's VMLS [0] which is (in my admittedly horrible opinion) even more clear and practical and serves as an incredibly good and basic introduction to both linear algebra and basic optimization (via least squares). It assumes nothing more than pre-calculus level math and some slight familiarity with derivatives.

Honestly, I really, truly highly recommend reading it, even if you're already familiar with linear algebra. It's a joy to flip through the pages and do some of the problems (both theoretical and practical!).

-----

[0] http://vmls-book.stanford.edu

> I study math for a living

You get paid for studying math?

It is what Maths researcher do. Relatively common thing, usually found in universities.

Yes! Well, at least, for the most part.

I am also coincidentally paid whenever I discover math and explain it to an audience, but this case is much rarer than the former.

> If you like Strang's new book, I think you'll be quite partial to Boyd's VMLS [0]

I actually had that on my list until I forgot about it! Thanks for reminding me. I'm learning Julia as well and there's a companion.

Thanks! I just took Linear Algebra and Regression Analysis class, and this book seems to be a perfect bridge between these two courses.

Less focused but enjoyed Eddie Woo's and 3blue1brown's more recent videos on Youtube

Also challenged by Tom Duff's trigonometry page - http://www.iq0.com/notes/trig.html

Enjoy learning how to derive identities and equations from first principles - quite like differential of x^2 => 2x and the quadratic eqn via completing the square

I'm working my way through Ivan Savov's "No bullshit guide to math & physics" - https://www.goodreads.com/book/show/22876442-no-bullshit-gui...

Would recommend it to anyone whose maths needs a bit of a brush up, or anyone who's interested in basic mechanics!

Oh and I'm using Khan Academy for extra practice, which I can warmly recommend as well.

Great book, too bad it's got such an offensive title. The book could be used more widely without it.

How is it offensive? Because of the word bullshit?

Yes. Can you imagine this used in a professional or collegiate setting?

You don’t have to show it to your professor.

Yes, I don't see why not? Maybe this is a cultural thing, but I don't know anyone who would be offended by the word bullshit, professionally or otherwise.

To be honest, I think the book is used widely because of the title. There are many (hundreds, perhaps even thousands) of good math books out there with standard titles, and from what I have seen, most learners will not bother with them, even though they are totally capable of understanding math AND the books would do a decent job at explaining.

The irreverent title gives me just one extra moment of attention with the cold-reader, and if I can capture their interest in that moment (the first section also has a swearword in it), and get them to bring their "math guard" down for fifteen minutes, then I can "convert" them to the beautiful subject.

You're right that the same "feature" of wow-this-book-seems-to-be-different is also a turn-off for most people who are already pro-math. This cannot be serious math if there is a swearword in the title. Some readers have said they get the sleazy "learn math in three-easy-steps feeling!" when they see the title. That's just life—you win some you lose some ;) I am less worried about people who already like math, since they, having already overcome the fear of math, can learn all the other good books out there.

Thanks for the plug elric!

Anyone interested in seeing what the book is about can check the (slightly outdated) preview here: https://minireference.com/static/excerpts/noBSguide_v5_previ... (a more descriptive title would be high-school math, mechanics, and single-variable calculus three-in-one)

I also have a followup book on linear algebra, which you can preview as PDF here: https://minireference.com/static/excerpts/noBSguide2LA_previ... or as spoken tutorial + SymPy notebooks https://github.com/minireference/noBSLAnotebooks#no-bullshit...

Other goodies and free stuff on the website: https://minireference.com/#freestuff

Thanks for writing the book! It's a fun read, and it packs a big punch. It's making me regret not spending more time on maths before.

I'm working through Oscar Levin's "Discrete Mathematics" - http://discrete.openmathbooks.org/dmoi3.html

I studied philosophy in college and am hoping several years of programming experience since will shed some new and interesting light on one of my favorite topics.

The predicate calculus. I regularly review Predicate Calculus and Program Semantics[1] to increase my fluency in the techniques. I also recommend A Discipline of Programming[2] as a gentler introduction to the subject for those who do not consider themselves particularly mathematically inclined. For me it was a natural progression from doing TDD. I still code test first, but now the structure of those tests and programs is guided by a better understanding of program semantics, greatly increasing my code quality.

[1] https://www.amazon.com/Predicate-Calculus-Semantics-Monograp...

[2] https://www.amazon.com/Discipline-Programming-Edsger-W-Dijks...

There's lectures for that book if you're interested (I don't own the book yet) https://ocw.mit.edu/courses/mathematics/18-065-matrix-method...

In my free time I attempt to work through The Nature of Computation by Stephan Mertens & Cristopher Moore. Edit: Forgot to add, there's lectures for the TCS book too in this playlist specifically 'CS Theory Toolkit' https://www.youtube.com/channel/UCWnu2XymDtORV--qG2uG5eQ/pla...

Nice, thanks!

That sounds interesting. Sadly, I have a huge backlog already...

I'm interested in fractional calculus.

It all started with an argument I had with my high school math teacher about whether something like a half derivative is a thing. Turns out fractional calculus is a real thing and shows up in many applied areas of math. The Fractional Calculus by Oldham and Spanier I have lying around treats its applications to diffusion problems, for example. As an EE student fractional PID controller design and fractional signal processing are interesting as well.

For a quick peek into that subject I would recommend watching Dr Peyam's videos on half derivatives[0].

[0]: https://www.youtube.com/watch?v=eB3OUl5TVSY

I am reading "The Art of Statistics" by David Spiegelhalter.

I just finished that. Great book!

I'm currently working through books on Linear Algebra (Friedman), Real Analysis (Bloch) and Abstract Algebra (Pinter) with guided help from a math PhD I found via Reddit/Discord. It's going great, and I'm getting feedback on my proofs and learning quite a lot.

I studied physics in undergrad, and am now a math/science teacher, but I feel I missed my true calling in deciding on physics over math; it's just so much more fun, in my opinion. I'd love to maybe eventually do an online math bachelors and then get a masters in it later (or skip the bachelors and get a masters), but all that will depend on if I decide to shift out of teaching or not.

Can you please elaborate on how you found a tutor via Reddit/Discord? I would love to study with someone who can help me if I am stuck with a problem.

I saw the person advertising on Reddit for his own Discord server in a /r/math thread about learning after graduation, etc., so got quite lucky. I think he still lurks around /r/math and /r/learnmath if you go and ask if anyone can help.

Nice! I usually try Crossvalidate SE first. But I'll cross-post to those subreddits as well.

Related: do people have good recommendations of "casual" math books for more advanced topics? I.e. ones that an educated reader with a background in math can read through in one pass (unlike most textbooks) but nonetheless have real math in them? (And has fun exercises!)

I can start with some. Feynman's Lectures on Computing. Scott Aaronson's Quantum Computing Since Democritus (though it assumes some background knowledge of quantum computing). I think Colin Adam's "The Knot Book" (on knot theory in topology) as well.

Understanding Digital Signal Processing, by Richard G Lyons. It's one of the most accessible books on the topic since it's aimed at undergraduate juniors.

This one is amazing and not to be missed: http://www.dspguide.com/pdfbook.htm.

Visual Complex Analysis. I’d hesitate to call it casual, but it’s extremely accessible to someone with some basic mathematical knowledge.

Also, Visual Group Theory.

I used Visual Group theory as the basis for some workshops. It's a really good read.

Roger Penrose' book _The Road to Reality_ is pretty good in this respect. It's a little skewed towards geometry (instead of, e.g., Hilbert spaces and operator algebra), but hey, what do you expect from Penrose.

Heh, this one is only for casual reading if you are already well-versed in the material. (This is a weird book - the author tends to dwell on simple things and then to rush through complex topics. Not recommended for an uninitiated. His later book, Fashion, Faith, and Fantasy, is more accessible but is primarily focused on discussing physics.)

John Baez's blogs (Azimuth, This Weeks Finds) are pretty accessible.

Susskind's Theoretical Minimum series might be to your liking?

Not a bad choice really, even though this one is more about physics than math - but hey, what is theoretical physics if not math mostly, right?

The topics that resonate with me the most are graph theory and probability. Incidentally, they are also the most useful for my day job. Graphs are such a beautiful and intuitive concept, it is amazing; basic probability is unbelievably useful for general problem solving, making decisions, modeling the world; both have been written about a lot, yet there is so much more to discover and apply.

Things I'd love to read about if I had more time are: topology (knots are weird interesting things), meta-mathematics (Gödelization and all that, read Gödel, Escher, Bach if you'd like to wet your appetite), paraconsistent logic (how to contain inconsitencies in systems of logic so that they don't become arbitrary - as from contradiction, anything follows). Digesting maths requires a _lot_ of time, wish I could be a student again to sit in whatever lecture that sounds interesting.

As a kid, I loved reading about history of maths; many discovery stories made me become a scientist (applied computer science researcher), and I still enjoy reading about it (also biographies or even mathematically related fiction e.g. The Solitude of Prime Numbers).

Been going through Conway's (and Conway related) books since his unfortunate passing. His biography by Siobhan Roberts was a great starting point to ease into it (lots of direct quotes from Conway, which makes it a very easy read that still touches on the important concepts in his work, in his own words; also highly recommend all the Numberphile videos featuring him for that)- then:

Winning Ways for your Mathematical Plays is really fun to thumb through.

The Book of Numbers is fantastic and something I would gift to any mathematically curious, somewhat independent, child.

Knuth's Surreal Numbers is also a great read.

Got On Numbers and Games coming in the mail, and am trying to track down a reasonably priced copy of The Symmetries of Things.

I'm tempted to get the Atlas for my collection, but I don't think I'd actually get much from reading it (:

In non-Conway recommendations, The Princeton Companion to Mathematics is a huge brick of a volume, but is a very complete math encyclopedia that I love to keep on my desk and thumb through when I feel distracted. You always end up learning something new.

I'm a big fan of Schelling's ideas on game theory, and focal points in particular: https://en.wikipedia.org/wiki/Thomas_Schelling#The_Strategy_...

I'd love any further suggestions on complex/multipolar/iterated game theory.

For combinatorics, I highly recommend Miklos Bona's A Walk Through Combinatorics[0]. Combinatorics is intuitive and approachable to begin with, and this book is particularly accessible as far as math texts go.

[0] - https://people.clas.ufl.edu/bona/books/

My capsule reviews of what I've read in mathematics over the last 22 years. http://don.dream-in-color.net/books/archive.php4?iSubject=79

I always come back to playing with the Exterior (Multilinear) Algebra because it seems like there's some deep structure hiding inside of it that connects a bunch of different fields of math.

I'm currently looking at propagator networks https://github.com/ekmett/propagators

I'd love to raise awareness of https://www.stat.berkeley.edu/~aldous/Real_World/RW.html (Probability and the Real World) https://www.stat.berkeley.edu/~aldous/ Professor David Aldous. This is a wonderful resource. I hope you all enjoy.

I tried to learn Galois theory on Coursera a few years ago (the course was in French, which is my mother tongue, but probably few HNers can understand it), and failed the class partly due to lack of time. Since then I've been trying from time to time to read about it, which refreshed knowledge of Group theory, but so far I haven't gotten to the point where I understand Galois's idea to prove that some polynomial equations are not solvable by radicals.

You can try this little essay by Stillwell: http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwel....

I filled a Kindle with just about every paper related to the proof of Fermat's Last Theorem and chip away at it when I can. When I get stuck there, I switch over to trying to understand Ono's closed form solution of the partition function. Both subjects provide hours and hours of diversions into areas of math I never got to learn studying physics.

There is loads of great maths stuff on YouTube. For some reason this is my favourite channel: black pen red pen. The author solves unusual algebra or calculus problems. I find it quite relaxing!

https://www.youtube.com/user/blackpenredpen

I find this girl's analysis of Fibonacci series and the Golden Ratio concepts and seeing how it applies to plants to be fascinating:

https://www.youtube.com/watch?v=ahXIMUkSXX0

I wish she would edit the video and slow it down a bit (don't like that rapid fire style of tutorial that so many people YouTubers adopt lately) but the presentation is amazing.

I’m obsessed with the Mandelbrot Set, so a lot of the mathematics I read recreationally branches from that: fractals, complex numbers, Riemannian and Hermitian manifolds, and related topics.

As a software developer, I explore lots of computer/data-science related topics as well, e.g. cellular automata, dynamics, and some statistics.

I failed Linear Algebra in college because I was more interested in partying, it wasn't essential to my degree (Poli Sci), and I incorrectly assumed I could drift through it and get a C like my other hard science requirements. I am currently working through Friedberg's textbook on it.

I've enjoyed this series of lectures on Youtube:

"Lectures on Geometrical Anatomy of Theoretical Physics" [0]

[0] https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv...

Can someone recommend an accessible book to learn multilinear algebra,tensors with focus on applications

Mostly crawling through ml papers on arxiv. Also going over "The Theory That Would Not Die", though this is in the realm of popular science books but it's an enjoyable getaway from it all.

Edit: How to by Randall Munroe for the math-comedy realm.

Common Core because my kids are in school. I've heard a lot of parents complain, but I actually love it. I've always had a knack for doing arithmetic in my head and common core is teaching all the stuff I do intuitively.

This: https://arxiv.org/abs/math/0608040v4.

Orbital mechanics and related mathematics like Stumpff series and Universal variables. Got inspired from playing too much Kerbal Space Program and Orbiter.

I've wanted to do this for a while. What's your process like? I have a few books (Introduction to Space Dynamics, the BMW book, etc) but I'm not sure if I should read it cover to cover or try to design a curriculum etc.

Do you use any simulation software or just pen and paper?

I started with Bate, Mueller, White: Fundamentals of Astrodynamics, which is US Air force material. It contains two courses, so you should skip some of the chapters at first.

Then I read a bunch of research papers and more in depth literature. Richard Battin's work for example.

I've been hacking on a two body orbital mechanics library in C with SIMD extensions, and an interactive sandbox with OpenGL. I wrote both from scratch.

Presently working through Harvard's online Abstract Algebra course and adjunct to that Bartoz's notes on category theory and some type theory.

In high school I read Vašek Chvátal - Linear Programming after buying it second hand for a few dollars thinking it was a computer text book.

I liked it at the time.

certainly some statistics thoery, when following any MachineLearning core.. (not big on DeepLearning here, all the other ones) Small bits of 'Understanding Machine Learning: From Theory to Algorithms'

Some clustering theory.. some computer vision components, including segmentation methods

Some "data mining" approaches, which are sets and stats, basically..

I've always been partial to Galois theory.

Sigmoid / logistic and bell curves to try and predict Covid-19 progress.

Discrete most recently but I am looking into category theory.

I study no kind of math in my free time.

Ditto

Topology

(Points at username)

Trigonometry.

homotopy type theory!

What are you reading/watching? Are there any self-contained material on this topic?

Not sure what you mean by self-contained, but there is the Homotopy Type Theory book:

https://homotopytypetheory.org/book/

Applied statistics.

Reply on news.ycombinator.com