For those of you who don't know: this is a companion to the Princeton Companion to Mathematics, which is excellent but focuses mainly on pure math. Also, the editor of this new volume was a highly respected applied mathematician named Nick Higham, who sadly died a few months ago.
Oh no... I used his work on nearest positive semidefinite matrices and liked to read his blog. In my opinion, he was very good in explaining his research.
In my case, I am applying this in the calibration of instantaneous correlation matrices. His relevant papers can be found at [0].
It was really helpful that he also published MATLAB code of most of his algorithms.
I’m an encyclopedia lover who works on a variety of applied math problems as part of $dayjob at a national lab.
I thought the book might be fun to browse around in, so I purchased it. I knew the companion book (…to Mathematics) had a very good reputation.
It didn’t work for me…long story short is, the articles were written at too high a level of sophistication to serve as an introduction for a curious outsider. It was more of what someone in a nearby field might want to get up to speed on what is known, when their background in “that kind of thing” is already quite strong.
I was surprised for various reasons - I know Higham’s technical work, enjoy his blog, and I have a decent math background. (Of course, he’s the editor, not the author - it’s an enormous book.)
How should one use a book like this? Is it to get an overview of a topic before diving in? I don’t think I’ve ever learnt any mathematics from reference works, so I’m curious as to their intended audience.
You use it like a conceptual dictionary. Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with (e.g. a submanifold). You’d look up “submanifold” and see that it is “ subset of a manifold that is itself a manifold, but has smaller dimension.” Okay, that seems to fit the intuition of a “sub”-something. But I don’t know what a “manifold” is. So I’d look that up.
“A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in R^n)”
At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done. If not, you either keep digging deeper into the various terms or you start seriously considering reading one or more of the curated reference books listed at the end of each entry.
Over time you develop the “mathematical maturity” that you don’t need to do a deep dive into the books and can mostly just use the reference.
> At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done.
I'm not sure. I only have a rather rudimentary understanding of topology, so I do understand the definition of a manifold on a technical level, but I don't know any interesting examples or theorems about them so it wouldn't be immediately clear to me why something being a submanifold is worth mentioning.
Similarly, I don't think that just reading the definition really gives you a good understanding of groups. You probably want to work through some examples of groups, and arguably, the importance of groups doesn't really become clear until you've encountered group actions.
You skipped over the second sentence of what you're responding to:
> Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with
In such a case you're not interested in either manifold or sub-manifold or group in and of itself. So a lack of familiarity with theorems isn't an impediment.
The historical notes are a great strength of this book. As for learning the material, from what you've written, you would likely be better off with the sort of books used in first and second year university.
A way to find good ones is to look at some university webpages, to see what books they use in 1-level and 2-level classes. (Of course, start with 1-level.). Those textbooks will be more expansive, with interesting diagrams, problem sets, and so forth. And they will use fancy typesetting patterns, like insets in boxes for subtopics, etc.
I suspect quite a few purchasers will be university teachers who want to have this on their shelves, for when students come by and ask for a book to borrow overnight to brush up on a topic.
Outside of the other suggestions in this thread, this book may also be helpful to someone interested in studying applied mathematics in college, but unsure of what that means either in terms of topics or career. I've only flipped through the book, but it seems to do a good job at giving a high level overview of various topics and applications. If one were to like what they see, then perhaps one should investigate further.
In a similar topic, if someone is considering a career in mathematics, I like the book, "A Mathematician's Survival Guide: Graduate School and Early Career Development." It applies to both pure and applied mathematicians, but it does a good job of walking through undergraduate studies all of the way to being a professor. Not all mathematicians end up in the professoriate, but the graduate school information is still valuable.
I wouldn't use a book like this for foundational learning. It's more a precis of existing information on a topic. Looking at one of the entries for Numerical Weather Forecasting, it presupposes at least a solidly-established understanding in Applied Math or Math Physics. If you're approaching that topic without a basic knowledge of what a divergence is, what vorticity is, what a gravity wave is, or the difference between implicit and explicit FD equations, etc. it's probably not going to teach you much. But, if you do have the background it's a great resource - a really super resource. It's a bit like Wikipedia, I suppose. Super helpful at some level, but not at others.
The most advanced math I had were the 1st and 2nd year multivariable calculus and linear algebra courses in college.
I am interested in visualizations/simulations of physical systems as a way to learn advanced math. Is there any books or resources that take that approach?
When looking into simulations of physical systems, you'll run into partial differential equations, but be careful about learning resources that don't put numerical methods front and center. The article on Numerical Weather Prediction in the post has a good description:
> "Analytical solution of the equations is impossible, so approximate methods must be employed. We consider methods of discretizing the spatial domain to reduce
the PDEs to an algebraic system and of advancing the solution in time."
Given Python's popularity in scientific computing, a lot of the available materials on the topic are in that language, using libraries like numpy and scipy a lot. I've been playing around with custom ChatGPT here - you can construct a workflow that takes a description of a common equation, generates the LaTex expression for it, translates that to a sympy expression, and then from that generate the numerical method code using numpy and then the code to plot the behavior over a given range in matplotlib. Bonkers, we're living in the future.
Take a look at this playlist [0] by MathTheBeautiful. It says it's about PDEs, but it starts with ODEs.
I think most differential equations courses are too focused on symbolic solving techniques.
To me, understanding a (physical) system is mainly understanding the differential equation (system) itself, not it's solution. MathTheBeautiful really excels at this approach.
Solutions are of course important as well, but i think that's what computers are for.
Absolutely love the book “Learn Physics with Functional Programming”.
Uses Haskell, and teaches it so knowing Haskell is an unnecessary prerequisite, though familiarity with programming in general would be very beneficial, and focuses on 2D and 3D visualisation.
I recently bootstrapped myself from your level to algebraic geometry — enough to read an annotated version of Einstein's paper. If I had to do it again, I'd just skip to the parts that worked: (1) hit up Michael Penn on YT, but work all the problems he works; and, (2) learn GA, I like MacDonald's "Linear and Geometric Algebra". I'm going to be honest: unless you do the homework, you can't learn the math.
I learned Penn because it's "real" (mainstream) math & lets you read in the same language; I learned GA because it's designed to proselytize to other mathematicians & deliberately presents serous ideas in very approachable ways.
Isn't general relativity framed in differential geometry, not algebraic geometry? All of Einstein's papers on special relativity use high-school algebra and calculus.
Yes, GR uses differential geometry; however, a good (mechanical) understanding of any exterior algebra helps when working with tensors; and, I found GA much easier to understand than algebraic & differential geometry, when starting out. The idea is to familiarize yourself with learning math. The actual math of the GR paper is pretty straight forward: it's the physics which is so mind bending.
I joked at an estate sale once that everyone has 3 bookshelves: the books they read, the books they want to read, and the books they want other people to think they read.
In fairness this one’s predecessor has some pretty accessible articles that I’ve actually read. It’s more like an encyclopedia. Some things I have the background for, others not so much.
