For the 2018 A-to-Z I spent some time talking about a big piece of thermodynamics. Anyone taking a statistical mechanics course learns about the Nearest Neighbor Model. It’s a way of handling big systems of things that all interact. This is really hard to do. But if you make the assumption that the nearest pairs are the most important ones, and everything else is sort of a correction or meaningless noise? You get … a problem that’s easier to simulate on a computer. It’s not necessarily easier to solve. But it’s a good starting point for a lot of systems.
The restaurant I was thinking of, when I wrote this, was Woody’s Oasis, which had been kicked out of East Lansing as part of the stage in gentrification where all the good stuff gets the rent raised out from under it, and you get chain restaurants instead. They had a really good vegetarian … thing … called smead, that we guess was some kind of cracked-wheat sandwich filling. No idea what it was. There are other Woody’s Oasises in the area, somehow all different and before the pandemic we kept figuring we’d go and see if they had smead, sometime.
The Summer 2015 A-to-Z was the first I’d done. Its essays tended to be shorter and narrower in focus than what I write these days. But another feature is that they tended to be more practical, like, something that you could use to read a mathematics paper with better understanding. N-tuple is an example this. N-tuples are ordered bunches of numbers, and turn up in many places. They’re not quite vectors and matrices. But the ordinary use of vectors and matrices we represent with n-tuples.
One of the pieces I wrote for the Leap Day 2016 A-to-Z was to explain the Matrix. Matrices turn up a lot in mathematics. They’re nice things. They organize a lot of information economically. They’re vector spaces, so that a lot of really nice tools come along with their existence. They don’t have much to do with what I’ve been writing about this year, but, so what? I can bring back from obscurity pieces I just liked, too.
All my A-to-Z pieces, from every year, should be at this link. And all of the 2020 A-to-Z pieces should be at this link. Also please let me know if you have ideas for the letters P, Q, and R. I’m also still eagerly looking for Playful Math Education Blog Carnival-ready pieces. Thank you.
I avoided most of the technical talk when I discussed the Möbius strip the other day. You — well, a mathematician — could describe the strip as a non-orientable compact manifold with boundary. The boundary part is obvious. The non-orientable bit is easy enough to understand, when you remember that thing about it being a one-sided surface. Compact is an idea worth its own essay sometime. In this context it amounts to “there aren’t any gaps inside it”. Manifold, too, is worth an essay, and I wrote one in 2018 about it. Thanks for reading.
I did not mean my archive pieces this week to all be patching up stuff omitted from my Leibniz essay. But that essay touched a bunch of mathematical points, some of which I already had essays about. One of them was published in the 2018 A-to-Z. We really, really want to use the idea of an infinitesimally tiny change to understand how calculus works. But we can’t do that with logical rigor. (Unless we redefine what “real numbers” are, and some mathematicians go in for that.) Ah, but: what if we could get all the things these infinitesimals give us without having to do anything weird? And that is the modern idea of the limit, which we sorted out about 150 years ago and are pretty satisfied with.
While talking about Leibniz, who isn’t the inventor of calculus — but is the person I’d credit most with showing us how calculus could be — I made some speculations unsupported by evidence about whether he looked into optimization problems. This because of the philosophical work that he’s famous for among lay audiences, the proposition that God’s will implies this must be the best possible universe. (I don’t know what he’s most famous for among professional philosophers.)
I don’t have an essay specifically on optimization theory, as mathematicians see it. Not exactly. But last year I did write about linear programming, which is a particular type of optimization problem. It’s a kind that’s often the best we can do in a complex circumstance. And it lets me introduce you to the word “simplex”, which is fun to say.
If complex numbers aren’t the dominant theme of this year’s A-to-Z, then biographies are. I’ve written biographies for past series, though. Here, from 2018, is a tiny slice about William Thompson, Lord Kelvin, and one of those bits of his work that is mathematical and important and had a huge effect on the world. But it’s also become invisible. So please consider a couple hundred words about that.
In writing about K-theory I mentioned the “kernel”. I didn’t have space to describe what that was, and I failed to link to the essay I wrote in late 2016 about what kernels were and why we care. Let me fix that now. We’re introduced to kernels in group theory, where, for the simple groups, they seem like a complicated way to talk about “zero”. But we also see them in linear algebra. And between those, we get them in analysis. That then leads into not quite all of modern mathematics. But a lot of it.
In my first A-to-Z I wrote a good number of pieces about the kinds of functions there are. For example, jump, a particular kind of discontinuity in functions. This is useful because there are a lot of pieces of functional analysis where we know things are true for continuous functions. And if a function has a jump discontinuity? Usually we know the thing is true except at the discontinuity. There’s more rules, of course. And, like, Fourier series will get strange around jump discontinuities.
I’d have written the essay a bit different today, but I am in awe of a time I could wrap up the point within six hundred words. That never happens anymore.
I can’t guess where this year’s A-to-Z series will lead. Often a theme develops. Complex numbers look like they’re trying to be it. So let me share something from last year’s A-to-Z, and which relies on complex numbers. Julia sets, which are some of the best-known fractals, are calculated by working out functions on complex numbers. By iteration, particularly. That is, start with some number. Evaluate a function where the independent variable has that number. This gets you some (probably) different number. Evaluate the same function again, but using this as the independent variable. This gets you (usually) another number. Evaluate the same function again, with this third number as the independent variable’s value.
You’ve done this sort of iteration when playing with a calculator and hitting the square root or the square or the sine or whatever other function key over and over. These usually end up pretty boring, at 0 or 1 or the calculator reading INF. Put in a slightly different function? You get something beautiful.
Although I did a lot of my functional-analysis-definitions for 2015 I certainly didn’t let that pass from the stage. My letter-I choice for the End 2016 A-to-Z was Image. This is useful when you think about functions as ways to turn one set (the domain) into another set (the range). A lot of definitions and tests for conditions, such as continuity, become much less technical and fussy when you start looking at images. It’s one of those changes in perspective that makes work easier.
My 2015 A-to-Z was before I asked for topic nominations. So what topics I did cover tended to focus on my own particular interests, and the courses I remember liking. So there was a good bit of functional analysis, or things that go into functional analysis. Into covers one of those topics. And you get some bonus coverage of “onto”. The terms describe ways that a function can use its range. To an extent it’s taxonomy, distinguishing functions that use the whole range versus those that use part. But it gives you vocabulary to discuss what cases you have to cover in proving a thing about a function. So it’s one of those small but useful terms.
Sorry to be late. We discovered baby fish in a tank that we had thought was empty, and that needed some quick attention. The water seems nearly all right and we’re taking measures to get the nitrate back in line. Fish-keeping is a great hobby for someone with tendency towards obsessions and who likes numbers because there is no end of tests you can start running and charts you can start to keep.
So I’ve seen two baby fish, one about the width of a fingernail and one about half that. We’re figuring to keep them inside until they’re large enough not to be accidentally eaten by the bigger goldfish, which means they might just be in there until we move fish inside for the winter. We’ll see.
Back to my archives, though. The hypersphere is a piece from the first A-to-Z I ever did. I could probably write a more complicated essay today. But the hypersphere is a good example of taking a concept familiar, as circles and as spheres, and generalizing it. Looking at what’s particularly interesting in a concept and how it might apply in different contexts. So it’s a good introduction to a useful bit of geometry, yes, but also to a kind of thinking mathematicians do all the time.
While looking through my past H essays I noticed a typo in Hamiltonian, an essay from the 2019 A-to-Z. Every time I look at an old essay I find a typo, even ones I’ve done this for before. Still, I choose to take it as a sign that this is an auspicious choice.
The Hamiltonian is one of the big important functions of mathematical physics. For all that, I remember being introduced to it, in a Classical Mechanics class, very casually, as though this were just a slightly different Lagrangian. Hamiltonians are very like Lagrangians. Both are rewritings of Newtonian mechanics. They demand more structure, more setup, to use. But they give fine things in trade. So they are worth knowing a bit about.
I’d like today to share a piece from 2017. Gaussian Primes are a fun topic, as they’re one of those things that steps into group theory without being too abstract. And they show how we can abstract a familiar enough idea — here, prime numbers — into something that applies in new contexts. In this case, in complex numbers, which are looking likely to be the running theme for this year’s A-to-Z.
Later in 2017 I talked talk about prime numbers in general, and how “prime” isn’t an idea that exists in the number itself. It exists in the number and the kind of number and how multiplication works for that kind of number.
And I’m still eagerly taking nominations for topics for J, K, or L Please leave a comment at this link. Thank you.
If you looked at my appeal for A-to-Z topics for the letter G, when I posted it a couple weeks back, you maybe looked over a bunch of essays I quite liked. I still do; G has been a pretty good letter for me. So one of the archive pieces I’d like to bring back to attention is Grammar, from the Leap Day 2016 A-to-Z. It’s about how we study how to make mathematical systems. That you can form theorems about the mechanism for forming theorems is a wild discovery, and the subject can be hard to understand. At least some of its basic principles are accessible, I hope.
And if you’d like me to discuss more topics in mathematical logic, or other fields of mathematics that start with J, K, or L, please leave a comment at this link. Thank you.
There are important pieces of mathematics. Anyone claiming that differential equations are a niche interest is lying to you. And then there are niche interests. These are worthwhile fields. It’s just you can get a good well-rounded mathematical education while being only a little aware of them. And things can move from being important to niche, or back again.
Continued fractions are one of those things I had understood to have fallen from importance. They had a vogue, in Western mathematics, where they do some problems pretty neatly and cleverly. But they’re discussed more rarely these days. The speculation I’ve seen is that they don’t quite have a logical place, as being a little too hard when you’re learning fractions but seeming too easy when you’re learning infinite series, that sort of thing. My experience, it turns out, was not universal, and that’s an exciting thing to learn in the comments.
My impression, not checked against evidence, is that my recaps here feature the 2019 series more than any other. Well, I really liked the 2019 series. I don’t think that’s just recentism. On rereading them, I often feel little pleasant surprises along the way. That’s a good feeling.
So here was my ‘F’ entry for 2019: Fourier series. They’re important. They’re built out of easy pieces, though. And they’re full of weird bits. You can understand why someone would spend a career studying them. And I almost give enough information to actually use the things, if you have enough background to understand how to use them. I like hitting that sweet spot.
If it does turn out that P equals NP we would, at least in principle, have wrecked encryption as we know it. So let me take this chance to mention my essay on Encryption Schemes, from last year’s A-to-Z. And that discusses some of what we look for in encryption, which includes both secrecy and error-free transmission.
There’s one past A-to-Z essay for the letter e that’s compelling after I looked at the exponential function on Thursday. That would be the number that’s the base of the natural logarithm. It’s a number that was barely mentioned in that piece, because I ended up not needing it.
But a couple years ago I wrote a piece that was all e, including points like how curious a number it is. I hope that you enjoy that piece too.