Modular forms, the ‘fifth fundamental operation’ of math
quantamagazine.org
quantamagazine.org
But of course, fundamental research does not give results that look shiny for the next quarterly report. That's why the current trend of running universities like business is so awful.
Steam engines came first, and thermodynamics was partially invented to study them. Later on, thermodynamics helped make better steam engines.
Vacuum science is also a critical predecessor to the steam engine, developed by Galileo in the 1630s and demonstrated as artificial vacuum and the barometer in the 1640s.
About Vacuum Science: I grew up around and in Magdeburg, which is famous for her (vacuum) hemispheres. https://en.wikipedia.org/wiki/Magdeburg_hemispheres
Firefox for Android.
When it doesn’t work, is when column width is too wide to fit on a vertical screen at once, which isn’t a problem here when zoomed in.
It doesn't need to go to the opposite extreme (as you described it :-)) to be more readable by default on different screen sizes.
Regarding content, it started as a database of modular forms, indexed by various specialized numerical data; since then, it has grown to a repository of all kinds of number theoretic data, including number/function fields, elliptic curves, various special functions and much more.
So many nerdsnipes for one video. If I could spare a decade or two, I'd love to read about all of these until they make sense!
At this time of year? At this time of day? In this part of the country? Localized entirely within your kitchen?
It's just a analytic function on the moduli space of elliptic curves.
The collection of equivalence classes of elliptic curve (torii of the form C/lattice) has the structure of a complex space (it's not a complex manifold, but rather a complex moduli stack). Modular forms are just analytic functions on it. That's all.
This dumb article doesn't help matter by presenting a brazen lie in the headline. Fifth fundamental operation, my butthurt ass.
First, they can be differential forms, not only functions. Second, there's an important note that we don't look only at things over C. For example, specifically in the context of Fermat's Last Theorem, we need Hida's theory of p-adic families of modular forms. Much of the arithmetic of modular forms comes from the modular curves being algebraic and (almost) defined over the integers.
If modular forms are (global?) sections of the structural sheaf of the moduli space of elliptic curves, the differential forms view will just be the standard construction of sheaf of 1-differentials. Similarly, since elliptic curves are easily defined over arithmetic fields, arithmetic modular forms will just be same thing, but over C_p or something like that.
I actually might be totally off in the above, but I doubt I am: that’s the power of Grothendieck approach, where everything just falls into its natural place in the framework.
There is rich structure in this area of maths that goes well beyond just sections of some sheaf, or at least this is what Serre, Deligne, Langlands, Mazur, Katz, Hida, Taylor, Wiles and many others seem to think.
Now make M to be a Mobius transformation, so that f((ax + b)/(cx + d)) = (cx + d)^k f(x), where the coefficients are in SL(2,Z) ie. integers with ad - bc = 1. Mobius transformations like this are common.
That's more or less it as far as I'm aware. There's some growth rate condition too, I don't think it's as important to an intuitive understanding as the transformation law.
Disclaimer; my training was in mathematical physics not mathematics. To me, what l wrote here was enough for me to feel like I understand what they are, at least to some basic level.
let f(ax, ay) = a^k f(x,y) be a homogeneous function in the plane x,y for a scalar a, then f can also be written as:
f(x,y) = f(y*x/y, y) = y^k f(x/y,1)
and the symmetry by a matrix transformation in the plane
x-> ax + by and y-> cx+dy then transforms the function f as:
f(ax+by, cx+dy) = (cx+dy)^k f((ax+by)/(cx+dy),1)
now introduce z=x/y and f(x/y,1)=F(z) then
F((az+b)/(cz+d)) = (cz+d)^k F(z)
Projective spaces are cool ;-)
just my 2 ct
I forgot to add to my post that, SL(2) is the symmetry group of the Riemann sphere. So since you often use this as your space in complex analysis, this symmetry transformation is everywhere. So that's one reason why you might expect to see modular forms in lots of places. (At least, that's what I understood, maybe a mathematician would tell you differently.)
A bit of repetition to avoid relying on a difficult-to-understand mathematical construct is often a good tradeoff when writing code for others to read. It's similar to how adding a dependency on a powerful library to do something trivial isn't a good move.
rotate45(rotate45(0deg)) = 90deg
sqrt2 * (sqrt2 * (1)) = 2
Length 2 in 90deg direction is 2i
Things don't really turn up, though. We use to say, say, that "addition is a group over the integers", but really what we mean is "we are allowed to see addition as a group over the integers, because it obeys the rules we've made for that abstraction".
There are many other fruitful abstractions, and even more not-so fruitful abstractions, that we can use as a lens to view things through.
And conversely, you don't have to view addition as a group over the integers if all you're doing is counting apples. Talk about overkill.
Modular forms are a fruitful lens to view many things through, apparently. Though, from the number of very educated people in the thread who have never really learned about or used them, they're apparently not so commonly useful as to get trivial. It's a niche abstraction, which has been used to get a grip on some problems where nothing else has worked - famously, Fermat's last theorem. Not what mathematicians reach for in everyday matters, but with a big wow factor when someone successfully does so.
Take tetration, for example, it fulfills all criteria of being "fundamental" as multiplication. but isn't listed as such
A mathematician would probably talk about addition, multiplication and inverse being fundamental operations.
You can't, in general, emulate multiple with addition (nor with subtraction).
Yes, for integers you can view multiplication as repeated addition. But that's not true for general structures. Have a look at eg https://en.wikipedia.org/wiki/Tropical_semiring for some inspiration. You can apply tropical semirings to shortest path problems in graphs: https://cs.nyu.edu/~mohri/pub/jalc.pdf
Just kidding, but not that stretched from what can easily find in contemporary mathematical publications.
log(a*b) = log(a) + log(b)
Also the log function comes very natural from the human senses, e.g. exponential scale of the musical tones, sound amplitude measured logarithmically, the eye works logarithmically too.
just my 2 ct