What I find intriguing here is that in the past century, breakthroughs in physics (relativity, quantum mechanics) followed from earlier insights in mathematics (linear algebra, stochastics). Yes, mathematics can be very, very abstract and sometimes it appears to be more like poetry then science. But when an insight allows to generalize and simplify many different problems it's quite possible that we will see the fruits of that insights for decades to come.
It has always been like that. Fundamental research is necessary for applied research. And the needs of applied research give purpose and intuition for fundamental research.
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.
Fundamental research can also exist without looking at applied research. Understanding the world and finding the beauty in its fabrics is a legit goal per se, we're humans, not ants or traders.
Stephen Hawking worked at the Cambridge Department for Applied Mathematics and Theoretical Physics. The interface between the two fields is so fuzzy that it isn't always clear where one ends and the other begins.
I don't understand what I am seeing on https://www.lmfdb.org, but frontend developers take note: See how quick informative websites can be even with server side rendered templates? And still the website looks consistently styled and actually kind of neat.
On my phone the default navigation text size is too small and the front page only uses half of the vertical height of my screen in portrait. Some of the input boxes stick out past the header bar on the right hand side of the layout.
Yup, a couple of months(weeks?) ago, I submitted that website as an example of good information density, responsiveness and ease of navigation.
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.
This was the content in a few of the generals-stories following the Tao generals link a few days ago. I feel pretty smart right now not knowing a thing what this is about. (A few minutes later…) What a great read OP is!
As one Putnam Fellow to another: I took a graduate course on modular forms and I still don't really feel that I know what they are. I can't help feeling that they're the mathematical analogue of quantum mechanics: "If you think you understand modular forms, you don't understand modular forms."
Wow! That's a lot. But are they something that it's intrinsically difficult to understand or is it more like "yeah sure I can follow the definitions and such and maybe a theorem or two, but why is this important at all?"
I published a graduate level textbook on Modular Forms, and I also sometimes think I just barely know what they are: https://www.wstein.org/books/modform/
The accounts in primitve terms obscure it's true meaning.
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.
The title paraphrases a pithy quote of Martin Eichler (I believe), an influential modular forms researcher. It’s not just some pretentious editor trying to be clever.
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.
Imagine you have a function which transforms nicely under scaling, ie. f(Mx) = M^k f(x) for some b. Then if you have systems where you use x -> Mx regularly, you'll find these functions turn up because they're the ones that respect the symmetry.
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.
That's hilarious to me -- I took two courses in Complex Analysis, 4 courses in Algebraic Geometry, can prove Riemann-Roch theorem, and also still don't know what modular forms are. They have always been just beyond the horizon, right after the next hill you need to very laboriously climb.
I have a BS and a PhD in Math (about 20 grad math courses) and I've published about 15 math papers, but I've never been a professor. I don't know what modular forms are. (I specialized in numerical analysis.) I also bet that my PhD advisor who has published around 80 papers does not know what they are.
Maybe, but not directly, and it could easily be done badly. Relying explicitly on abstract symmetries tends to make code more mathematical, harder to read, and harder to explain. (See the many attempts at explaining monads for a relatively accessible example.)
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.
Readit News
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.
https://www.damtp.cam.ac.uk/
Firefox for Android.
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.
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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.
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
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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.
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