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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.

Applied research can also predate fundamental research. They both feed back into each other.

Yep! And engineering advances can trigger breakthroughs in both. It is marvelous to see basic research, applied research and engineering forming a virtuous complex.

Thermodynamics is my go-to example here.

Steam engines came first, and thermodynamics was partially invented to study them. Later on, thermodynamics helped make better steam engines.

Not accurate. Early thermodynamics (in the broad sense, including gas laws) like Boyle’s 1662 Law and others led to Papin’s 1679 invention of the steam digester and related inventions (steam release valve) which directly led to Savery’s 1698 steam engine.

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.

Yes, nowadays we systematize and include Boyle's law and others within thermodynamics.

About Vacuum Science: I grew up around and in Magdeburg, which is famous for her (vacuum) hemispheres. https://en.wikipedia.org/wiki/Magdeburg_hemispheres

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.

https://www.damtp.cam.ac.uk/

Let me take this opportunity to link again to the modular forms database: https://www.lmfdb.org/

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.

Firefox for Android.

I probably don’t fall into the majority on this one, but I wish more websites basically defaulted to a zoomed out default and let me pan around them on mobile. I would much prefer it to modern non-dense information sparse screens with tons and tons of scrolling.

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.

> I would much prefer it to modern non-dense information sparse screens with tons and tons of scrolling.

It doesn't need to go to the opposite extreme (as you described it :-)) to be more readable by default on different screen sizes.

The default android browser used to let you just double tap to reflow text to the current zoom level, mostly like reducing the window width on desktop.

Same on Chrome for iOS.

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.

Very nice website although I don't understand why it needs googleapis spyware access. It is something I'd never allow.

Very good observation; I will ask them.

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!

Very much related. I recently stumbled upon this excellent YouTube channel in which he explains modular forms https://www.youtube.com/@PeakMathLandscape/videos

A field with one element? Replacing abstract algebra entirely? The Riemann Hypothesis? The Langlands Program?

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!

> A field with one element? Replacing abstract algebra entirely?

At this time of year? At this time of day? In this part of the country? Localized entirely within your kitchen?

A series of lectures by Richard Borcherds on modular forms:https://youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xy...

I find it somewhat discouraging that I took about 8 courses of graduate mathematics and still don't know what modular forms are.

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?"

It's more that they're kind of magical. You think you have a handle on how they behave, then you see a theorem and them and you're like "how the hell do they do that?"

For automorphic forms, absolutely.

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.

Sounds like it's a pretentious researcher trying to be clever.

Well, they're not _just_ that, right?

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.

The above definition (analytic function on a moduli space of elliptic curve) actually extends in a natural way. I haven’t known what modular forms were before the parent comment, but I know algebraic geometry, and so it is natural for me to extend above definition for cases you mention.

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.

This definitely fits with Grothendieck's philosophy: he basically ignored all work in this area, implicitly claiming it was trivial, while some of his closest friends and most famous student made huge strides with actual hard work - not quite things falling into place. In fact, the paper most famously proving the Weil conjectures has as an explicit target the coefficients of a modular form, uses an inspiration from automorphic forms theory, and is infamously Grothendieck's greatest disappointment.

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.

Oh, I did not meant to imply that the framework necessarily makes it so that the results open like a softened, rubbed nut, as Grothendieck said; I don't quite agree with that. For me, the benefit is rather in building a mental framework, which facilitates understanding, and putting seemingly disparate things into one coherent whole. The actual hard thinking and insights are still necessary, it ain't no royal road.

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.

Modular forms can also be viewed as functions in the 2d projective space with a special symmetry:

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

Oh yeah that's a nice observation! Protective space in 2D is the same as the Riemann sphere, which is the complex plane with one point added at infinity.

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.)

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.

then you took the wrong courses.

There is a lot of repetition in programming. Could this more abstract reasoning about symmetries inspire language research?

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.

Assuming I do not want to prove Fermat's last theorem, what are the real world application of modular forms?

Modular forms also have applications to representation theory and physics in conformal field theory https://researchers.ms.unimelb.edu.au/~dridout@unimelb/semin...

Correct me if I'm wrong but (1+i)^2 = 2i is false? I thought it is = (i^2)+2i+1

i² = -1, so it all works out

(1+i) = scale by sqrt2 and rotate by 45deg

rotate45(rotate45(0deg)) = 90deg

sqrt2 * (sqrt2 * (1)) = 2

Length 2 in 90deg direction is 2i

If its a fundamental operation, what is 5 <modular form> 4.4, e.g.(5*4.4=22)?

The guy who said it meant it as a joke, because they turned up in so many areas of mathematics.

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.

Addition, multiplication, and division can be emulated with subtraction. so how are they fundamental?

Take tetration, for example, it fulfills all criteria of being "fundamental" as multiplication. but isn't listed as such

Talking about addition, subtraction, multiplication and division as fundamental operations is just to capture the imagination of the layman. Modern mathematics talks more abstractly.

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

Meh, don’t you had the memo that it’s all about the endogenic structure of the identity function within a toroidal network topology plunged within the diagonal of a transcendental mesh?

Just kidding, but not that stretched from what can easily find in contemporary mathematical publications.

I wonder how a society would have developed if they would have invented the log function before multiplication.

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