6 ms·

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

by lacker 3y ago

xyzzyz 3y ago
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.
cperciva 3y ago
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."
- nhatcher 3y ago
  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?"
  - cperciva 3y ago
    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?"
- macrolocal 3y ago
  For automorphic forms, absolutely.
deleted 3y ago
[deleted]
jksk61 3y ago
then you took the wrong courses.
- deleted 3y ago
  [deleted]
joe__f 3y ago
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.
- fxj 3y ago
  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
  - joe__f 3y ago
    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.)
deleted 3y ago
[deleted]
psacawa 3y ago
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.
- henrydark 3y ago
  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.
  - xyzzyz 3y ago
    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.
    - henrydark 3y ago
      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.
irchans 3y ago
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.
williamstein 3y ago
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/ https://www.wstein.org/books/modform/

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

by lacker 3y ago

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

xyzzyz 3y ago

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.

cperciva 3y ago

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."

nhatcher 3y ago
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?"
- cperciva 3y ago
  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?"
macrolocal 3y ago
For automorphic forms, absolutely.

deleted 3y ago

[deleted]

jksk61 3y ago

then you took the wrong courses.

deleted 3y ago
[deleted]

joe__f 3y ago

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.

fxj 3y ago
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
- joe__f 3y ago
  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.)

deleted 3y ago

[deleted]

psacawa 3y ago

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.

henrydark 3y ago
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.
- xyzzyz 3y ago
  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.
  - henrydark 3y ago
    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.

irchans 3y ago

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.

williamstein 3y ago

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/ https://www.wstein.org/books/modform/