I never did math past linear algebra/real analysis, so the only concept of sizes I have are countable/uncountable infinities.
Apparently the crux of this proof was showing that "the space of all modular forms with bounded denominators" and "the space of all congruence modular forms" were the same size.
I wonder what kind of expression "size" is here. Presumably not some finite integer, nor one of the simple infinities, since their first step was showing one is "a bit bigger" than the other. I wish this article went into more detail on that.
I definitely remember nerding out about modular forms via Andrew Wiles as a younger self.
If I understand the intro correctly, the "size" they're referring to is the growth rate of a sequence, where the sequence is counting the dimensions of certain subsets of bounded denominator modular forms.
Let BDMF = bounded denominator modular form.
They show congruence BDMFs grow at least N^3, but all BDMFs grow at most N^3*log(N). (The latter bound is the hard part of the proof.) To get the contradiction, they show a hypothetical noncongruence BDMF example would imply additional counterexamples that (just barely) get over the N^3*log(N) bound.
So is this a kind of result akin to the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem) where you have things that are asymptotically guaranteed to be/remain very close to one another?
Not sure if you're being sarcastic or not, but 99.9999% or possibly even 100% of this kind of abstract mathematics will never have any meaningful impact on day-to-day software engineering.
I keep hearing this about abstract mathematics "never" having an impact because it is too abstract and relates to pure mathematics. It's not true. Mathematics is a formal system that provides insights on surprising patterns. Surprising patterns can almost always be applied outside their intended area. And not surprisingly, non-Euclidean geometry and even the inability for mathematics to find certain proofs related to primes has resulted in breakthroughs in other areas. Surprising patterns take time to have effect mostly because they are not generally known until some genius is able to apply them outside their intended area.
I am 99.999% certain that you are wrong to say that "this kind of abstract mathematics will 'never' have any meaningful impact on day-to-day software engineering." I would be less certain if you replaced "never" with "will probably not have a significant impact in the short term".
I understand that there are sometimes deep connections between seemingly unrelated fields of mathematics. However, are there examples of applications of really pure abstract mathematics in the fields of say, symplectic geometry or differential topology (or any other very abstract fields), that have come to impact the lives of a significant number of software engineers?
Most programmers (myself included) spend their days using IDEs building a new API in a CRUD app or implementing a new UI widget in a mobile app. I may be unimaginative but I have a hard time seeing modular forms, fiber bundles, or exact sequences making a breakthrough in my life and impacting my programming.
Agreed. That's like hearing about early group theory and disregarding it as abstract nonsense. Turns out it's pretty useful everywhere now, but at the beginning it was not so obvious.
I would also change the statement to "never have a direct impact on programming". Indirectly in 100 years this may lead us to a better understanding of physics (modular forms have some weird hypothetised connections to physics that I don't really understand), which may let us construct better computers, which will obviously have an impact on programming.
- "I'd like to talk to you about producing a similar visualization for modular forms. These are inherently "just complex functions", but they're nontrivial to compute. But in two recent papers, I study how to compute modular forms and various visualizations of modular forms."
- "I'm knowledgeable about various 2d plotting, but I don't actually know anything about 3d plotting. I'm aware that blender exists and that shaders exist, but that's the extent of my knowledge. This is a major aspect of complex function visualization that I'm missing."
Maybe in a hundred years one of these discoveries will unlock time/FTL travel or wormholes or something... maybe a jailbreak for the simulation we're in.
Apparently the crux of this proof was showing that "the space of all modular forms with bounded denominators" and "the space of all congruence modular forms" were the same size.
I wonder what kind of expression "size" is here. Presumably not some finite integer, nor one of the simple infinities, since their first step was showing one is "a bit bigger" than the other. I wish this article went into more detail on that.
I definitely remember nerding out about modular forms via Andrew Wiles as a younger self.