What are some nice, relatively unknown, unsolved problems in

>>8846239
Homological Mirror Symmetry Conjecture.

Relatively unknown in the sense it is so niche that very little people who aren't explicitly working with it actually even understand the problem.

Generalizing the hyperoperation.

Addition, multiplication, and exponentiation are all part of the hyperoperation sequence. There's an infinite number of operations that come after them but their behaviors are almost completely unknown beyond how they act like iteration of the previous operation(when using positive integers). So all hyperoperations above 3(exponentiation) need to be defined for everything up to the complex numbers, then the hyperoperation itself needs to be defined for as many sets of numbers as possible(it might not actually be possible to extend this beyond natural numbers but if it is that would be rad).

>>8846239
Fermat's last theorem is pretty neat, because it's so easy to state:

[math] x^p-x=0 [/math] (mod p)

for any integers x and p.


And yet, after 300-some-odd years, no one knows whether this is true or not!

>>8846281
Whoa, that sounds interesting.
Exponentiation is a number multiplied by itself x number of times, so wouldn't the next in the sequence be a number exponentiated to itself x times?

>>8846287
:^)

>>8846291
https://en.wikipedia.org/wiki/Tetration

>>8846281
>>8846291
I assume you mean exponentiation to an integer?

>>8846291
Yes. [math]3\uparrow\uparrow 4[/math] means [math]3^{3^{3^3}}[/math]. The problem is that we don't know what something like [math]3\uparrow\uparrow \dfrac{1}{2}[/math] means.

[math]3\uparrow\uparrow 4[/math] can also be written as [math]\mathbb{H}_4(3,4)[/math], with [math]\mathbb{H}_4[/math] standing for the forth hyperoperation. So then that brings up the question of what something like [math]\mathbb{H}_\frac{3}{2}(3,4)[/math] means.

>>8846332
Don't know why latex broke there, it works in the preview. Lets try [math]\mathbb{H}_{\frac{3}{2}}(3,4)[/math]

If that still doesn't work, then what it's trying to show is [math]\dfrac{3}{2}[/math] in place of 4 in [math]\mathbb{H}_4[/math]

>>8846287
:3 you made me the day

>>8846344
>The problem is that we don't know what something like 3↑↑123↑↑12 means.
What's wrong with the following "natural" approach?

3^^(1/2) is the evaluation of the real-valued function f(x) = x^^(1/2) at x=3, such that f(f(x))=x

>>8846281
damn I'm glad I made this thread, I would have had no idea about this
>>8846344
I'm not sure I understand. Is the problem 3 tetrate 1/2, or the hyperoperation between H1 and H2

>>8846376
Both of them are problems, but defining the operation between 1 and 2 is a much more difficult one.
The trouble with defining [math]3\uparrow\uparrow \dfrac{1}{2}[/math], on the other hand, is that tetration doesn't have handy identity like exponentiation does that lets us find the definition algebraically. [math]4^{\frac{1}{2}}[/math] is defined because we have [math]a^x \cdot a^y=a^{x+y}[/math], which we can work backwards from.

Finding an identity is much easier than figuring out what the operation halfway between addition and multiplication does.

>>8846281
The problem is that, for example the 4th operation in your sequence, which I know as "tower", it's not clear how it should be defined for nonintegers.

tow(2) = 2^2
tow(3) = 2^(2^2)

what should be tow(2.5)?

With addition, multiplucation and exponentiation, we come up with some definition for nonintegers that "makes sense" at least because all the rules still work if you define it that way. For tower this isn't clear how to do

>>8846287
2/10 tickled me

I'm interested in characterizing all planar domains for which the spectrum of the dirichlet laplacian over the domain determines congruence.

Before you say anything, yes, I've read Zelditch.

I'd like to know whether the Euler-Mascheroni constant is algebraic.

Abstract: probably not.

bamp

I'd be interested in a structuralist framework of stochastic processes. Many objects, transformations and different modes of convergence are begging to be put into a categorical description, but seemingly nobody has done this yet.

It's well known but the Riemann hypothesis

0.999...=1

>>8847508

same here,

also would like to see the Toeplitz conjecture proved.

>>8848373
How would this clarify anything? People working in the subject have no problem forming a coherent "framework" of it.

>>8846332
Sounds like a stupid approach, but draw a graph and smoothly connect the dots? Should give you at least an idea of what noninteger operations are supposed to yield.

>>8848600
Are you talking about the first thing, or the second thing?

The first doesn't work because the numbers get too big to analyze. The second doesn't work because the operations create too different of graphs.

>>8846332
Umm isn't 3^^(1/2) the solution to x^x=3

>>8848373
I don't see why this should be hard at all: a random variable is a measurable function, so an arrow in the category of measurable spaces.

A stochastic process is, classically, simply a T-indexed family of random variables (T a time set), which translates into a T-indexed family of random variables. The appropriate categorification would be a T-shaped diagram in Mea, and (I only know rudimentary category theory so this may be inaccurate) the natural filtration for the process can be given by the universal pullback, and from there developing the rest of the theory should simply be an exercise in arrow-chasing.

>>8848803
I'm guessing you tought something along the lines of :
Since (3^^1)^(3^^1)=33=(3^3)=3^^2=3^^(1+1)
3^^(1/2) ^ 3^^(1/2) = 3^^(1/2+1/2)=3^^1 = 3
So 3^^(1/2) solves x^x=3

It has several problems tho ;
- has x^x=3 a unique solution ? If not you should add another condition (as for x^(1/2) for example, which has two solution before we define it as exp(x*ln(1/2)), ect...)
- how do you expand it to non-rationnal, or non-algebraic numbers ?
- ^ being non-commutative, non-associative, you'll have problem to extend this method beyond simple cases... to be faire i'm not even sure of (a^^b)^(a^^c)=a^^(b+c)...

>>8846239
bampp