Mathematics

12:00 AM

But it seems to use various generalization of the Fourier transform so there’s that

you're right that PDEs come up with complex varieties, but the Weil conjectures are about finite fields, so it would be very surprising if PDEs are applicable in any way

Zee

Ya , that makes sense

I need to emphasis to my haters here : the work of Grothendieck is what I consider honost catagory theory . Am mainly arguing about the catagory theory stuff that is not directly related to good ol classical math

And to each his own , that’s the nice thing about math , you study what you like , but don’t come and try to take over my spot

MatheinBoulomenos

I can tell you that the stuff I was discussing with Leaky is related to good ol classical math: Algebraic groups, Lie groups, group schemes, topological groups. It provides a different perspective (you probably don't like) on each of those

maybe it's more useful for algebraic groups and group schemes, but for that it can definitely be helpful

Leaky Nun

don't drag me in your food

@MatheinBoulomenos do you have other applications of Yoneda?

MatheinBoulomenos

@LeakyNun there are a lot, but I'm going to sleep now

Leaky Nun

12:11 AM

ok

gite nucht

MatheinBoulomenos

gute nacht

@LeakyNun let me just mention one thing: if you think of universal properties as statements about hom-sets, then the fully faithfulness of the Yoneda embedding tells you that two objects with the same universal property are isomorphic. And if you look closely e.g. at the proof for uniqueness of the tensor product you can see that it's related to the proof you gave for the fact how fully faithful functors relfect isomorphismic objects and the Yoneda argument

Leaky Nun

I see, thanks

Zee

On the other hand , we need a unfying theory of mathematics . We can’t just have a bunch of disconnected small areas with small problems

Catagory theory and specially topos theory provides that

For example :one of the most geometric areas of mathematics

Complex geomtry , has deep ties with catagory theory

As in oka Gromov principle and model catagories

So the connections are there

You can also do sifferebtialgwomtry using natural bundles and catagorical arguments

This provides a much nicer and honestly coordinate independent formulation

@MatheinBoulomenos @Daminark

loch

12:26 AM

@LeakyNun Not really 'an application' - but I may have mentioned about functor of points before (or maybe you guys did in the above) - specifically about how one could view a scheme as a generalisation of solving equations. For example if I take the scheme $X=\operatorname{Spec} \mathbb{Z}[x,y]/(x^2+y^2-1)$, then I can consider the functor from the category of affine schemes to the category of sets mapping each scheme $Y$ to the set $\operatorname{Hom}_{Sch}(Y,X)$. Yoneda tells you that your scheme is determined by this functor -

Leaky Nun

I see

loch

12:51 AM

sometimes the only thing we know about a scheme is that it represents a certain functor, and understanding the functor allows us to say some things about the scheme.

Leaky Nun

@loch right, something like $A^\times = \operatorname{Hom}(K[\Bbb Z],A)$

loch

apart from the fact that i've never seen people write $K[x,x^{-1}]$ as $K[\mathbb{Z}]$, sure

but i was thinking more in the lines of say - computing dimension of its tangent space /or giving a bound of its dimension at a point

but yes probably alg group gives you plenty of examples of these things too

Leaky Nun

you just seen one

I wonder what the prime ideals of $K[\Bbb Z]$ look like

will you be there tomorrow?

(today)

loch

well you can understnad what the primes of K[x,x^{-1}] look like

no

Leaky Nun

right, the ideal correspondence theorem for quotient ring

and then the dimension of K[X,Y] is 2

loch

1:00 AM

probably easier to think about this as inverting $x$ in $K[x]$

Leaky Nun

who am I kidding, the primes of K[X,X^-1] is just the K points

Simply Beautiful Art

:) Glad to see chat is rolling along (perhaps a tad bumpy as of recently though)

Leaky Nun

i.e. K^\times

am I right

maybe not

loch

no - if $K = \mathbb{R}$, then $(X^2+1)$ is a prime in $K[X,X^{-1}]$

and its residue field is $\mathbb{C}$

Leaky Nun

interesting

then what does K-points mean?

loch

1:03 AM

$K[X,X^{-1}]$ is just inverting $X$ in $K[X]$ - and you know how prime ideals correspond under localisation!

Leaky Nun

oh right

prime ideals in K[X] that do not meet X

since K[X] is PID, that means irreducible polynomials that do not divide x^n

i.e. irreducible polynomials that do not divide x

or 0

@loch am i right

loch

yes

Leaky Nun

oh so you're just removing (X)

that's right

that corresponds with the classical picture of removing 0

right?

loch

@LeakyNun yes!

A $K$-point of a scheme $X$ (say defined over $k$) would just be a morphism of $k$-schemes $\mathrm{Spec}(K) \rightarrow X$.

There's no need to only consider fields etc. but anyway the idea is the same.

Now let's suppose I take the scheme $\operatorname{Spec} k[x,y]/(x^2+y^2+1)$, and say you take $K$ to be a field extension of $k$. Then a morphism of $k$-schemes $\operatorname{Spec}(K) \rightarrow \operatorname{Spec} k[x,y]/(x^2+y^2+1)$ corresponds to a ring map $k[x,y]/(x^2+y^2+1) \rightarrow K$ fixing the underlying field $k$.

Simply Beautiful Art

While I'm here, if anyone wants to talk about large numbers/fast growing functions, feel free to ping me.

Leaky Nun

1:12 AM

@SimplyBeautifulArt I feel like I have a lot to learn about fgh

@loch I see

Simply Beautiful Art

:P

loch

none of the assumptions made there are necessary (field extension blablabla, scheme being finite type etc.) - but uh it's an example!

Simply Beautiful Art

I came up with what might actually be a decently simple thing that grows decently fast.

relatively to the things people normally deal with ofc

Leaky Nun

@loch oh and I think I found a generalization of separable extension

loch

?

Leaky Nun

1:16 AM

$L/K$ is separable if $\overline K \otimes_K L$ is reduced

loch

anyway im off to bed now
maybe i'll read what you say tomorrow / later this week lol

Leaky Nun

reference (5.3)

ok

Simply Beautiful Art

1:44 AM

Hi @XanderHenderson

>.>

$$(I_0f)(n)=f^n(n)=\underbrace{f( f(\dots f(}_nn)\dots))$$
$$(I_{k+1}f)(n)=(I_k^nf)(n)=\underbrace{ (I_k(I_k\dots(I_k}_nf)\dots))(n)$$

<.<

Xander Henderson

2:23 AM

HELLO!

How's things, @SimplyBeautifulArt?

Famous Michael Wang

what is this about?

Xander Henderson

2:37 AM

At a guess: very large (but finite) cardinals

that is a thing that SBA gets into

Simply Beautiful Art

2:49 AM

@XanderHenderson fine. Reading up on No Game No Life light novels :)

@FamousMichaelWang I tend to be that person who makes very fast geowing functions for no particular reason. Seeing as no-one is occupying the chat, we can play a game involving such functions, if you're up for it.

Secret

Currently trying to show via cantor diagonal argument of decimals that the number of new element counting function grows quicker than any computable function

Simply Beautiful Art

3:24 AM

@Secret not entirely sure what you mean but ok

Symposium

3:41 AM

Any computable function?

student

4:08 AM

what does a "computable" function require?

given, of course, the standard math definition of a "function" :-)

Secret

4:26 AM

Computable function

Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all...

1000000000000

0100000000000

bleh too long

$Mathematics$ Rambles

For all my maths rambles that are not even qualified to fit in...

student

thanks @Secret

Secret

5:55 AM

Preliminary result of attempting to count the set of all binaries with countable places:

Leaky Nun

I misread it as bananas

Laotse

Hello. I don't understand the difference between tensor product and Kronecker/outer product. Here (en.wikipedia.org/wiki/Outer_product#Tensor_multiplication) it is said that tensor product of two tensors of order 2 makes a new tensor of order 4. But how is Kronecker product still "tensor product" and it produces still a matrix?

Leaky Nun

I just woke up

student

good morning :-)

Secret

The number of $\omega$ places increases in the sequence 1,1,2,2,5,5,12,12,...

The total number of entries per units of $\omega$ increases in the sequence:

1,2,3,5,7,12,17,29,...

Therefore given any $\alpha$th iteration there is at least $f(\alpha)$ elements where $f$ computable, and the next iteration will produce something such that $f(\alpha + 1) > f(\alpha)$ for all countable $\alpha$

Therefore the lower bound of the number of reals by "counting" is $\omega_1^{CK}$ the supremum of all computable ordinals

The proof is complete if the following conjecture is true:

Given $\omega$ uncomputable binary sequences that are distinct. Place them into a table. Then the elements obtained from reading each columns are always unique

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Rambles

Transcript for

$Mathematics$ Rambles

$Mathematics$ Mathematics