/sci/ - Science & Math

To motivate why prime numbers are of interest, or more genrally "the concept of a prime", you should think of primes not just as numbers coming after and before some other number in the ordering of natural numbers, but as generators of prime ideals for the ring Z:
For any number x, think of the grid {x*n | n in Z}, which it defines.
So the number 6 defines the grid
{...,-18,-12,-6,0,6,12,18,24,30,36...}
on the integer number line.
Some grids are subgrids of others. For example the number 3 generates the grid
{...,-9,-6,-3,0,3,6,9,12,15,18,21,...}
and this grid contains the grid generated by the number 6.
The prime numbers are exactly the numbers which generate the grids containing all the other grids.
And whenever you multiply a number n with a number m from two grids, the result n*m is contains in the n-grid and in the m-grid again.
And the set of numbers corresponds to the set of grids. You can in large parts replace studying numbers with studying those collections, the grids. Btw. loop up ring theory.

Now the general concept can be used to do a shitload of math. For example, take R^2 and consider the collection F of functions of 3 variables which are zero at the point (3,-5). The if you take any function g and multiply it with a function f from F, the value at (3,-5) is g(3,-5)*f(3,-5) and this is again zero because f is. So g*f is in F again, just like multiplying with numbers from a grid gets back to the grid.

You can now go on and replace a space with function spaces.
It's part of algebraic geometry.
RIP Grothendieck.

Firstly, the "usefullness for the world" of a mathematical subject shouldn't be a general factor for the validity of people persuing it in the first place. But okay, that conversation has been had.

Now to motivate why prime numbers are of interest, or more genrally "the concept of a prime", you should think of primes not just as numbers coming after and before some other number in the ordering of natural numbers, but as generators of prime ideals for the ring Z:
For any number x, think of the grid {x*n | x in Z}, which it defines.
So the number 6 defines the grid
{...,-18,-12,-6,0,6,12,18,24,30,36...}
on the integer number line.
Some grids are subgrids of others. For example the number 3 generates the grid
{...,-9,-6,-3,0,3,6,9,12,15,18,21,...}
and this grid contains the grid generated by the number 6.
The prime numbers are exactly the numbers which generate the grids containing all the other grids.
And whenever you multiply a number n with a number m from two grids, the result n*m is contains in the n-grid and in the m-grid again.
And the set of numbers corresponds to the set of grids.

Now the general concept can be used to do a shitload of math. For example, take R^2 and consider the collection F of functions of 3 variables which are zero at the point (3,-5). The if you take any function g and multiply it with a function f from F, the value at (3,-5) is g(3,-5)*f(3,-5) and this is again zero because f is. So g*f is in F again, just like multiplying with numbers from a grid gets back to the grid.

You can now go on and replace a space with function spaces.
It's part of algebraic geometry. RIP Grothendieck.
Also look up ring theory.

>>6697060
>excuses.

Also, you don't need much breath anyways. Everyone can lift, pic related.

>>6598584
Does the Yoneda lemma for a natural trafo from hom(A,-) to F simplify somehow, if we know that F is actually representable by F? Then we know that F(A) is nonempty and more importantly, there is the inverse from F to the hom-functor. Can we use this to write down a new relation between the two beyond nat(hom(A,-),F)==F(A)?

Let's have thread about topos theory.

Tell you opinions or speak about interesting frameworks of study.

Ask basic question if you don't know what it is.
Here is a relatively related motivation:
http://arxiv.org/pdf/1212.6543v1.pdf
And here is the wikipedia article
http://en.wikipedia.org/wiki/History_of_topos_theory

pic related, engineers at work

I have an sufficient understanding of predicate logic and some ideas about computer science. Could someone elaborate:
What are the barriert to solve "P=NP?" ?

related, I'm watching this atm.:
http://techtalks.tv/talks/1301/

Let G be a group, N a finite normal subgroup, and H a subgroup of finite index. Also gcd(|N|, [G : H]) = 1. Prove that N < H.

How do I do this, /sci/? I've reduced it to showing that the natural group action of G on the set G/H is trivial when restricted to N, but how do I use the relatively prime condition to do this?

>>5728847
>Grothendieck will never hoist you with his strong arms and broad shoulders