/sci/ - Science & Math

[Anonymous No.8445339]

What determines whether an abstract "space" is geometric?

There are various types of spaces used in geometry, and so far the only common factor I can find is some notion of topology and descent.

[Anonymous No.8446219]

is there even an agreed-upon definition of geometry in higher math?
algebraic ""geometry"" is like 50% commutative algebra with a few digressions to topological "geometry"
arithmetic "'geometry"" has literally no classical geometric interpretation

[Anonymous No.8446260]

>>8446219
>is there even an agreed-upon definition of geometry in higher math?

That's pretty much what I am asking.

[Anonymous No.8446266]

"Geometry" is based on the primitive notions of a "point or location in some space"

[Anonymous No.8446280]

>>8446266
Yes but there are many abstract "spaces" in mathematics. When are those abstract spaces geometric?

[Anonymous No.8446319]

>>8445339
Pulling the lever would kill infinite people, not pulling the lever would save 1/12 lives.

[Anonymous No.8446331]

>>8446319
[math] \zeta(0) [/math] is actually equal to [math] -\frac{1}{2} [/math]

so by pulling the lever, you would be saving a half of a life instead of a twelfth of a life

[Anonymous No.8446337]

>>8446331
have we finally found the physical interpretation of zeta regularization?

[Anonymous No.8446341]

>>8446331
If you follow up by running beside the train and slitting the throats of everyone on the other track, you'll save even more people.

[Anonymous No.8446347]

>>8446337
>have we finally found the physical interpretation of zeta regularization?

It has already been physical "proven" via QFT experiments.

[Anonymous No.8446350]

>>8446280
When the points in your space and the sets you can construct out of those points demonstrably follow the axioms of a geometry.

For example, can you define lines in your set?

Great. Now if by the definition you gave it follows the axiom of 'There exists a line that contains any two points' then you may be in a geometry.

Then keep studying your definition to see if it implies an euclidean geometry or some other kind.

[Anonymous No.8446357]

A geometry is a topology space with a metric. A metric is a function that determines the distance between two points in the space.
Of course, a metric is often not enough to do interesting things. You will also want some notion of an inner product/orthgonality to have well defined angles.

[Anonymous No.8446358]

A topology (T,X) on X is just a way to view parts of X.
From my feeling I'd say it's considered geometry as soon as you have a topology on X and care for a function space A^X. That is, either as soon as you may want to think of masses and fields living in X, or if you use coordinate functions to construct intersections within your space that aren't made to understand the topology, but for themselves or for a "geometric" (as in cone sections) problem.
Then again, this definition might make differential topology a geometric theory, which one may disagree with.

[Anonymous No.8446362]

>>8446350
That is a far too restrictive definition. I mean look at the open points of any scheme.

[Anonymous No.8446364]

>>446357
I'm (>>8446358) who just posted.
Yeah definitely, if you have a metric then it's geometry. But the issue is that many mathematicans speak of geometry in settings where metrics isn't cared for, e.g. algebraic geometry for number theory - even you can of course put some metric on spaces if you'd really want to

[Anonymous No.8446368]

>>8446350
Following up this idea, consider the set of natural numbers.

Define lines as:
Given two natural numbers a and b such that a<b the line ab = {x element of N such that x>=a and c<=b}

That satisfies the axiom I mentioned. Here is another axiom: Any line contains at least two points.

Let L be a line. Suppose it contains just a point. This can only happen if a=b and that contradicts the definition of a line.

Suppose a line has no points. This could only happen if there existed two points such that they have no point inbetween (possible) and are also not equal to themselves (because if they were then those points would be in the line)

The existence of such a natural number is ridiculous, therefore a line cannot be empty.

Therefore this definition of line also follows this second axiom: Any line has at least two points.

[Anonymous No.8446371]

>>8446362
But a geometry is only a geometry because it follows the axioms of a geometry.

What kind of geometry will even work if it does not follow the axioms of geometry?

I mean, you could make up new axioms and call them anon geometry but when geometry is already so well established, odds are your new geometry will be shit.

[Anonymous No.8446382]

>>8446371
Well, something like
https://en.wikipedia.org/wiki/K%C3%A4hler_manifold
is called geometry everywhere, and can be curvy and high dimensional as fuck. If by "axioms of geometry" you mean whose that at best capture hyperbolic geometry, then you miss more than you don't miss.

[Anonymous No.8446396]

>>8446382
But hyperbolic geometry is not the only geometry.

What you don't get is that a kahler manifold is an abstract definition and then people found spaces that align with that definition. The sames goes for everything else.

[Anonymous No.8446405]

>>8446396
Manifolds barely qualify as abstract compared to some spaces used in geometry.

I mean Stacks aren't even sets.

[OHP No.8446408]

All of the geometric spaces I have seen are defined as some sort of higher stack. I think descent is the common factor. The point is, we have points, paths, homotopies, et cetera, which are compatible with local structure and agree coherently on intersections. The local bits constitute the "local geometry" and a space is obtained by gluing local bits together.

[Anonymous No.8446430]

>>8446405
>Manifolds barely qualify as abstract compared to some spaces used in geometry.

But think about logic. Manifolds have a definition that you can think of the axioms of manifolds. Now, lets say you want to prove the properties of manifolds. You have two choices:

1) You first look for a topological space that follows the axioms of manifolds and then start proving theorems on this manifold

2) You say "Let M be a manifold" and then prove things using the abstract manifold M and the axioms it, by definition, follows.

Mathematicians always choose 2 because if you choose 1 then you may find theorems that are specific to that space.

>I mean Stacks aren't even sets.
That just means stacks aren't real.

[Anonymous No.8446446]

>>8446430
>That just means stacks aren't real.

But they are. DM-Stacks are even used to model parts of actual space in some special cases.

[Anonymous No.8446468]

You're replying to someone else, but getting back to it - if you say there are "axioms of geometry" then you should state those axioms, that seemingly are inclusive enough to make complex and Riemann manifolds into geometries.

And if you say "well the axioms of Riemannian geometry", then I come back to what I said above:
E.g. a 2-dim plane in 3 dimensions given by f(x,y,z)=0 is surely viewed as geometric space in many theories, but -while it's possible to consider some induced metrics on that surface- there are no metrics used in playing with those geometries. So there is nothing that could satisfy the axioms of Riemannian geometry.
If I take some random set to reason about injective functions e.g. for programming, then just because you can put the discrete topology on every set doesn't make every set into the study of topologies.

[Anonymous No.8447079]

>>8446468
>-while it's possible to consider some induced metrics on that surface- there are no metrics used in playing with those geometries.

Well the usual idea is just to glue together pullbacks of the euclidean metric using partitions of unity. And that work work fine, assuming f(x,y,z) smooth.

If you are talking about algebraic surfaces, then that is a different story.

But the idea of building a more complex space by gluing together simpler spaces seems to be the main unifying thing found in all geometric spaces. i.e. manifolds,varieties,schemes,stacks,etc.

[Anonymous No.8447102]

>>8446371
>But a geometry is only a geometry because it follows the axioms of a geometry.

Wrong.

Having a topological space seems like a good definition of a "geometry" but even that is too restrictive. Being able to do some sort of useful cohomology certainly means you are doing geometry, and grothendick showed you can do this for things that have very coarse topologies or are not quite top spaces (G-topologies for example).

[OHP No.8447764]

>>8447102
I think that cohomology is too broad a characterization of spaces. Cohomology is in general just the calculation of connected components of internal homs, but this is very broad and applies outside of geometry. For example, homotopy in a homotopy category of some sort of space is just cohomology in the dual category of algebras. Even group cohomology, which can be seen as cohomology of loop spaces, is not enough to tell you that you are working with loop spaces, as it can be defined independent of the external homotopy stuff going on.