Mathematics

12:06 AM

Anyone familiar with the discrete method of defining a distance on a metric space?

12:23 AM

@Thorgott I don't think it's annoying. Give an atlas consisting of charts either contained in the interior of the two sides, or consisting of charts obtained by gluing a boundary chart on one side to a boundary chart on the other via f.

OneCold Ruben

Why do math texts ask you to prove things instead of telling you how to prove things?

Semiclassical

@user2103480 dunno, i don't deal with fluid mech that much, especially not involving the stress tensor

Thorgott

@MikeMiller I think writing the glued charts is annoying. First of all, you need to restrict the domains so that they actually glue together to an open set in $M\cup_fN$ and then you also want the charts to have image a box or something in $\mathbb{H}^n$ (because you need to propagate the twist by $f$ through the entire chart) and both conditions need to be fulfilled simultaneously. At least I don't see a better way.

@OneColdRuben cause math is something you ultimately only learn by doing yourself, not by watching others do it

of course, any sensible text will provide you with the tools necessary to prove the things it asks you to prove

user2103480

@Semiclassical ok ok

OneCold Ruben

I find a complete lack of explanation, in math explanations at a higher level. Just read the book, Just read the book carefully, Just read the book 6 times and more carefully.

user2103480

12:31 AM

@leslietownes I just noticed the guy with the bleached hair wears a tracksuit bearing the logo of a german football club lol

Mike Miller

Yeah that's what I had in mind Thorgott.

Notice that the resulting atlas is canonical, though ugly

Thorgott

actually, after second thought, I'm not sure if that works at all

why is gluing two boundary charts going to be smooth at the boundary

Mike Miller

Definition chase, to say that a map from a half-space is smooth means it extends to a smooth map on a slightly larger open space

That's definitely the correct recipe

It's the atlas consisting of charts contained in both sides and those whose intersection with $\partial M$ is a chart and intersection with $M, N$ are half-space charts

Thorgott

but these extensions may not agree with the other chart

what I'm saying is that a smooth function on upper and lower half-space respectively don't always glue together to a smooth function on all of Euclidean space

this atlas definitely works in the topological category, but I'm not convinced it does in the smooth one

Mike Miller

Why doesn't your collar idea run into the same issue

Global choice so no compatibility problems?

(I agree that this is an issue)

OK, yeah, I think it's because of this global choice thing.

Your collar idea is fine

One is then reduced to showing that collar neighborhoods are unique up to isotopy, though I think all you actually need is that their germs are unique up to isotopy, which I believe is very straightforward and analagous to what you did for discs

Thorgott

12:40 AM

Taking union of two collars give me an open subset homeomorphic to $\partial M\times(-1,1)$. This, together with the embedded copies of the interiors of $M,N$ gives an open cover of $M\cup_fN$ by three open subsets that carry compatible smooth structures by transport, hence they give a smooth structure on $M\cup_fN$

Mike Miller

Yeah

Definitely not the language I would use but it's surely the same and the atlas junk is just unpleasant to talk about

Thorgott

it's a global choice of straightening out the boundary, so to say

Mike Miller

yeah

Thorgott

This explains why Lee does the explicit atlas construction in his topological manifolds book, but uses collars in the smooth manifolds book

I thought the atlas would still work and he was just lazy the second time around, but there actually is a substantial difference

Mike Miller

My collar isotopy argument: Fix a "standard collar". Take any other collar, shrink it into this standard collar, isotope it so that it is linear with a derivative argument. Then observe that the space your derivatives take values in is contractible

(I'd like to say "the derivatives are all diagonal block matrices which are all-1s except the last entry, which is positive", so that the derivatives live in the space R_{>0}, but that's not quite right; the matrix need not be diagonal; the argument is still fixable)

Then use that to isotope to your standard collar

Thorgott

12:45 AM

what's a "linear collar" in this context?

it's not like it lies in one chart

Mike Miller

Dunno, whatever the proof forces it to be

What is written above might actually isotope the collar to be fiber-preserving

your standard collar is M x [0,1) so the fiber factor is a line and then linearity makes sense

robjohn

@TedShifrin either would be an interesting statistic

@OneColdRuben $d(x,x)=0$, $d(x,y)=1$ when $x\ne y$

Thorgott

yeah ok, I buy that doing this fiberwise works

sweeping uniformity issues under the rug, but it ought to work

Mike Miller

Yeah I get more paranoid if something is noncompact.

Ted Shifrin

1:00 AM

@robjohn i think I skew the mean age in chat up about 4-5 years on a typical day. ;)

robjohn

Combined, we do more than that

leslie townes

1:30 AM

then i show up and lower the mean with my extreme youth

Karim Mansour

2:19 AM

Hi

leslie townes

howdy

Karim Mansour

I haven't studied deformation theory yet but does anyone know if you start with smooth projective variety. In particular it is a manifold. What if you vary the almost complex structure associated with S. Do you get a family of smooth projective varieties ?

I think you might actually leave the projective category if you do that

anyway I know next to nothing about this but yeah worth to know if anyone is willing to discuss.

leslie townes

i don't even know what a projective variety is. i mean, i could guess, but i missed that by a mile in my own study.

Karim Mansour

@leslietownes oh np. Consider first is $\mathbb{P}^n = (\mathbb{C}^{n + 1} - \{0\}) / \sim$.

After that consider homogeneous polynomials whose zero lie in $\mathbb{P}^n$ and those define projective varieties.

Hi, @TedShifrin

Mike Miller

2:54 AM

There's no reason to even believe that the deformed ACS is even a complex structure. But even if you're deforming among integrable complex structures the resulting complex manifolds need not be projective.

Look into the moduli of complex structures on complex tori and on K3 surfaces.

Karim Mansour

I should say subject to integrable condition. But yeah you escape projective category.

Do you suggest a book about this ? I want a deformation say that is not algebraic that capture a lot of properties in families and doesn't escape projective.

Thanks for the suggestion about K3 surface I have Huybrecht has a book on K3 surfaces also found his book about complex geometry does what I am saying so I will read both @MikeMiller

OneCold Ruben

3:49 AM

@robjohn This was regarding an interval and and the function F(x) = x^2 trying to find the preimage of the interval [-1,0).

robjohn

4:25 AM

@OneColdRuben okay, I guess I don't understand where the discrete comes in.

OneCold Ruben

4:35 AM

That was a different question.

robjohn

I was responding to

5 hours ago, by OneCold Ruben

Anyone familiar with the discrete method of defining a distance on a metric space?

Mike Miller

6:05 AM

No, I never read a book on this material @KarimMansour, since it's not really my specialty. I'm just aware of results.

I probably learned the story about complex tori from stackexchange or just from conversations. The K3 story I learned from Joyce's book on manifolds with special holonomy. I love that book but I strongly suggest against it for you, since it points in a very very different direction then you are looking in (a lot of it is about very hard analysis of PDE but you are more interested in the complex and algebraic-geometric side).

Huybrechts good might be good, I do onot know.

Ted Shifrin

I already pointed out Morrow-Kodaira and, indeed, Kodaira's published papers, as good sources. The key thing to get started is to understand that the infinitesimal deformations come from $H^1(X,\Theta_X)$, which is the algebraic geometer's notation for the sheaf of sections of the tangent bundle.

Karim seems not to have looked for any of it, though.

Mike Miller

Amusingly you see variants of the same idea in gauge theory or really anywhere you'd like to deform something.

Ted Shifrin

6:26 AM

Sure. Kodaira rules :)

Matthew Christopher Bartsh

6:58 AM

I posted an answer in math.stackexchange.com/questions/65760/… a few hours ago. My answer discusses binary and hexadecimal notation and pronunciation and proposes a new unified set of names for all powers of two. Do you know of a forum where I could post it and thus have some discussion of it?

The name of any power of two can be deduced from a small set of rules, so memorizing the new names is unnecessary.

fido9dido

9:59 AM

I want to know the meaning of the symbol ^, r in random number uniformly distributed

Leaky Nun

@fido9dido "and"

fido9dido

thx

porridgemathematics

11:00 AM

Is it true that $\{(x,y,z) : x^2 + y^2 + z^2 = 1 \} \cup \{(0,0,t) : -1 \leq t \leq 1 \}$ is homotopy equivalent to a wedge sum of $|\mathbb{R}|$ many circles?

via collapsing the straight line connecting the antipodal points