Mathematics - 2016-10-12 (page 3 of 4)

Balarka Sen

5:00 PM

I second that you should.

Danu

What else is new?

Mike Miller

I'be heard a number of people describe their papers as exercises they worked out.

Balarka Sen

5:22 PM

Hi @Ted.

Ted Shifrin

Hi @Balarka, g'night @MikeM

Mike Miller

morning.

Ted Shifrin

@Balarka: In case you're interested, I used projective geometry to come up with that parametrization of the hyperboloid. Basically, it's projectively equivalent to the saddle surface $z=xy$, where the $x$- and $y$-curves are of course rulings.

Mike Miller

I think it's probably time to give up on what I've been stuck on this week. Something to work out for another paper.

Ted Shifrin

Sometimes letting your subconscious contemplate is a good idea, @MikeM.

Balarka Sen

5:25 PM

@TedShifrin Ah, I see.

Ted Shifrin

Just shows how horrible it is to give a parametrization by rulings in both variables :)

Mike Miller

@Ted I'm skeptical this is a subconsciously solvable problem. More of a "work at it for a month" problem.

Ted Shifrin

Well, fair enough.

Danu

Hey @Ted!

Ted Shifrin

Hey @Danu ... How's "vacation"?

Danu

5:27 PM

I'm writing up my presentation on section 4 of Milnor-Stasheff

So much easier than Huybrechts :D :D

Balarka Sen

@Danu Which sections are you going to present?

I think you told me, I forgot.

Danu

4, 9

4 early November, 9 on the 7th of December

Balarka Sen

@TedShifrin I am onto the next section, by the way. I liked the conformal parameterization problems from the previous, though they were easy.

Ted Shifrin

Most of this stuff should be easy for you, @Balarka. But that's ok.

Did you figure out the geometry of #10?

Balarka Sen

I was about to ask you. On polar coordinates, isn't that just "move faster radially, but slower angularly"?

Mike Miller

5:31 PM

@Ted Not to mention there were n long-awaited papers that came out yesterday. Sigh.

Ted Shifrin

But how would you explain what the mapping does to a high school kid, @Balarka?

Balarka Sen

@Danu Nice, I don't know the Euler class in full generality yet (that's probably the content of 9, from what I gather from the title).

Ted Shifrin

Is $n$ a large cardinal, @MikeM?

Danu

@MikeMiller which categories on arxiv do you check out?

Balarka Sen

@TedShifrin I think it's mapping to a cone.

Building a cone out of a pacman region by gluing the sides of the missing slice?

Ted Shifrin

5:36 PM

Sure, but can you explain what the mapping does to the pacman in a concrete way?

There you go.

Mike Miller

@Ted It's larger than 2.

Ted Shifrin

You can actually do that with paper. In fact, I always had a big pacman to illustrate this in class and office hours.

Mike Miller

@Danu I'm subscribed to AT, DG, GT, and SG. Occasionally I pop into AP.

Balarka Sen

@TedShifrin Yep, I certainly have done it before :)

Ted Shifrin

Cool, @Balarka. Pacman appears numerous times in exercises later (particularly in discussing parallel translation and holonomy). So I realized I should add this exercise only recently.

Balarka Sen

5:41 PM

I see.

Mike Miller

Cones are good.

Ted Shifrin

Even without ice cream?

Danu

@TedShifrin This sounds like fun ^^

@MikeMiller Hmkay :)

Balarka Sen

I don't like ice cream cones.

Mike Miller

@Danu Should I say what those stand for?

Balarka Sen

5:43 PM

I'd like to know what AP is.

Mike Miller

I'll take two guesses first. Are the other four really so obvious? I'd have thought at least one of them would be tough. But no spoilers for Danu.

I really enjoyed writing this email:

Balarka Sen

I think I can guess all of those 4.

Mike Miller

Hi Mike,

Can you let me know if anyone takes notes for this talk? I'd like to see them.
By the way, your new paper with Mike looks exciting!

-Mike

Balarka Sen

lol!

Ted Shifrin

Goldilocks and the Three Mikes

Mike Miller

5:47 PM

If only one of my fellow grad students cared about that talk, I could have asked him to find notes on behalf of Mike and myself.

hahaha

Ted Shifrin

There are two Mike H's?

Balarka Sen

I wonder if any of those is the famous Mike H.

Danu

@MikeMiller No worries, I know them.

Mike Miller

@TedShifrin Only one at UCLA.

Danu

AlgTop, DiffGeo, GeomTop, SympGeom, AnalPDE

Mike Miller

5:50 PM

(Deleted the second message because it was private correspondence I don't know that I have permission to share, even though it's harmless.)

@Balarka There are two famous Mike H.'s.

Balarka Sen

Ah, fair.

Mike Miller

But yes, this is they.

Ted Shifrin

I know a famous Mike U.

Danu

lol

Balarka Sen

Oh, wonderful.

Ted Shifrin

5:51 PM

No comma.

Danu

I'm right about the abbreviations, aren't I @MikeMiller? :P

Ted Shifrin

And a more famous Mike A.

Not to mention Mike S.

Mike Miller

Yes, @Danu.

Have I said that there are three Mike Millers at UCLA? One works with one of the academic employee unions. Another teaches math. A third works with an academic employee union and teaches math.

Danu

What category would complex geometry stuff go in? Just diffgeo?

@MikeMiller God

Best of both worlds ;D

Ted Shifrin

What a Venn Diagram the Mikes make.

Isn't there a separate complex geometry category?

Mike Miller

5:52 PM

algebraic geometry, differential geometry, or complex variables, depending

most of the last is analysis rather than geometry

Ted Shifrin

I wouldn't put it in complex variables ... the other two are fine. Maybe several complex variables or geometric analysis.

Mike Miller

Depends on the paper. Sometimes people are doing local existence etc to complex PDEs of geometric origin.

Danu

Hmm, I'm having some good thoughts about the tautological line bundle... I only now see how it sort of inevitably leads to linear equations

Mike Miller

Or (pseudo-, sub-, whatever-)harmonic analysis.

Balarka Sen

@MikeMiller I presume those three aren't you?

Mike Miller

5:54 PM

@BalarkaSen One of them is.

Danu

The last one should be our Mike

Balarka Sen

Gotcha

Mike Miller

It always confuses me when I see Mike Miller in my contacts list.

Ted Shifrin

I'll use descending induction and remove you all.

Balarka Sen

I need to stop slacking and do computations to get used to the IInd fundamental form.

Mike Miller

6:01 PM

You're close to knowing more Riemannian geometry than I do.

Balarka Sen

Heh, I doubt.

Mike Miller

I never learned the second fundamental form very well.

Ted Shifrin

My favorite symmetric 2-tensor with values in the normal bundle.

Danu

Quick: How many do you know?

Ted Shifrin

(That's why I changed from "one of my favorite" to ...)

It works in projective geometry (no metric) as well.

Danu

6:15 PM

:)

Balarka Sen

@TedShifrin Because it's just the differential of the Gauss map, isn't it?

Ted Shifrin

Not quite, @Balarka. That's what is standardly called (with a minus sign) the shape operator. A quadratic form and a linear map aren't the same, of course.

Balarka Sen

I don't understand. The differential is a symmetric matrix - you can make it act on a tangent space by multiplication. That's what I'd call the shape operator. But I can also think of it as a quadratic form on the tangent space because it's symmetric.

Mike Miller

I think i learned an iota of it from G&H. It's probably important.

Balarka Sen

But the matrix of both are the same, aren't they?

Ted Shifrin

6:18 PM

No, read on, MacDuff.

Mike Miller

I'm still waiting for Balarka to learn about curvature of surfaces so i can talk about that.

Ted Shifrin

He's getting there, @MikeM.

But not with $D^2$.

Although when he reads section 3.3, he'll know how to do it with moving frames and Cartan.

Mike Miller

That's fine. I want to talk about GB. And the lovely theorems of Kazdan and Warner.

Ted Shifrin

GB is next chapter. He has a ways to go.

Mike Miller

I really need to read those papers - they're some of my favorite theorems.

Balarka Sen

6:21 PM

@TedShifrin I don't think I know what "MacDuff" means :)

Ted Shifrin

It was a paraphrase of a line from Hamlet :)

Balarka Sen

ah, I don't know much of Shakespeare at all.

Ted Shifrin

Lead -> read ... I thought it was cute, actually :D

0celo7

@MikeMiller G&H?

Balarka Sen

which, speaking of, I have to decide on a story to translate for my language project.

maybe something of Borges. I don't know of too many microstories.

Danu

6:23 PM

Which to which language?

Balarka Sen

English to Bengali.

Danu

Do a short story by Roald Dahl

Mike Miller

Borges was already translated once. Stick to something that was first written in English.

Ted Shifrin

Good advice, that.

Danu

Or if you want a challenge go to E.A. Poe :D

Balarka Sen

6:24 PM

@Danu Poe is too long.

Danu

What?

His short stories are super short

Like 2-3 pages type short

Mike Miller

Translate hemingway's six word story.

Balarka Sen

I tried to do Tell-Tale Heart. It was not as short.

Danu

I know, becuase I found a discarded copy of a collection of his short stories in Trieste

I've been reading some of the stories

Balarka Sen

I have read his short stories

He's one of my favorites

Danu

6:26 PM

So then you should know plenty short ones

Ted Shifrin

Maybe O'Henry ... pretty short.

Or a chapter of Winnie the Pooh :)

Balarka Sen

That's what I thought, @Danu.

Ted Shifrin

You'd have fun with the woozles.

Danu

@BalarkaSen Do you want me to tell you some? :P

Mike Miller

It's sad, but I can't think of any short story authors I really admire.

Balarka Sen

6:27 PM

@MikeMiller I have an example translation of Chekov, so it doesn't seem like that's a requirement.

Mike Miller

Normally when I'm looking for a reference I look at my bookmark bar. But that's all math.

Danu

Did you read Roald Dahl though? :P

Balarka Sen

@Danu Sure, go ahead. I'll warn you I have tried to translate some and it got long quickly.

Danu

Balarka Sen

I haven't read Dahl's shorts. But I have read his novel, BFG.

Ted Shifrin

6:28 PM

@Balarka, @MikeM, @Danu: Care to comment?

(Sorry, back to math.)

Danu

> Thom isomorphism

:'(

Mike Miller

I don't much care to comment, but hopefully one of the other two does.

I am chickening out.

Worthless twits.

I would, damnit

6:30 PM

lol

Danu

You guys are constantly making me feel inadequate lately :P

Ted Shifrin

Now watch Danu remove that, just like he removed my last twit yelling spree.

Danu

Anyways @Balarka Ligeia, The Masque of the Red Death

@TedShifrin Do you have an issue with that?

Ted Shifrin

No.

I just will start ignoring more people here.

Mike Miller

@Danu Let $E$ be a rank $k$ oriented vector bundle over $X$. Then $H^{*+k}(DE, SE) \to H^*(X)$ is an isomorphism, where the map is moderately fancy.

Danu

6:31 PM

Okay. In case you ever have an issue with my moderation (I try not to do much...) you can always bring it up

@MikeMiller DE,SE

Germany, Stack Exchange?

Ted Shifrin

Disk bundle, sphere bundle

Balarka Sen

@Danu Masque of Red Death is like 5 pages

Ted Shifrin

(coming from vector bundle $E$)

Danu

@BalarkaSen Sure

Ted Shifrin

5 pages is pretty short for a short story.

Danu

6:31 PM

How short exactly are you looking for?!

Balarka Sen

Tale-Tell Heart is the shortest I could find, like 3 and it got long.

Danu

^^that

Balarka Sen

@Danu Microstory, 1-2 pages.

Mike Miller

Why don't you just translate Finnegans Wake?

Ted Shifrin

Stop whining and just translate.

War and Peace would be better, @MikeM.

Mike Miller

6:32 PM

I actually prefer the first. But I'm a well-known madman.

Danu

Did any of you ever read anything by a Dutch author? ^^

Ted Shifrin

I've actually not read either, @MikeM.

@Danu I'm not sure. I know I've seen some Dutch films.

Mike Miller

Don't bother with the first unless you're a snob. And I prefer Dostoevsky to Tolstoy.

Ted Shifrin

Googles "famous Dutch authors"

Balarka Sen

I haven't read anything of Joyce.

I should.

Danu

6:33 PM

@TedShifrin Turkish Delight?

Balarka Sen

I prefer Dostoyevsky to Tolstoy too.

Danu

W.F. Hermans has amazing books

Mike Miller

@BalarkaSen Be warned that it's hard work.

Frankly, I haven't had the patience for the sort of hard work involved in the books I like since I started grad school. I've read maybe two non-math books in their entirety. Sad, huh?

Danu

I've only read Metamorphosis

in the past years

Balarka Sen

I didn't find that too exciting, unfortunately, @Danu.

Mike Miller

6:37 PM

Kafka has better.

Or you could skip the Kafka and just try to work with university administration.

Danu

It wasn't exciting at all, but I thought it was a good book

Balarka Sen

Want to give some recommendations?

Danu

lol

Mike Miller

The Trial is the canonical choice.

Danu

@TedShifrin in case you're interested, famous Dutch books are mostly about the second World War

Ted Shifrin

6:38 PM

Yeah, I don't think I've read any.

Danu

Hermans has several nice books

Balarka Sen

Thanks, I'll try it out.

Danu

The most critically acclaimed one, though, is not about WW2

Pichi Wuana

Is it correct to say that $$\lim_{x\to -\infty} e^x = \lim_{x\to \infty} \frac{1}{e^x}$$?

Danu

So what are the left and right hand sides equal to?

Balarka Sen

6:45 PM

(Irrelevant: From a historical point of view I like Dostoyevsky (and Gogol) instead of the other writers of that age partially because his works seem to reflect his forseight about the next age in the history of Russia, of terror and blood, instead of looking back at the previous. /rant)

Mike Miller

@Danu I don't think that's the explanation you want to give - those two are equal for better erasons than "when you evaluate both sides you get the same thing".

Danu

@MikeMiller I think it is, because I assume the person understands perfectly well why they should be equal, and is just afraid because it's about limits

So I was trying to make it more concrete

(I do think about things, occasionally :D)

Mike Miller

I am not patient enough to argue pedagogy, but I am patient enough to wrestle you to the ground about your opinion. Meet me by the flagpole at 6PM.

Danu

Ill rek you bro

Mike Miller

7:56 PM

Hi @Pax.

How's Northwestern?

Ajax Edm

8:07 PM

anyone here able to proof this math.stackexchange.com/questions/1965764/…

Ted Shifrin

8:32 PM

@Pichi @Danu: So you're wanting to say that $e^{-x} = \dfrac1{e^x}$, which of course is valid. The missing step is $\lim_{x\to -\infty}e^x = \lim_{u\to\infty} e^{-u}$.

I agree with Mike that you want to understand the principles (i.e., algebra and symbolic substitution) and not evaluate the limits to see they're equal.

Pichi Wuana

@TedShifrin Why to use $u$?

Mike Miller

He's just changing variables. $u = -x$. You could just as easily rewrite the first limit as "$\lim_{-x \to \infty} e^{-(-x)}$". That's all he's doing.

PVAL-inactive

Let $b$ be a positive integer. Are there infinitely many fractions $p/q$ with $p$ prime and $q \ne 1$, so that all the terms in the contiued fraction expansion of $p/q>b$ for any $b$?

Mike Miller

Do you want $0 < p/q < 1$?

PVAL-inactive

Ya

All the numbers here are positive integers

Mike Miller

8:38 PM

Sure, but $p$ could be significantly larger than $q$ (which would make this construction easy)

PVAL-inactive

Whys that?

Oh sorry

p/q does not need to be less than 1

Mike Miller

Consider $p = (b+n)q + 1$, where $n \geq 0$. Dirichlet's theorem guarantees there are infinitely many primes of this form

It's probably still true if the slope is less than 1 but it would be harder to construct

PVAL-inactive

Ah i see

Mike Miller

I guess I know I'm a topologist when I refer to rational numbers $p/q$ as slopes

PVAL-inactive

so that has expansion b+n, q

JeSuis

8:43 PM

@TedShifrin Salut

Right

@MikeMiller Oh ok

and hi

Evening! :) Wh is the uploads button grayed out? Am I untrusted?

Mike Miller

You need like 30 rep or something.

Oh you have that. I dunno.

Moberg

8:50 PM

It turned blue! =]

OK lets see, I have 6 cards (labeled A-F), divided each in 4 rows and two columns.
When we align two cards next to eachother, for example AB, the top row will now have 2 bluw squares. This is not allowed. By sliding the A card down 1 row in relatin to B, it is possible to place them next to eachother in a legal position.
Does this make any sense? :)

If we take two D cards (DD) there will be a blue collision in row 1, and red collision in row 2 and 3. By sliding one of the D cards up or down, is it:
1) possible to have a blue collision without having a red collision? (answer is no)
2) possible to have a red collision without having a blue collision? (answer is yes)

Does anyone have a take on how tackle this? I want to find out in what cases it is possible to have a blue collision i without a red collision (or vice versa).

What kind of mathematics would this involve? If I were to post a question, I would have no clue how to properly tag it.
BR

Mike Miller

@Moberg Maybe combinatorics. I don't know.

@PVAL Does that suffice for you? Why is this interesting?

in general I think you can probably get very strong existence results for what you want. (eg restricting the numbers to lie in a given open interval and demanding that the continued fraction series be longer than a given length) I don't have a construction in that generality though.

PVAL-inactive

@MikeMiller I think I may have messed up the condition I need.

Moberg

9:11 PM

@MikeMiller yeah, that's my closes guess as well :P

PVAL-inactive

10:08 PM

@MikeMiller I guess I don't mean the standard continued fraction expansion. I want p/q=a_0-1/a_1-1/a_2 etc.

Mike Miller

@PVAL You can probably do that too. Does the same approach not immediately work?

PVAL-inactive

This is continued fraction you get when you look for an integral surgery description of L(p,q)

Mike Miller

Right

PVAL-inactive

I think (b+n)q-1=p works

or

Mike Miller

Probably should

PVAL-inactive

10:09 PM

i guess thats still true via dirchelt

Mike Miller

My strong existence claim above should still hold for this kind of CF I think.

PVAL-inactive

Was I supposed to learn Direchlet at some point?

Mike Miller

Only if you did number theory as an undergrad

PVAL-inactive

Is it in undergraduate number theory courses?

Ah

Mike Miller

I think it's usually just a little bit beyond that. I never took one, I just read a lot of number theory.

Where do I go to learn about lim^1?

PVAL-inactive

10:11 PM

what

Mike Miller

its a functor that apparently shows up when you try to take homology of limits of chain complexes.

PVAL-inactive

Is it like a graded inverse or direct limit or something?

That would be my guess.

Is it in homological algebra books

Mike Miller

I guess I should just look at Weibel. I don't know much homological algebra.

It's the "first right derived functor of lim" which is one step better than meaningless for me.

It's in Weibel. I guess I'll just read it.

PVAL-inactive

So I guess if everything works out on this elementary number theory exercise, I can prove that if p/q satisfy the conditions I stated for b approximately 10ish then L(p,q) admits a virtually overtwisted tight contact structure with a unique Stein filling up to homeomorphism.

Mike Miller

surely virtually overtwisted here is equivalent to saying the universal cover is overtwisted on S^3 which is easy to check isn't it?

PVAL-inactive

10:20 PM

in this case it just means it isn't the one induced from the tight contact structure of S^3 quotiented about by the group action.

So its just another contact structure besides the most natural one.

Mike Miller

k

i forgot the tight structure on S^3 was unique.

PVAL-inactive

I can probably prove its true for all contact structures on some restriction of L(p,q).

Mike Miller

how do those uniqueness proofs normally go? some invocation of freedman?

PVAL-inactive

My proofs are very different than the normal proofs (usually they work up to diffeo.)

Mike Miller

i know there was a recent theorem one of the students here is trying to extend that was only up to homeo for the cotangent bundle of surfaces

looks freedmanesque.

PVAL-inactive

10:27 PM

I construct an exact symplectic cobordism to something which is known to have a unique filling (which is negative definite). This implies H_2 and hence signature are unique and H_3=0. Then I use the fact that pi_1 is surjective and the argument Ruberman gave me as an answer to my recentist mo question to conclude pi_1=0 and then invoke Freedman.

@MikeMiller I think some of those cases you can't even prove homeomorphic as surface groups aren't known to be "good".

That's probably one of the better known open topological 4-manifold questions.

Mike Miller

Yea I just looked at the paper. They get a relative s-cobordism and can only conclude for the Klein bottle.

IIRC the student I'm thinking of wants to upgrade it to diffeomorphism.

I have a mild suspicion Danny knows everything.

PVAL-inactive

I think he knows everything that is known at least.

Mike Miller

That's definitely true. The question is whether he knows everything unknown too and is just not publishing it because he's nice to us kids.

PVAL-inactive

I think the main thing is that he knows algebraic topology so much better than most geometric topologists.

He's very good at those kind of technical algebraic topology arguments that people need. My understanding is that his and LS's recent collaboration arose out of him being able to do exactly that for her.

Mike Miller

He knows a lot. He told me about an approach to a question about the Dirac operator on hyperbolic manifolds via geometric group theory. But I've forgotten it now.

Ya I remember that paper. That sounds about right.

PVAL-inactive

10:36 PM

I should probably work out all the cases where I can prove L(p,q) has a unique Stein filling (for any contact structure).

Mike Miller

His brand new paper is what I'm reading now and need to look up homological algebra for.

PVAL-inactive

Its inverse limit isnt it?

Mike Miller

Yea I get what's going on now.

PVAL-inactive

Maybe I should write two papers instead of one.

Mike Miller

I had a new idea and my advisor told me it should be a second paper, since two papers is better than one, and long papers have a harder time getting published.

PVAL-inactive

10:39 PM

I have a very difficult time getting my adviser to give me any real advice on anything other then proving theorems.

Mike Miller

I've heard he does that... Do you talk to anybody else like TP?

PVAL-inactive

I talk to CG about general advice, but he doesn't really understand my work or the significance so things like this are harder.

TP probably doesn't either.

Mike Miller

Ah yeah that'd be a problem. I thought TP would be closer.

PVAL-inactive

TP actually turned me down as a student (I've probably said this before), and I've been under the impression that he doesn't have time for me.

Mike Miller

I don't think I'd heard that. That sucks.

PVAL-inactive

10:45 PM

Nearly all of my adviser's students have been really successful, and it seems that I have really good results at this point too, so its sort of difficult to say hes not doing his job well.

Mike Miller

Meh i get your frustration.

I don't know how common turning down students is in general. I don't think my advisor ever has.

He normally just asks a lot of students and the ones who don't keep up don't try to work with him after a point.

PVAL-inactive

I am glad it happened at this point.

My adviser is a lot smarter than he is.

Transcript for

$Mathematics$ Mathematics