Mathematics - 2016-03-08 (page 1 of 3)

Semiclassical

00:14

heya

Jessy Cat

00:35

2

Q: Without Cauchy Integral Formula or Residues: $\int_{L}\frac{dz}{z^{2}+1}$

I am currently working on the following: Prove that $\displaystyle \int_{L}\frac{dz}{z^{2}+1} = 0$ if $L$ is any closed rectifiable simple curve on the outside of the unit disc, i.e., $L$ is contained in the region $|z|>1$. Show that the equality is in general false for arbitrary closed rectifia...

The only answer anyone's provided is HORRIBLE

And has been deleted...

robjohn

@JessyCat why do you want to compute a complex integral without any of the tools for computing complex integrals?

Jessy Cat

00:52

@robjohn, because my professor asked me to ;P

@robjohn, actually, we can use Cauchy's Theorem or Cauchy's Theorem for Multiple Contours, but that's it.

And I now understand the second part of the problem, but not the first part.

@robjohn, if you can offer any help, that would be nice

Jessy Cat

01:14

Or anybody??!!

Please??

user19405892

01:30

hi

Jessy Cat

Hi

user19405892

how do i minimize $\frac{1}{\sin(A)}+\frac{1}{sin(120^{\circ}-A}$?

how do i minimize $\frac{1}{\sin(A)}+\frac{1}{sin(120^{\circ}-A})$?

Jessy Cat

Take the derivative. Find the critical points. Plug them into the original function, see which point gives you the minimum value.

robjohn

@JessyCat If you can use Cauchy's Theorem, then integrating around two circles and using Cauchy's Theorem makes it doable.

FTem

My professor today said if you take a double torus and consider X/A where A is the loop going around one of the outer loops (usually drawn vertically) then the space is isomorphic to S^1 V T^2... Where does S^1 come from?

Jessy Cat

01:45

That's what my professor said. But I don't understand how you're allowed to do that when the region we're given where the function we're integrating is analytic is |z|>1 and those two circles (not even their boundaries!) are not inside that region.

They're not even around the outside of it

Actually, it's analytic except for those two points, but it's not the region we were given

Mike Miller

@FTem: There's a picture of this in Hatcher somewhere. So what you get looks like a sphere with two of the points glued together. (Do you see how to get this?) Now this is homotopy equivalent to a sphere with a line drawn outside the sphere from the north pole to the south pole.

FTem

For example, if you let A be the loop that "divides" the double torus into two tori then it makes sense that X/A = T^2 V T^2

Mike Miller

Which, drawing a path on the sphere from the north pole to the south pole, is homotopy equivalent to collapsing that path. Which gives a sphere wedge a circle.

That's what you get if you take T^2 and kill a circle. Now connect sum a T^2 away from all the action.

FTem

I guess I'm not seeing where the sphere is coming from for the double torus

If I understand correctly taking out A above looks like a torus with two horns glued together at a point

Mike Miller

Ignore the double torus. Just work this out on a torus first.

Torus mod some circle.

math.cornell.edu/~hatcher/AT/ATch0.pdf

p11 ex 0.8

FTem

01:52

Right, this makes sense now

Okay, thanks!

Mike Miller

@Ted: I think I'm going to do the Lie group / Lie algebra correspondence tomorrow.

Ted Shifrin

Hmm, ok ... Probably can't prove the hard theorems.

Mike Miller

Can describe the actual correspondence and see how to go from Lie algebras to Lie groups (integrate!)

Subalgebras, I mean.

PVAL

We had a geometric group theory/hyperbolic stuff speaker who kept drawing examples on his "favorite genus-2 surface" obviously making the point that in his mind there were a lot of such things.

Ted Shifrin

@JessyCat: Be careful. Your function is analytic on $\Bbb C - \{\pm1\}$, not just on $\{|z|>1\}$. Perhaps that's causing your confusion?

heya @PVAL :)

Mike Miller

01:55

@PVAL: I don't really understand the genus 2 surfaces.

Jessy Cat

Maybe. But L is supposed to be on |z|>1

PVAL

@MikeMiller Which one?

Ted Shifrin

Sure, that's fine, @JessyCat.

Mike Miller

Beat you to it.

Jessy Cat

Argh. It's so confusing.

I'll have to let it percolate some more.

Ted Shifrin

01:56

But you can make a region by taking the stuff inside your curve $L$ and outside circles of radius $\epsilon$ centered at each of $\pm 1$.

@MikeM: I'd like to see you do those integrals. :P

PVAL

Theres an arxiv paper on sg claiming to prove NLC for T^2

Ted Shifrin

Of course, we're back to Frobenius again.

Mike Miller

@PVAL: NLC?

PVAL

It still is open for a genus 2 surface

Jessy Cat

I'm actually working on something else right now. Going bakc to that one later

PVAL

01:57

nearby Lagrangian conjecture

Ted Shifrin

OK, @JessyCat. I just wanted to try to make that comment before I disappear for the evening.

Mike Miller

How new is the paper? I thought this was true for some reason.

Jessy Cat

Sounds good @TedShifrin

PVAL

So apparently when you asked me if it was still open for T^2 the answer was yes, and is now probably no

Last week I dont know the day. They reprove it for $S^2$ it seems

as well

Jessy Cat

Different question: If you are evaluating the integral of a funciton with singularities at z=0, z=i, z=-i over the curve |z-i|=R, and you can avoid values of R for which the denominator turns to zero, does that mean you avoid values of R that would enclose those points? Or values of R that would make those points be on the boundary of the curve?

Mike Miller

01:58

I don't know how I missed this paper... Maybe it was on the day the arXiv didn't email me

PVAL

Well it wasn't crossposted to GT

I sometimes forget to look at sg

Mike Miller

I'm subscribed to sg

user19405892

hi

Ted Shifrin

On the actual curve, @JessyCat. They can be inside. Just like what I just did with your other problem.

user19405892

hey guys i need help with this question

Ted Shifrin

02:00

I'm heading out. Have a good night, all.

user19405892

Mike Miller

I'm thinking of subscribing to ap but I figure that will end up like dg, a massive number of boring papers and an interesting one once a month that someone else tells me about anyway.

Jessy Cat

Thanks @TedShifrin

PVAL

Is ap analysis of PDEs?

Mike Miller

Yea.

A while back I was trying to answer an MSE question and it came down to "A non-injective map $\pi_1(\Sigma_g) \to \pi_1(\Sigma_h)$ has a simple closed curve in its kernel". This was apparently a longstanding open problem that was solved by Gabai in '85.

I'm glad I didn't spend long trying to prove it.

PVAL

02:01

That sounds obviously hard to me.

user19405892

can anyone help with my question

Mike Miller

@PVAL: It sounds obviously true to me.

PVAL

Is it doable for geometric maps?

Mike Miller

Does geometric mean induced by a continuous map? Because no, because they're $K(G,1)$s every homomorphism is induced by a continuous map.

It's doable when $g=1$ iirc.

PVAL

I believe I could do it for $g=1$/

Mike Miller

02:04

I actually believed it for any codomain. But that's false.

It's true for any codomain when $g=1$ iirc.

PVAL

The Hw I have to grade is literally 20 row reductions zzzzzzzzzzzzzzzzz

Mike Miller

Give them all As or all Fs.

Your choie which.

Alternatively use the stairs method.

PVAL

I found the solutions using mathematica and if a student asked me I'd tell them to do the same

If I had to do this homework I might drop/withdraw from the class jeez

Mike Miller

I found the paper on Lagrangian isotopy. Did you read it?

PVAL

No I just glanced at it.

Mike Miller

02:11

Do you know how the proof goes or know someone who does?

I don't want to read it but I want to know how they do it.

PVAL

At first I thought it was trash because the S^2 result is well known, but then I realized they are doing other things too which clearly aren't.

Mike Miller

I don't know how to prove the S^2 result either.

PVAL

Idk something uniruled, 2-dimensional makes some moduli space easier for some reason beyond me.

easier n ot either

Semiclassical

let me find the tedious problem my students had last week

Mike Miller

I'm just going to email Ko and ask if he knows how they do it.

Semiclassical

02:13

actually, it's simple enough to say. just a real bear to solve by hand.

Rudy the Reindeer

Interesting. I leave a comment on one of BD's answers and hours later I get a downvote on one of my top answers (40+ upvotes)... Coincidence?

PVAL

Think its probably close to incomprehensible less you already know a lot of J-hol curve theory.

I recently downvoted a 30 upvoted question. I don't think it was yours.

Rudy the Reindeer

Nah. And it's probably coincidence.

shrugs

Semiclassical

this is the problem + a solution. link

PVAL

It was something like heres an easy to understand basic definition. I think this is cryptic can you explain in baby words for me :)

Semiclassical

02:16

main thing to notice is the four-by-four linear system that shows up on the second page below figure 1

Mike Miller

Yea it's completely incomprehensible.

I wonder if given an hour they could explain it to me.

Semiclassical

the idea being, solve that by hand and verify that the answers you get for $|E|^2/|A|^2$ and $|B|^2/|A|^2$ are correct

PVAL

Goodman's Eliashberg's student I think.

Semiclassical

which is pretty miserable.

Mike Miller

I might ask Ko to invite them if he thinks they could explain it in an hour.

PVAL

02:18

Someone in a similar school (of mathematics) recently gave a talk here and they literally said nothing that wasn't completely elementary.

Mike Miller

That's a shame.

PVAL

and they stated their big interesting thms without a hint of how they were proved.

well I guess they did that at least

Mike Miller

Our hour seminars here are really 45min because Ko asks questions for a quarter of the talk.

So that's also bad for getting through a whole paper.

PVAL

I feel like a lot of the talks here, I couldn't imagine someone like Ko not having seen everything 100 times before.

At least the talks I pay attention in..

Mike Miller

I'm pretty sure he knows everything he asks about.

PVAL

02:26

I think most of the people here who drag on with questions do it precisely when they are completely ignornant of every word the speaker has said.

They usually quiet down (sometimes interjecting with comments) when someones talking about stuff they know well.

Mike Miller

lol

Idk most of the talks we have here are interesting.

Next quarter the seminar is doing cluster algebras though which I'm not yet convinced I care.

PVAL

Does Edwards ever attend talks.

?

I'd imagine hes that way too

I watched a lecture series he attended and he talked half the time.

Mike Miller

He does come but he's quiet and just listens.

Well he does come sometimes.

This quarter's participating seminar was about isntanton homology and the KM approach to 4-colors. He didn't come to that.

PVAL

I think cluster algebras are my main motivating reason to become a topologist.

(since I don't have to learn about them)

Mike Miller

lol

Maybe I should ask Ciprian to give me 8 units of research credits next quarter so I don't have to give a talk.

PVAL

02:32

Is there not graduate seminars where people just talk about whatever they want for an hour.

??

That's the norm here (and we dont get any credit for it unfortunately).

Mike Miller

There's a general grad student seminar for all grad students that I run but no topology one.

The participating seminar has talks given by students but the theme is picked by faculty.

If we ran a topology-specific thing like this my money is three people would go.

PVAL

The graduate seminars here are a good chance to practice your thesis/candidacy exam as well. People usually give a practice talk on these.

Well you could have a topology-geometry one.

I think it would be easy to make instanton stuff relevant to Ko's students.

and contact/symplectic stuff relevant to Ciprian's

Mike Miller

They're both in the topology group.

It's not that it wouldn't be relevant it's that k know the precise 3 people who would go every week.

But still I'll float the idea by some people.

PVAL

You could try and pull postdocs. We don't usually do that here, but one postdoc comes pretty consistently.

Mike Miller

It'd be worth doing if only so I can talk to people about the garbage I like.

PVAL

02:39

Ya its useful for that, and if you want to give like a research talk somewhere you have like a built in practice one in front of people you know.

It's also a chance to get undecided 1st or even 2nd years interested in your field.

Mike Miller

I don't want that. Ciprian already has thirty students.

L33ter

can somone tell me why point at infinity act as the group identity in an elliptic curve?

geometrical reason

I don't need complicated argument

PVAL

@MikeMiller Aren't you exaggerating a bit.

Mike Miller

6+2 students doing reading with him. One is going to graduate.

Jessy Cat

0

Q: Evaluate $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)}dz$ as a function of $R>0$

I need to evaluate the integral $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)} dz$ as a function of $R>0$. I may omit values of $R$ for which the denominator turns to $0$. Now, using partial fraction decomposition, the integral can be split up as follows: $\displaystyle \int_{|z-i...

complex-analysis complex-numbers complex-integration

Different question

PVAL

02:50

@MikeMiller Well I guess that's more than 1

Mike Miller

How many does Bob have?

PVAL

1

He had 3 two years ago

(not including me at that point)

Gordon has two

Luecke has one

Reid only has one (though hes got I think two or 3 reading with him)

Mike Miller

sounds like you have a big topology group which helps

Here it's Ko, Ciprian and Mike Hill. Mike doesn't have any UCLA students yet and Ko has 3 UCLA students and 2 USC students

There's also that most people who want to do geometry work with Ciprian or Ko too.

PVAL

I think all the algebraic topology people here have decided to migrate towards the geometry seminar.

Think were trying to hire more as Gordon and Bob probably aren't going to work for another 10 more years.

Mike Miller

What kinds of ppl are you trying to hire?

Oh I just remembered Sucharit Sarkar is coming next year which might help lower Ciprian's student count.

PVAL

03:00

two sympleticy people were given job talks with at least one given an offer.

Mike Miller

I think we're done hiring topologists for a little while since we have 4 now which is a nice number. I think PDE probably has next claim.

PVAL

Well were trying to get a package deal for a topologist and a number theorist I think.

The number theory department here completely fell apart

Mike Miller

Actually any number of people could have next claim. Combinatorics has 1 person, PDE has 3 but really 2 since 2 of those are an Oszvath-Szabo type package deal, number tbeory has 2.

I guess NT has four but one is effectively retired and the the other analytic and I wlways forget about the analytic number theorists.

I'm not counting Tao in these lists or he'd be in every group.

PVAL

I don't think combinatorics has any here and no one cares about creating that kind of a department.

How many students does Tao have?

Mike Miller

Idk mayve 3 and they all do different things.

PVAL

03:10

Wow I'm surprised hes not in higher demand.

There must be like 10 working under Cafferelli in some fashion.

Mike Miller

I think it's mostly that he's selective.

My impression is that most of his students have specific problems they want to work on with him. You also have to have passed the analysis qual and done really well on it before he'll talk to you.

I also have the vague impression that if you say Terry Tao without having a good reason for it on your statement of purpose you get autotrashed.

PVAL

I don't know of anyone here who judges people on prelim scores (besides pass/fail). I have no idea how I did on the ones I passed.

Mike Miller

Terry is the only person I've heard rejects students. Some people say "Sorry, I have too many" but that's fair. And some people have high bars before you can talk to them.

(To do a reading course with Totaro you apparently should have done most of the problems in Hartshorne.)

PVAL

Hey I've done like 40% of them I think

and probably forgotten the solutions to 95% of those.

Mike Miller

You're closer to a PhD with Totaro than I am.

PVAL

03:20

Bob just explained things in the language of GT until I started to understand what hes saying.

Mike Miller

Explained what things? GT = geometric topology? Idgi.

PVAL

Well he talked about things using defintions and theorems I didn't know.

for like the first 6 months of working with him.

Mike Miller

Oh, sure.

My reading course with Ciprian was a list of books on 3-manifolds, 4-manifolds, symplectic geometry, Morse theory, and knot theory. He told me to learn them and that he would see me in a few months.

I think this is what he does with all his new students.

PVAL

I am interested in this list

I'm guessing Monopoles and 3-manifolds was one.

Mike Miller

Lol. No.

Actually I thought about reading that and he told me not to because it's too long. I now agree with him.

PVAL

03:27

Okay was it like Roflsen/Gompf-Stipsicz

Mike Miller

Yea, it's all intro books.

The point being that his students should know all that language and elementary proof techniques.

PVAL

So baby Mcduff-Sala

?

I'm guessing the Morse theory book was Milnor's

Mike Miller

No, he knew I knew that. I mean Morse homology since that's all Floer homology is.

Semiclassical

reminds me, i tried to look up taubes' book on connections etc.

and it's checked out from our math library :/

Mike Miller

Yea basically first half of MS. Audin's book on Morse homology. 3folds: Saveliev, Hatcher, Hempel, Scott. Lickorish & Rolfsen. Gompf-Stipsicz.

PVAL

03:30

Hatcher wrote a 3-folds book?

Mike Miller

He has some nice notes online.

PVAL

Those are pretty short aren't they

Mike Miller

Yea. It's what I started with since they help elucidate the ideas.

After that he had me wander through stuff until I found something i liked. I learned a bit about HF and casson's invariant and monopole etc.

PVAL

I think I need to read more books on 3-folds

Mike Miller

Someone needs to write a post-geometrization, post-Floer book on it.

I'm not sure what would be in it but I want it.

PVAL

03:32

Im not sure the same people writing about geometrization should be writing about Floer stuff.

user116211

Can anyone tell me how $$\mathrm d \left(\frac{x}{y}\right)= \frac{y\,\mathrm dx- x\,\mathrm dy}{y^2}\;?$$

Mike Miller

@PVAL: Get two authors then.

Nathan Dunfield would probably be eminently qualified to write about both.

Semiclassical

@user36790 well, if you replace $\text{d}$ with $\text{d}/\text{d}t$, that's just the quotient rule

Jessy Cat

I've only got 11 views so far.

And I posted it 46 minutes ago

:(

Mike Miller

Idk I think there's probably interaction between the fields because there always is, we just don't know what it is yet.

Semiclassical

03:35

if you don't like that, it's also just the product rule.

user116211

@JessyCat Oh it only happens in Physics SE ;D

PVAL

I kinda found a lot of the stuff in McDuff Sala to never be any use for me.

At least the little book.

Mike Miller

I dont do symplectic stuff but it helped me form little pictures.

I learned symplectic quotients in practice, not from that book, and most things I actually use probably come from reading Seidel-Smith or Fukaya intros.

user116211

@Semiclassical yes! Thanks. I think I can find all the differentials by thinking of $\text{d}/\text{d}t$ and not just $\text{d}$

Mike Miller

I still don't know what a Lagrangian submanifold really is or what they look like.

PVAL

03:40

I think my main weakness right now is not being able to use the Giroux correspondence to prove anything. That's the next thing I really need to carefully learn about (think thats more important than any useful understanding of HF or c at this point).

Mike Miller

Have you read "Holomorphic discs and genus bounds"? That uses that result.

Semiclassical

@mikemiller what i find myself wanting to understand after our previous conversation is the ins and outs of vector flows, formally at least

if only so that I can translate that one problem we talked about into terms I fully recognize

Mike Miller

I guess it mainly uses it to construct the Giroux 2-handle but whatever.

Semiclassical

glancing about for the definition of a lagrangian submanifold, i see a MO answer with the following paragraph

"Occurences of lagrangian submanifolds are indeed manifold: they arise as semiclassical support for certain FIO's and can also be thought of as semiclassical version of states in quantum mechanics via the WKB expansion. This point of view is exemplified a lot in the nice booklet of Bates and Weinstein."

which sounds like something i really should recognize, even if just at the level of intuition

PVAL

@MikeMiller Isn

t a Giroux 2-handle just Legendrian surgery?

Mike Miller

03:45

@Semiclassical: My problem with them is they're very nonrigid. Just like most things in Symplectic geometry they're globally interesting, not locally. I can't get a good feel for them.

Semiclassical

of course, for that matter I also should probably have some sense of it just from what I know of Lagrangian and Hamiltonian mechanics

Jessy Cat

1

Q: Evaluate $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)}dz$ as a function of $R>0$

I need to evaluate the integral $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)} dz$ as a function of $R>0$. I may omit values of $R$ for which the denominator turns to $0$. Now, using partial fraction decomposition, the integral can be split up as follows: $\displaystyle \int_{|z-i...

complex-analysis complex-numbers complex-integration

Mike Miller

@PVAL: Probably. But this is how they define the contact invariant. The Giroux 2-handle, suitably stabilized.

PVAL

@MikeMiller Ya I think I'm ok with the non-HF parts of that paper. I mean the things that explicity work with open books to get bounds on tight contact structures.

in other words stuff that Ko does.

Semiclassical

looking at one of the answers, i kinda-sorta see it, at least in the simplest case.

user196973

03:47

Hi, if anyone could help with this problem it would be greahttp://math.stackexchange.com/questions/1687755/analyzing-spirals

user196973

*greatly appreciated

user196973

4

Q: Analyzing Spirals

I am in the process of studying spirals, where each successive length drawn is the same but every angle is increased at a certain increment. *Please note the angles selected are the exterior angles or the angle by how much the turtle turns by. Whenever the increment is one, no matter what the a...

calculus linear-algebra geometry trigonometry algebraic-geometry

Mike Miller

I like that paper a lot. It's where I learned most of the HF I know.

PVAL

I'll bookmark and see where I'm at when I finish my candidacy.

user19405892

hi

what does it mean to have a triple point of intersection

PVAL

03:50

Im trying to read the Stipsicz Lisca series of papers

Michael

Hey guys, quick question. Is it possible to find the integral of $1/x$ from 0 to 1?

PVAL

The area under the curve is infinite.

Semiclassical

it's possible to find that it doesn't exist, sure.

user19405892

what does it mean if a line has a double point of intersection or a triple point of intersection?

PVAL

Well it means it isn't a line

user19405892

03:52

like if a line has a triple point of intersection with the cubic y = x^3

Michael

oh How can I prove that the area beneath it is infinite?

rigorously

Semiclassical

simplest way i know (though not the most elementary) is to integrate $1/x$ from $\epsilon$ to $1$ where $\epsilon$ is some small positive value.

Michael

and say the value approaches infinity?

Semiclassical

show that, yeah. if you know how to integrate $1/x$ away from 0, that's fine. if you're looking for something that doesn't require you to know how to do, there's probably some obvious way to do that

by showing that the riemann sum doesn't converge, perhaps?

Soham Chowdhury

@user19405892 vaguely, it means something like this: if you perturb the line a little, there will actually be three piiunts of intersection. so $x=y$ intersects $y=x^3$ with multiplicity 3, or at a triple pint.

Michael

03:55

Yah I know how to do that, for some reason haha

Soham Chowdhury

(I think, someone who knows this stuff should be able to say better than I can)

Semiclassical

to make Soham's statement concrete, suppose you replace $y=x$ with $y=m x$ where $m>1$

and just be drawing that you can see that there'll now be three intersections (at zero, at a positive value, and a negative value)

Mike Miller

@PVAL: There's a lot of things I've bookmarked but will never read.

Semiclassical

same holds if you slightly change the y-intercept.

Soham Chowdhury

For +5 rigor, read this

user19405892

03:58

@Soha so what you mean is just multiplicity 3?

Soham Chowdhury

Intersection number

In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of x- and y-axes which should be one. The complexity enters when calculating intersections at points of tangency and intersections along positive dimensional sets. For example...

@user19405892 isn't that what you were asking for the meaning of?

user19405892

yes, and i was doing a problem where they were saying like the tangent line to the cubic y = x^3 has a double point of intersection, while the line y = 0 has a triple point of intersection

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