Mathematics - 2016-10-10 (page 7 of 7)

Ted Shifrin

22:00

Better to think about a saddle surface, @Danu, to see the two linear rulings, but OK.

Go to sleep, @Balarka.

Lozansky

What course would include Riemann–Stieltjes integration?

Ted Shifrin

(Undergraduate) real analysis, @Lozansky, although many courses I bet will skip it. You can find it in Rudin.

There's only time for so much.

Lozansky

Yeah we skipped it

Although I'm in engineering

Mike Miller

@Balarka, @Ted: I got a copy of the exercises associated to Taubes' book.

PVAL-inactive

I think my measure theory professor assumed we knew it without teaching it. So I learned it in measure theory.

Ted Shifrin

22:02

Oh, cool. If I knew the book at all, I'd ask for 'em, @MikeM. But ... meh

@PVAL: It all gets subsumed by Lebesgue for the appropriate measure, anyhow.

Danu

@TedShifrin Okay. So anything one-dimensional thing should at least intersect one of the generators of the $\Bbb P^1$'s

Mike Miller

@Ted It's a very good book.

Ted Shifrin

Yup. What does primitive mean?

I don't doubt it, Mike. Taubes is a classy guy.

Danu

That's what I'm trying to understand

in terms of geometry

I have no idea (yet)

PVAL-inactive

@TedShifrin I think that I have been asked to prove that for certain cases they agree. That was in another lifetime though.

Ted Shifrin

22:03

Hint, Danu. Go back to the formulation in terms of $L$, rather than $\Lambda$.

Danu

@TedShifrin Yeah, I'm just trying to use that $\Lambda$ is $L$'s dual

Ted Shifrin

Imagine how many lifetimes ago it was for me, @PVAL. And I took graduate real analysis junior year of college.

Doesn't Huybrechts give a concrete criterion, @Danu? He should. I mentioned it several days ago, I think.

Danu

Ain't no geometry in Huybrechts

Ted Shifrin

In terms of $L$, dammit.

Danu

Yeah, you said intersect with hyperplane

I got that much

Ted Shifrin

22:05

I'm gonna go look.

Danu

Lemme search the chat

Ted Shifrin

I'm gonna look in Huybrechts.

Danu

Ah, you did say something about geometric interpretation of primitive cohomology

Ted Shifrin

I don't find it explicitly in Huybrechts. Look on the back of p. 14 and p. 15 of my notes.

The point is that you can express primitive cohomology in terms of kernel of the appropriate power of $L$.

Danu

Because of $*L^k$ being some power of $L$ (up to a factor)?

Ted Shifrin

22:10

That's what tells you about intersecting null with appropriate-dimensional linear spaces.

I think it actually uses the representation theory.

I think Huybrechts may have it buried in there, but it's not clearly displayed.

Danu

In the earlier or later chapters?

Ted Shifrin

$\phi\in H^p$ is primitive $\iff p\le n$ and $L^{n-p+1}\phi = 0$.

Danu

Because I'm pretty familiar with the earlier ones by now.

Ted Shifrin

It may actually be back in 1 where he does all the representation theory set-up. I dunno.

Danu

Let's see, I know where to look.

Ted Shifrin

22:12

But it should fall out of Lefschetz decomposition, anyhow.

Danu

Yeah, prop. 1.2.30, item (v)

Ted Shifrin

LOL, yeah.

That's where you get your geometry!

Danu

or so they say...

Ted Shifrin

That's what I am saying.

Don't you dare contradict me.

Danu

You're plural by now

Ted Shifrin

22:15

Balarka already showed me I was a dumb idiot.

Danu

like the god that you are

Ted Shifrin

Oh shaddup.

Danu

I do find it regrettable that Huybrechts has like 0 geometry

Ted Shifrin

Well, he wasn't going to rewrite Griffiths/Harris.

Anyhow, now you can do my exercise. I leave you quiet for a while.

Danu

@TedShifrin So based on the above we have that something is primitive exactly (since $n=p=2$) when it doesn't intersect with the hyperplanes that yield the Kahler form

Ted Shifrin

22:27

Well, it can intersect set-theoretically, but the net intersection number of the divisor with the Kähler divisor (hyperplane class) is 0, yes.

Danu

Ah, I have to count orientation-dependent intersection numbers

Ted Shifrin

Everything is + in the complex world. But remember that divisors have integer coefficients.

Danu

What do you mean by everything is $+$?

Ted Shifrin

Any two complex submanifolds that meet transversely intersect in a + orientation.

Danu

Oh, right

because permuting two at a time is an even permutation

Ted Shifrin

22:29

Right, and canonical complex structure blah blah ...

Danu

I can live with that

Ted Shifrin

BTW, if you take $\omega$ as we said, what is $\int \omega\wedge\omega$?

Danu

I thought that $w^n\propto *1$

Ted Shifrin

Sure. So what? :) What's the volume, then?

Danu

Ah okay

Ted Shifrin

22:31

Note how geometrically you can do this with $(L_1+L_2)\cdot (L_1+L_2)$ ... which is ... ?

mick

Nice

Danu

@TedShifrin The $L_i^2$ terms should be zero, so $L_1\cdot L_2+L_2\cdot L_1$?

Ted Shifrin

Which is ... ?

Danu

In terms of intersections I just want to say $L_1\cdot L_2$ is one

But I'm generally pretty overwhelmed with this

Ted Shifrin

Indeed. So the answer is $1+1=2$.

Danu

22:35

I don't really understand how to do this intersection stuff

in any non-trivial-to-visualize case

Ted Shifrin

Note that this is the degree of the quadric surface you get embedding $\Bbb P^1\times \Bbb P^1$ in $\Bbb P^3$. (Think saddle surface.)

Draw it as the axes ... $x$-axis is one $\Bbb P^1$, $y$-axis is the other $\Bbb P^1$. Now it's obvious.

Or think of it on the saddle $z=xy$ with the obvious rulings.

Note that if you intersect a hyperplane in $\Bbb P^3$ that happens to be tangent to the surface, you get exactly the two axes, hence $L_1+L_2$.

I know you have to learn to visualize this a bit, but I'm trying to provide geometric intuition that you're not getting from reading.

Danu

Yeah, I really appreciate it

but I'm mostly terrified when you ask me this stuff :P

Ted Shifrin

You can put me on ignore, too. :D

Danu

Lol

Just switch off my mathematical light bulb

Ted Shifrin

OK, so what cohomology class or divisor should I write down that's killed by intersecting with $L_1+L_2$?

Danu

22:39

$-L_1+L_2$ or minus that

Ted Shifrin

Yup. That's primitive.

Danu

lol

You know how I feel right now

Ted Shifrin

And note that the primitive stuff plus the multiples of the Kähler class gives you all of $H^2$.

Danu

Ted Shifrin

OK, I'll leave you now. :P

Danu

22:40

So the #1 thing I'll try to bother Rui (my second private (behind you) though he doesn't know it yet) with is this geometric stuff

I was already happy that I actually managed to do an exercise today

Ted Shifrin

LOL ... OK. You can promote him ahead of me.

Danu

No, no Ted

Ted Shifrin

Well, you develop intuition by doing more exercises. Not that Mike and I haven't told you this.

Danu

Your soul has already been signed over to me by L.U. Cifer himself

@TedShifrin I'm going to push for a complex geometry topic and when I get it I'll go through all my typed notes (I'm at 70 pages now already) but while doing exercises

Balarka Sen

@MikeMiller Can you e-mail me a version?

I'd appreciate it.

Ted Shifrin

22:43

It'll be good to have a specific question to think about and then refer to the stuff you've sorta learned to make it make more sense.

G'night, @Balarka.

Danu

@TedShifrin Sorta :'(

On to Hermitian vector bundles and your supervisor's great invention

Ted Shifrin

Yup.

Balarka Sen

@TedShifrin You sound like an alarm clock, but one which tells to go to sleep, not wake up.

Ted Shifrin

I have my uses, @Balarka.

Danu

The best kind of alarm clock

Also I probably won't be able to work much on complex geometry over the next week

I'm going back to my hometown for 5 days starting tomorrow

PVAL-inactive

22:44

Writing is too hard.

Danu

I was planning to prepare my seminar talk

Ted Shifrin

It is not easy, @PVAL, I agree.

Danu

@PVAL-inactive Understanding is what's real hard

Ted Shifrin

Well, don't forget everything, @Danu.

PVAL-inactive

@Danu I understand everything fine.

Danu

22:45

@TedShifrin It'll be a nice warmup---the seminar is about characteristic classes :P

PVAL-inactive

Expressing that understanding in an acceptable way however...

Danu

@PVAL-inactive Then you've got the world at your feet

Balarka Sen

@Danu Aren't you going to learn char classes.

Danu

@BalarkaSen Yeah, I'll start now.

Ted Shifrin

@PVAL: Don't ever try to get it all right the first time. Just write. Then go back and rethink/revise later.

Danu

22:47

My first presentation will be on the very early stuff in MS

Half recap of (co)homology stuff

Half Stiefel-Whitney classes definition ish stuff

section 4

Balarka Sen

ok. I'm here if you want me.

PVAL-inactive

@TedShifrin That's what I am doing. Even doing that its going slow. I mean it took me like two hours to write like a page today (which is by no means final or finished).

Ted Shifrin

Do you make an outline first?

So you have a sketch in your head of what you want to say in what order, approximately?

Danu

The Beatles - I Want You (She's So Heavy)

►

Ted Shifrin

I think @Danu's flipped what's left of his lid.

Balarka Sen

22:50

Not a fan of Beatles.

Danu

It's 12:50 and I'm eating avocados and drinking whisky

Sounds about right @Ted

PVAL-inactive

I am sort of just writing everything I need to say down, and then figuring out the order as I'm doing it.

Ted Shifrin

I love avocados, but not with whisky.

Danu

@TedShifrin And right you are

Balarka Sen

It's 4:30 and I'm starving.

Ted Shifrin

22:51

4:30 AM, when you're supposed to be sleeping.

Have a little snack and go to sleep.

Danu

Holy crap @Balarka

Ted Shifrin

BTW, @Danu, I'll be AWOL for at least two weeks, so have fun without me.

Danu

dafuq are you doing up

@TedShifrin "fun" :D

Balarka Sen

I'm doing math.

Danu

No fun without the Tedster

PVAL-inactive

22:52

@Ted I wrote a short note to a few experts explaining what I was doing and asking if it was new and interesting. It got a few positive responses, but still seems a far cry from something anyone would feel comfortable posting.

Danu

Even I don't get to the point of being up at 4:30 doing math. I stop at 3, latest!

Balarka Sen

@TedShifrin When are you leaving?

PVAL-inactive

So I've just been expanding on that note and rewriting the parts that need rewriting.

Ted Shifrin

Well, the fact that they're interested is exciting and encouraging, @PVAL. Just have to make it into something presentable.

Nov. 1, @Balarka.

Danu

@Ted Weren't you leaving around Nov 1st, though?

Oh lol

PVAL-inactive

22:53

@TedShifrin I've set the entirely arbitrary deadline for myself of having it posted by christmas.

Ted Shifrin

Have another glass of whisky. It's almost my martini time, @Danu.

PVAL-inactive

I still think I can hold myself to that.

Ted Shifrin

OK, @PVAL. Good for you :)

Danu

@TedShifrin Nice.

Balarka Sen

@TedShifrin Coming back right when Trump wins the game? Nice.

Ted Shifrin

22:54

No, a week after, @Balarka ... Or driving to the moon.

Danu

I gotta stay up a few more hours for laundry to finish

PVAL-inactive

According to 538 Trumps chances are less than 20%.

Danu

Looked to me like Trump plunged in the polls

PVAL-inactive

I'm inclined to believe 538.

Ted Shifrin

I don't want to jinx us (or the world), but yes.

Balarka Sen

22:55

Good for everyone

Ted Shifrin

Everyone (except his mindless angry supporters) realized he's a blathering idiot.

And the video of his acting like a spoiled brat going to rape women is more than most can abide.

Amazing how all these bible-thumpers and people afraid for a transgendered woman to be in the bathroom with them make excuses right and left for this twit.

But I digress.

Danu

realclearpolitics.com/epolls/2016/president/us/…

Note the jump from +4 to +6 in the past 4 days

PVAL-inactive

It's kind of unfortunate to me that what seems to have brought him down is him joking to someone else in a private conversation. Considering he's been nothing more than a con-artist his entire career as far I can tell .

Danu

→ 18 messages moved to Trash

0celo7

23:10

Yikes

Mike Miller

@PVAL I need to write too. Do I know what your note is about?

@Balarka I will when I get home.

Balarka Sen

Thanks.

PVAL-inactive

23:31

@MikeMiller Kind of. I'll send you a copy if I have something presentable a non-trivial amount of time before I can post it. The thing I sent out is a note, however the thing I am writing is a paper!

Azmisov

23:46

Hey, is anyone here familiar with recurrence relations? I'm wondering if there are any math techniques/fields for converting closed form functions into recurrence relations. All the information I've found seems to only deal with the reverse, e.g., "solving" recurrence relations.

Transcript for

$Mathematics$ Mathematics