Hm.. unless you're leaving out details, I'd respond with a nicer form of "rtfm" if at all.

morning @Ted

Hi @Karl :) Happy holidays.

G'night, @Huy.

Howdy @Ted

Happy holidays

Tanx. G'night @MikeM ... hope you're feeling better.

Hi @Ted. Mostly, yes. Curling up by the heater with holonomy. Perhaps not catchy enough for a Christmas song.

@EwokNightmares: I have gotten plenty of emails from random readers.

LOL, you out in the boonies with your family, @MikeM?

No, I fly east on the 25th.

Ah.

Putting off packing is all i'm doing.

Thought I had a good idea last night; worked it out this morning and it was a pretty mediocre idea. So it goes.

I will be traveling 3 weeks, but a suitcase only holds so much ... particularly troublesome with so many different climate zones.

Yeah, I know the feeling. Luckily I'm only gone half that.

Also lucky that upstate NY isnot that cold right now.

I think Mike Artin also told me once that if 95% of one's ideas aren't junk, one isn't having enough ideas. @MikeM

Yeah, same for Michigan and Georgia. I'll have to look at the month-long forecast this weekend.

So far I'm batting 100, so I must be doing great from the POV of quantity.

I don't remember when I last had an idea. :)

@TedShifrin got like 3 inches of snow (my own estimate) so far this year

Since I have no idea where you are, @Karl, ... ok :)

San Diego, ofc.

I answered you, @user159870

@TedShifrin Hoping that the message I replied to was enough of a hint

@Karl: It could be midwest. I don't know if the east has got that much yet.

I thought the implication was that he's from Mich and doesn't entirely agree wiht your estimate.

I didn't have an estimate, @MikeM.

But it does appear that I won't get buried in multiple blizzards this visit, as I did last time.

Is the text I wrote above the justification why the sign changes because of the direction of $N$ ? @TedShifrin

MI is correct. 3 in. of snow by Dec. 23 is nothing.

I know, @Karl. Last visit they had about 2 ft in the week I was there.

@user159870: Yes, and I answered that last night.

@Ted: Was calculating canonical classes of log transforms earlier, after giving up on calculating the canonical class of the degree 5 hypersurface in CP3... I need to get around to reading G&H at some point.

@TedShifrin Yeah, ridiculously cold too iirc .

@MikeM: Learn the adjunction formula. It's very easy and very cool.

@Ted: Didn't want $c_1^2$. Wanted $c_1$.

Yeah, it was below 0 several nights, @Karl.

Right, @MikeM: If $V$ is a smooth hypersurface in $M$, $K_V = (K_M\otimes [V])\big|_V$.

One way to understand this is the Poincaré residue, but there are lots of ways.

Oh, yes, that I knew. But what I would have to do, @Ted, was figure out what the pullback of $K_M$ was. In context, I had already picked a basis for $H^2(V)$.

Or, rather, not even that. I knew there existed a basis and blah blah blah.

But it's easy to pull back the hyperplane class in $\Bbb P^n$ to a degree $d$ hypersurface, @MikeM.

Not sure what it means for that to be easy. I think we have different goals.

I have that the intersection form of $V$ is diagonalizable and of the form $(1,\dots,1,-1,\dots,-1)$. I want to pick such a basis and write down what $K_V$ is in this basis.

Well, we do that by undetermined coefficients, basically, I guess.

Not really sure I know what that means.

How do you understand the basis?

Oh, I don't.

The first step is to figure out what the hyperplane class is. For the quintic, how many 1s and -1s?

It's an algebraic trick: if I know that $H^2(S_d)$ ($d$ the degree of the hypersurface) has something with odd square, then because the form is indefinite I know it's diagonalizable by algebra.

Hmmm... 9 and 44. :)

What? $H^2$ is 53-dimensional for the quintic?

Ayup.

That doesn't ring any bells at all.

$b_2$ grows cubically in degree.

It's got to be big, @Ted, because the quartic is a K3 and that has $b_2 = 22$.

The interest was that there's a theorem: given a smooth algebraic surface (+ conditions) $M$, any diffeomorphism $f: M \to M$ must preserve the set $\{K_M, K_M^{-1}\}$. There's also a theorem of Wall that says that for a topological 4-fold $M$, there's a self-homeomorphism inducing any given automorphism of the intersection form. So one can combine these two facts to get homeomorphisms not homotopic to diffeomorphisms.

OK. So the real question is: What is the hyperplane class in that basis? I dunno.

Or, what the hell is the basis geometrically?

That is certainly known. Just not by me.

Right, the latter was why I gave up.

For lots of surfaces, one can understand this in terms of blowing down the $-1$ curves and working with that resulting surface. (I.e., understanding the surface as the blow-up of something understandable at a certain number of generic points.)

The conditions: general type, minimal (not a blowup, ie can't remove connected summands of CP2). So if I'm looking for examples in hypersurfaces I can't do anything until degree 5, which is when I'm of general type. $b_2$ is annoyingly big there.

I'm confused. The $-1$-curves can be blown down.

Huh? They're not holomorphic curves.

Or I'd agree they could be blown down.

Ohhh, duh.

Damn, I've forgotten too much.

Continuing with the previous story: so I gave up and went for the simplest general type surfaces I was aware of, Dolgachev surfaces, which one gets by log transforms on E(1). (Does the notation E(1) mean anything to a complex geometer?)

Anyway, these have $b_2 = 10$, so a human being can work with them.

It might. Is that one of the standard elliptic surfaces?

It's $\Bbb{CP}^2$ blown up at 9 points equipped with an elliptic fibration over $\Bbb{CP}^1$

(I don't need the C's in there. :) )

Right, one of the standard elliptic surfaces.

(To check my language by the way: general type is synonymous with $K_X^2 > 0$, right?)

Yes, that should be right.

Ok, the latter condition is the one I need for my theorem above. Which, BTW, I consider mind-boggling.

The smooth manifold itself remembers the canonical class, or at least the canonical class and its negative. That's crazy!

Go talk to Kodaira :)

@Ted: This is newer than Kodaira. Probably early 90s.

What're you referencing when you invoke his name?

That K3 surfaces are deformation equiv -> diffeo?

No, I was referencing all the classification of surfaces he did.

Oh, ok.

I'm not sure you're convinced the above is exciting yet. So I'll repeat it: this is exciting!

I'm sure that saying it again will convince you.

I hadn't really paid attention when you typed it up there. Who proved that thm about diffeos?

Would guess it's due to Witten. I can check.

Yeah, surprising.

Yeah, it's pretty exciting

goes back to the shadows

So, @Karl, what exciting courses did you take this fall? What's in store for spring?

@TedShifrin: A modified version of the full story: SW takes as input two bits of data: a (smooth, oriented) 4-fold and an element of $H^2(M;\Bbb Z)$. We say that $x \in H^2(M;\Bbb Z)$ is "basic" if $SW(x) \neq 0$. In addition, given a diffeo $f$, $SW(f^*x) = SW(x)$. So diffeomorphisms preserve basic classes.

The above result, then, is just a calculation: a minimal surface of general type's basic classes are precisely $K_X$ and $K_X^{-1}$. The minimality hypothesis shows up because whenever you blow up, you double the number of basic classes.

OK.

Sorry. :P

No need to be sorry. I asked!

So I can blame you for my excitement!

@TedShifrin Not incredibly exciting math-wise, though I did learn some optimization. Same goes for next semester, when I will have three upper level philosophy courses.

Are you majoring in philosophy, @Karl, I presume? Surely you're doing math as well?

Yeah, dbl majoring

awesome

Mainly catching up on the philosophy major

I thought you were still a first-year, but you've always confuzled me.

second year

Second-year in upper div courses is catching up?

OK ... surely in math you're ready for grad courses soon enough, although I don't know if you've taken the breadth of interesting advanced undergrad.

Right, @MikeM ... his notion of catching up differs from mine.

@TedShifrin My advanced lin. alg. class helped quite a bit (last semester).

Though I didn't learn any new concepts strictly speaking.

I think that's when I started taking upper div courses.

@Karl: Have you done stuff like single- and multivariable analysis?

@TedShifrin Nope, next year for sure

And even though you're a confirmed algebraist, I'd encourage you to take some undergrad diff geo and topology ...

@TedShifrin I'm leaning toward geometry these days, so those sound like good options.

@TedShifrin Nice hat.

You're supposed to be asleep, @Balarka, but thanks.

Well, always glad to butt in where I wasn't asked to, @Karl :)

@TedShifrin: He's slowly shifting his sleep cycle to coincide with ours, hoping we don't notice.

@MikeM: He has to function in his world.

"sleep cycle"?

heh

I wonder if the Poincaré-Bendixson theorem applies.

@Ted: I'd say he has a 1-dimensional personality, so it's even easier.

Simplifies the whole unstable manifold game.

What's the Poincare-Bendixson theorem?

lmgtfy

It's about dynamical systems and limit cycles, @Balarka ... I took a slight liberty with sleep cycle.

@MikeMiller ?

"Let me google that for you"

Oh, I thought the L was an I.

I might google that for you

@KarlKronenfeld: Not a good sign for U Mich's geometry group

imgtfytpn

though probably not?

@TedShifrin cool stuff.

@MikeMiller that's a valid interpretation

Not exactly accurate, @MikeM, although I don't know how active some of the people I know are. There's some serious algebraic geometry (David Speyer and I have interacted on here a few times, and I know he's teaching some diff geo). But Rolf Spatzier, Dan Burns, and Alex Uribe are all (diff) geometers of sorts.

Dan Burns does (did) several complex variables and geometry, but Uribe and Spatzier are more classic differential geometers.

@TedShifrin: It is not accurate. There are differential geometers listed under, say, topology.

I think Ruan is also a diff geometer, but I'd have to check.

@Karl: I'll actually be in Ann Arbor for a bit over a week in a week.

It depends how carefully you draw the line between symplectic topology and geometry.

Those guys don't take their web pages too seriously, sadly.

Alex Uribe was a grad student at MIT when I was a postdoc, so I actually know him.

Well, I know Dan Burns, too, from my grad student days. Agh.

All of these topology and geometry lines are pretty muddled nowadays, anyway, so let them put their (mostly nonexistent) webpages wherever they like.

Most of them have decent webpages one or two levels down. But not all.

Just googled Dan. Can't find his page.

One of my good friends (and a mathematical brother) from grad school days is at Michigan State doing geometry.

The easiest way to find out what he likes are to check out his students' thesis titles on the genealogy site...

Yup, Burns has no webpage to speak of.

!! Kutluhan is one of Burns's students

Maybe I'll go by the department when I'm there.

I don't know who that is, of course.

He (+ Yi-Jen Lee + Taubes) wrote the very, very long proof that HF=HM, whatever those symbols mean.

He and Taubes also have some serious results about when S^1 x M, M a 3-fold, can have a symplectic form. The sort of stuff I would care about.

Dan Burns was a very, very smart guy, so I'm sure he's had some good students.

Well, when one of your students is a frequent Taubes coauthor, you can probably say you've done well.

And that's just one. :)

Where is Kutluhan?

Buffalo, looks like

Webpage says he's at the IAS this year, which makes sense.

Oh, at IAS this year, Buffalo ordinarily. One of my good grad school geometer friends is at Buffalo. Last time I checked, he had mostly punted on research for his career.

Yup, you and I googled the same info.

IAS is doing a year on low-dimensional geometry, which is why it makes sense.

This looks like a decent course.

I am very fond of his reference #6.

Which one was that?

Morita, geometry of characteristic classes

Ah ...

Characteristic classes of fiber bundles (AKA cohomology of BDiff(M); M usually a surface), characteristic classes of foliations, characteristic classes of flat bundles.

First bit is pretty hip because mapping class groups are perpetually hip.

Hmm... I'll stop killing the conversation.

Well, I'm gone for now ...

@MikeMiller Problem 1a looks highly nontrivial to me (any problem which I cannot solve after positive amount of thinking = highly nontrivial for me). Do you want me to look at the geometry of the Hopf map for this, or try otherwise?

I don't know what the Hopf map has to do with anything here. I also wouldn't call this highly nontrivial, since if I gave you a small hint you'd be able to do it in minutes.

The attaching map of the 4-cell in CP^2 is precisely the Hopf map.

Ok, I still don't know what the Hopf map has to do with anything here. :)

I had an idea, but it looks pretty much impossible to work it out.

I'll keep thinking.

?

(I am glad to shoot ideas down. I won't give ideas, though.)

:P I'll post the idea after formalizing it. Though I don't believe it's going to work either way.

Hello!!

Suppose $\gamma$ is a curve on a sphere of radius $r$. Which relation does $\gamma$ have to satisfy?

@OFFSHARING Really? That would be something since the series without any negative terms is known only to be irrational.

@robjohn Yes, you know I never say stories about my math, and in this case it's even hard, very weird to invent such a story. :-) The series looks in a certain way. :-)

@robjohn But there is a minus sign for each 4th term.

+ + + - + + + - and so on

Guys if I have a continuous function $f$ from $X$ to $Y$, and then I take the same function, but consider it as a function from $X$ to $Z$, with $Y \subset Z$, it's still continuous right?