Mathematics - 2016-12-04 (page 4 of 8)

7:00 PM

You don't seem to like my style, either :D

Not that my lecture notes are meant for public consumption.

(Except for the diff geo stuff I put on the web.)

Oh, I really don't know. I haven't read too much of your stuff---the complex geometry notes are hard to deal with because I just can't find anything :(

Maybe I'll give the diffgeo a shot once I'm tutoring on surfaces.

Semiclassical

[insert obligatory -stuff i should write notes on- comment here]

Ted Shifrin

Oh, right.

Danu

Hatcher was okay, not spectacular but fine. Nice pictures (though it ain't tikz!)

Ted Shifrin

Hatcher is a weird combination of chatty and too terse.

Semiclassical

7:02 PM

Hatcher seems like a good entry point into things, but not the right book to master the subject from

Danu

Yeah...

@Semiclassical Huge entry point though.

Lots of material

Semiclassical

True.

Ted Shifrin

But I'll take him over Spanier and more formal/algebraic treatments any day.

I think one can master quite a bit. His exercises are SUPERB ... infinitely better than anything else out there.

Semiclassical

Sort of like an overly talkative prof doing an two-semester intro course :)

Danu

I hope that Hirzebruch's book on topological methods in alg.geom. is nice. I think I could really dig it if he writes nicely.

Ted Shifrin

7:03 PM

It's typical European terse style.

But lots of info in there.

Danu

Hmmm... That bodes... Not well?

Ted Shifrin

I haven't looked at it in decades ... and no longer own it.

Danu

What about the book on Spin Geometry?

Ted Shifrin

It's hard. Lawson is a great expositor.

Danu

I think I have some kind of broad idea of what kind of topics I'd like to learn about now

Ted Shifrin

7:04 PM

I don't have that one any more, either.

I love reading articles Chern wrote, but most people can't read them ... because of all the differential forms :P

Semiclassical

Spin Geometry is a title which makes me gulp a bit

Probably because spinors are something which I appreciate at the level of SU(2) but not much beyond that.

Alessandro

sorry @Ted, had to run for dinner earlier so I missed your answer, I'm not sure we're thinking about the same thing then

Danu

I'd like to learn a bit of Morse theory and slowly work towards gauge theory on the one hand (also a bit of knot theory if possible; perhaps more symplectic geometry to be able to use it?), and on the other hand work towards complex geometry and the theory of vector bundles (slowly catching up on algebra in the background)

Ted Shifrin

Not sure what you're thinking of, @Alessandro ;D

Semiclassical

@Danu Have you heard of Picard-Lefschetz theory?

Alessandro

7:05 PM

what do you mean with dual projective plane?

Danu

@Semiclassical Naw

Null

Grunch. Can giving $0^0$ the value 1 lead to problems?

Danu

Oh, like complex Morse theory?

Semiclassical

Picard–Lefschetz theory

In mathematics, Picard–Lefschetz theory studies the topology of a complex manifold by looking at the critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for complex surfaces in his book Picard & Simart (1897), and extended to higher dimensions by Lefschetz (1924). It is a complex analog of Morse theory that studies the topology of a real manifold by looking at the critical points of a real function. Deligne & Katz (1973) extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of the We...

Ted Shifrin

@Alessandro: If the plane is $\Bbb P(V)$, then it's $\Bbb P(V^*)$. :P

Semiclassical

7:06 PM

Yeah.

Danu

Sounds neat***

Semiclassical

I always wanted to understand some of that, but it seemed like to steep a hill to climb.

Danu

But I honestly would really like to learn Floer homology stuff---this is the main reason I want Morse theory for now :P

Ted Shifrin

A point of the dual projective space is a hyperplane in the original projective space, and vice versa.

Semiclassical

Fair 'nough

Ted Shifrin

7:07 PM

Yes, @Null.

Semiclassical

I am a bit proud of understanding a special case of the Picard-Lefschetz formula they give there, though :)

...namely, the $k=1$ case :P

meow-mix

@Ted all about what? :P

Ted Shifrin

@meow-mix Huh?

Danu

@Semiclassical NOW APPLY INDUCTION YES?

Semiclassical

lol

Ted Shifrin

7:11 PM

@Null: What is $\lim\limits_{x\to 0^+} x^{1/\log x}$?

Null

if $\sum\frac{1}{a}$ converges, then $\sum\frac{1}{a+1}$ converges surely? (a is some polynomial)

Semiclassical

But I only understand the base case :(

Ted Shifrin

I hate that notation, but think about comparison, @Null, assuming everything's positive.

Semiclassical

Is $a\geq 0$?

Null

@Semiclassical oh, i didn't think about that. but yeah for our purposes. I see the problem if not.

Semiclassical

7:13 PM

@TedShifrin Not going to lie, my first reaction to that was just "ugh." but then I actually thought through how I'd do that and saw that's easy. :)

Astyx

Why is prep school physics so boring ?

Ted Shifrin

@Semiclassic: Probably the easiest example of the failure of $0^0=1$. :P Now you can create uncountably more.

Danu

@Astyx Same for the math?

Astyx

I find math more interresting

Semiclassical

Because it's not being taught by people who care about physics?

Ted Shifrin

7:14 PM

@Astyx: Even college mechanics is boring unless you have a great book, like Kleppner & Kolenkow was.

Null

@TedShifrin so x approaches 0 from the positive side?

Danu

Looking back now as a grad student, college physics really looks shitty hehe

Astyx

My prof came out first of her class in ENS Ulm, so I doubt she doesn't care about physics

Ted Shifrin

I said that, yes, @Null.

Alessandro

If I describe points with homogenous coordinates, read them as coefficent of a linear equation and map points to the zero set of this equation don't I have a map sending points to hyperplanes? (this depends on a choice of basis of course and there is no canonical one) @Ted I'm probably misremembering things from last year's geometry course since we never used the name "dual space" explicitely

Ted Shifrin

7:15 PM

Are all the students as strong in math as you, @Astyx?

Semiclassical

Looking presently at the intro physics course I'm TA'ing for (as well as my previous experience in the subject) I -know- that college physics is pretty shitty

Danu

@Astyx I came first in plenty of physics classes---look at where I am now :P

Ted Shifrin

@Alessandro: There is no map though. It's a correspondence.

Astyx

As much as I hate to say this, not quite @TedShifrin

Danu

I'm tutoring quantum mechanics @Semi :P

Semiclassical

7:15 PM

Oh, fun :)

Ted Shifrin

@Astyx: You need solid math skills to do interesting physics :)

Danu

Next semester it's gonna get real though

Ted Shifrin

See if you can find the book I referenced up there ^^^, though, @Astyx.

Danu

topology & geometry of surfaces

Astyx

@Danu Yeah, but my point is that my prof is a very good one, it's just that what we are taught isn't interresting (not really her fault)

Alessandro

7:16 PM

I'm confused

Semiclassical

The problem with intro physics to me, though, isn't the textbooks or the lectures.

Astyx

@Ted I will

Gotta go now, brb

Null

@TedShifrin mmh, that limes doesn't exist. and since the exponent approaches 0, $0^0$ is best left undefined for the general case

Danu

@Astyx My point is grades don't matter :P

Semiclassical

It's that the workload is ridiculous.

Ted Shifrin

7:17 PM

It does exist, @Null. Try again.

@Alessandro Per que?

Null

@TedShifrin oh, i did read $x^{log(x)}$ instead of a fraction ;)

Semiclassical

An intro physics class is a big machine with a lot of moving parts, and like any machine it really doesn't have any time for subtlety or nuance.

Ted Shifrin

I wanted it to be of the $0^0$ form, @Null.

Semiclassical

Plus there's a lot of stuff we do which isn't done for pedagogical reasons but for institutional ones

Null

@TedShifrin is it 10?

Semiclassical

7:19 PM

$\log x=\ln x$ or $\log x=\log_{10}x$?

Adeek

Hey @TedShifrin what do you think of the following proof.

Ted Shifrin

No, @Null. Oh, I'm writing $\log$ for natural log. I guess you write lg in Europe?

Semiclassical

For instance, why do we assign lab reports? At the end of the day it's not for any pedagogical purpose; it's because it means that the class counts as 'writing intensive' and therefore people are more likely to take it to satisfy that requirement.

Null

@TedShifrin ah ok, then you meant ln probably ;)

Ted Shifrin

Mathematicians always write $\log$ for $\ln$, Semiclassic, unless we're teaching engineering calculus :P

Semiclassical

7:20 PM

I am very very cynical about the intro physics labs at this point, at least the grading part of it.

Null

@TedShifrin well then its e

Semiclassical

well, I know that :)

Ted Shifrin

The labs at UGA seem to be decades behind the pedagogy in the classes, @Semiclassic, although my friends in Physics are trying to work on it.

Adeek

Let X denote a normed space and $Y \subset dual(X)$. Denote $Perp(Y)= \{ f \in dual(X) : f|_Y = 0\}$. I wish to show that $Perp(Y)$ is closed.

Ted Shifrin

Right, @Null. And now you can cook up any positive number as a limit.

Semiclassical

7:20 PM

The pedagogy of the labs themselves is fine, if a bit stupid in places.

Adeek

suppose that $\{f_n\} \subset Perp(Y)$ such that $f_n \rightarrow f$ We want to show $f \in Perp(Y)$

Ted Shifrin

@Semiclassic: My point is that the lectures have changed some, but not the labs.

Semiclassical

But the lab reports are done purely so that we can claim Physics 1,2 as writing-intensive courses.

Null

@TedShifrin so $0^0=1$ makes only sense in specific contextes?

Semiclassical

Ah. Ours are more receptive in that regard.

Ted Shifrin

7:21 PM

Right, @Null. Since you can get all sorts of different limits, it's best not to define it to have a meaning.

Alessandro

@Ted because I don't see why it's not a map (I'm probably missing something very very obvious)

Semiclassical

I think they'll be rewriting the lab manual for the course I'm TAing right now, for instance, within about a year?

If I'm around still I might see if I can be a part of that.

Ted Shifrin

@Alessandro: A map $f\colon X\to Y$ sends a point of $X$ to a point of $Y$, no?

Alessandro

it does

Ted Shifrin

You are thinking of a map $X\to \mathscr P(Y)$?

Adeek

7:23 PM

Fix $y \in Y$ and consider the constant sequence $\{y\}$. Since $f_n \rightarrow f$ this means that $f_m(y) \rightarrow f(y)$. Since each $f_m$ is continous we have $f_m(y) \rightarrow f(y)$

Ted Shifrin

I'm saying a point of $\Bbb P^2{}^*$ corresponds to a line in $\Bbb P^2$. How is there a map?

Adeek

right ? @TedShifrin ?

Alessandro

I was thinking about a map from projective subspaces of $X$ to projective subspaces of $Y$

Ted Shifrin

Karim: I can't think about that now

Adeek

oh no wait I think something is wrong.

Alessandro

7:24 PM

but I see what you meant now

Ted Shifrin

Yeah, @Alessandro, if you have a linear map $V\to W$, you get an induced map $W^*\to V^*$, hence inducing one on the projectivizations. But this is not what we were doing.

user379685

F is continuous in 0 and n->inf (f(1/n)-f(0))/(1/n)=1. Does that mean f'(0)=1?

Semiclassical

How does that correspondence work, out of curiousity? For instance, if I've got a point $[x,y,z]$ in $\Bbb P^{2*}$, what's the corresponding line in $\Bbb P^2$?

Alessandro

Ah, I see, the fact that if $2$ subspaces are incident in the projective space their duals will be incident in the dual space is true though, right? And if $A\subseteq B$ in the projective space this inclusion is reversed when I consider their duals

Ted Shifrin

@Semiclassic: If you make your point $[a,b,c]\in\Bbb P^2{}^*$, then the line is $ax+by+cz=0$ in $\Bbb P^2$.

Semiclassical

7:28 PM

Gotcha.

Ted Shifrin

@Alessandro: if you have two lines in $\Bbb P^2$, they intersect in a point, and So projective duality says that the point corresponds to the line through the corresponding points in $\Bbb P^2{}^*$.

I.e., the point corresponds to the pencil of lines through that point.

Alessandro

ok, that makes sense

Null

If $F$ is a field, $V$ a $F$-vectorspace and $A,B\subset V$, then the following holds: $\langle A\cap B\rangle _{K}=\langle A\rangle _K \cap\langle B\rangle _K$

My counterexample would be: R as the Field, Q as the vectorspace, and A={1}, B={2}.
The span of $A\cap B$ would be the 0-vector. The span of A would be Q, the span of B would be Q. And the intersection of Q and Q is Q itself, which is not the 0-vector.

Ted Shifrin

@user379685: Assuming $f$ is differentiable at $0$, yes. But that limit could exist without that being true.

Null

^ or do i have to take Q as the field, and R as the vectorspace?

Ted Shifrin

7:30 PM

Huh? @Null. What is $Q$?

arctic tern

rationals

Null

$\mathbb{Q}$

Ted Shifrin

Is $\Bbb Q$ closed under multiplication by real scalars?

Null

nope

i see

user379685

@TedShifrin Without assuming it's differentiable why can/can't i say that f'(0)=1?

Ted Shifrin

7:31 PM

Because that particular limit could exist and have $\lim_{h\to 0} \dfrac{f(h)-f(0)}h$ still not exist, @user379685. Continuity of the function won't do it.

Are you in a calculus class or an analysis class, @user379685?

user379685

@TedShifrin analysis

Ted Shifrin

Cool. So try to construct an example where you get a limit with $h=1/n$ but not in general.

Semiclassical

You've also got $f$ continuous at zero, right?

Ted Shifrin

Yup, $f$ is continuous.

Semiclassical

Think I see an example, then, though I'll keep that to myself

Ted Shifrin

7:35 PM

Yes, make @user379685 figure this out.

Semiclassical

nods

Ted Shifrin

I have in mind making $\lim\limits_{h\to 0^+} \dfrac{f(h)-f(0)}h$ not exist, in fact.

But that remark gives a cheap way out.

Semiclassical

hm, i was going for the cheap way then.

Astyx

Couldn't find the book you referenced up there ^^^ @Ted

Danu

Wait... Are there more (isomorphism classes of) complex than holomorphic line bundles over $\Bbb P^n$, @Ted?

Ted Shifrin

7:37 PM

@Astyx: Not in French libraries, huh? It's called An Introduction to Mechanics.

Astyx

No I meant I couldn't find the reference

Semiclassical

though I don't see how you could get that right-limit to not exist when $f$ is continuous at zero and $n(f(1/n)-f(0))\to 1$ as $n\to\infty$.

Ted Shifrin

@Danu: Topologically (or smoothly) they're classified by Chern class. Holomorphically, there are lots with the same Chern class.

Astyx

I've been watching the lectures of Leonard Susskind, which I found very good

Ted Shifrin

Then try harder, @Semiclassic. :P

Semiclassical

7:38 PM

Pffff

Ted Shifrin

$n\in\Bbb N$ here.

Semiclassical

Is the point that it's continuity at zero, not continuity near zero?

Danu

@TedShifrin I don't see what exactly you mean. I proved already that the Picard group of $\Bbb P^n$ is isomorphic to $\Bbb Z$ (so they're classified by first Chern class).

Ted Shifrin

At least, I always interpret it that way.

@Semiclassic: I can give you a continuous counterexample.

Semiclassical

Huh.

Ted Shifrin

7:39 PM

Oh, @Danu, right, on $\Bbb P^n$. No difference.

Sorry.

Danu

@TedShifrin What were you thinking of?

Semiclassical

If it's continuous, I'm not seeing why N versus R makes a difference.

Danu

Line bundles in general?

Ted Shifrin

General compact complex manifold, even algebraic. Yeah.

@Semiclassic: Because you only have control at the points $1/n$, $n\in\Bbb N$.

Danu

Okay. How do I prove that complex ones are still classified by Chern class?

Ted Shifrin

7:41 PM

Cohomology exact sequence, @Danu.

Semiclassical

Sure, but if it's continuous in the vicinity of 1/n...

Ted Shifrin

Just not with $\mathscr O^*$ in there.

Danu

By the way, I always thought there had to be more bundles if less data is required...

Ted Shifrin

No more discussion, @Semiclassic, or we'll do @user379685's homework.

Semiclassical

Fiiine.

Danu

7:41 PM

But I guess the bar for isomorphism also goes lower.

user379685

@TedShifrin It's not my homework :C

Ted Shifrin

@Danu Right.

Semiclassical

I could bug you about 1-forms on Riemann surfaces instead? :3

Ted Shifrin

What is it, @user379685?

user379685

@TedShifrin 4.1.6b wit.uber.pl/pliki/ID%20Semestr%20I/Analiza%20Matematyczna/…

@TedShifrin a book i'm doing on the side

Semiclassical

7:42 PM

(or I could go back to grading quizzes, ugh)

Danu

@TedShifrin Right... So everything follows from the fact that the sheaf of smooth/continuous functions is soft?

user379685

@TedShifrin not required, and the example is hard (with a star)

Semiclassical

Still homework, then, just not assigned :)

Ted Shifrin

@Danu: I think I have a different word, but I guess so.

Danu

Is there any more geometric way to argue that kind of stuff, btw?

@TedShifrin [insert one of 100000's of criteria that make $H^{k>0}$ vanish] ;)

Ted Shifrin

7:43 PM

You can concretely use partitions of unity to trivialize the $1$-cocycle, @Danu.

Danu

Yeah, sure. Like Forster does in his book.

Ted Shifrin

I guess you'll say pfeh.

Danu

I saw that one example, nothing more

I trusted it can be done in every case I care about :P

Ted Shifrin

@user379685: Try drawing a picture of the graph of the function. What do you know and what do you not know?

Danu

It seems so easy in that example that it's probably not worth it to generalize :D

Ted Shifrin

7:44 PM

But a cheap way out is to mess up the limit on the left side, @user379685. I already gave that hint.

Semiclassical

I guess the way to make the problem harder would be to assume that $-n(f(-1/n)-f(0))\to1$ as $n\to\infty$ as well.

Ted Shifrin

Right. :)

Semiclassical

I don't know how to do it in that case :)

user379685

@TedShifrin 1/n when n->inf isn't the same as h when h->0 because with n we can't get to 0 from the left?

Ted Shifrin

twiddles fingers knowingly

@user379685: First of all, when you or the author write $n$, you intend that to be only positive integers.

Semiclassical

7:47 PM

That's one difference. It's sufficient for the task at hand, but it's not the only difference.

Ted Shifrin

So you only know what's happening at $1/2$, $1/3$, $1/4$, $1/5$, etc., not at all $h\to 0^+$.

Semiclassical

Right. That's what leads to the subtler version of the problem (which I don't know how to do, but evidently @Ted does.)

Ted Shifrin

Ted has the advantage of having taught Calculus with Theory 14 or 15 times in his life. :)

Semiclassical

Point.

Null

do you think a question box and a "what i have tried"-box would make asked questions on MSE better?

Semiclassical

7:50 PM

Well, you can submit answers to your own questions

And you can even do so at the same time as you submit the question itself.

That said, people don't really notice that and so don't take advantage of it.

Ted Shifrin

I don't like the whole thing of submitting one's own answer. That should be an edit to the question, unless, I suppose, it's a community wiki.

Null

@Semiclassical i mean to filter the questions where the question is some excercise, without any try

Ted Shifrin

@Null: There are some thoughtful questions that don't fall into that two-box format, though.

Semiclassical

I like that a user who figures out their problem can submit a solution

Null

@TedShifrin i agree

Ted Shifrin

7:52 PM

Semiclassic, I've had plenty of people whom I've helped with a sketch (as an answer) and then further hints. They then write up a solution and accept their own answer. That annoys me.

Semiclassical

That said, I think one has to distinguish between "what I've tried so far" with "what I think the resolution of my problem is"

Ted Shifrin

I like that they wrote it up, but I spent a half hour helping :)

Semiclassical

Eh, not much point in helping out in that case. I don't really see how else that's supposed to end.

Oh, wait. You mean you submitted an answer, and then they did their own?

Ted Shifrin

Right.

Semiclassical

That's annouying, yes.

Ted Shifrin