Mathematics - 2016-06-08 (page 2 of 2)

8:02 PM

@robjohn hey, you've become rare here. :-)

@robjohn btw, these days I obtained some fantastic results, I cannot call them otherwise (that was after a period in which I was less productive).

Oh, I see also Akiva had something to say ...

@AkivaWeinberger $\rm\tiny{Because~I~can}$

@Semiclassical are you in a better mood today?

Akiva Weinberger

OH, OKAY.

user 1618033

@AkivaWeinberger My math makes me yelling pretty often, I mean a real yelling, and this is great from all points of view.

Especially when you meet that kind of stuff that at first sight seems impossible, and then ... with a couple of clever hits, the problem is down.

K.O.

Akiva Weinberger

What are you working on?

Ted Shifrin

Heya DogAteMy :)

hi @Danu, @MikeM

Akiva Weinberger

Oh hi!

Ted Shifrin

8:16 PM

Your school year has pretty much wound down, I imagine.

Hiroto Takahashi

Hi

Akiva Weinberger

How's life been treating you? Gloriously knocked out any problems on the field of battle (see user$\phi$) lately?

Hiroto Takahashi

someone know what is the defintion of plane conic?

2

Q: Show that any smooth projective curve of genus zero over a field $K$ is isomorphic to a plane conic over $K$

I have the following question: Show that any smooth projective curve of genus zero over a field $K$ is isomorphic to a plane conic over $K$. Assuming that a plane conic is a conic cut by a plane, but I don't see how can I get a hyperbolic by cutting the conic. Is the definition of plane con...

Mike Miller

Hi @Ted.

Akiva Weinberger

I had my last classes of the year yesterday, and my first final tomorrow.

History.

Ted Shifrin

8:17 PM

@Hiroto: The zero locus in the projective plane of a homogeneous quadratic polynomial.

Mazltov, DogAteMy ... What are your plans for summer?

Akiva Weinberger

MathCamp, in Maine this year.

(They move around.)

Ted Shifrin

I'm surprised you didn't apply to PROMYS or the Ohio State program. Or did you?

Mike Miller

Math camp isn't a bad program.

Ted Shifrin

I know nothing about it; my statement wasn't intended to suggest it wasn't fine.

Akiva Weinberger

Honestly, what happened was that three years ago my dad Googled "Is there a good math camp" and it showed up

Ted Shifrin

8:19 PM

Wow ... three negatives in that sentence.

Akiva Weinberger

What's PROMYS?

Ted Shifrin

Google PROMYS and Ross Program (OSU).

Akiva Weinberger

@TedShifrin That's not nothing, isn't it

Hiroto Takahashi

@TedShifrin I don't know too much about that.. I saw that question and then I wonder myself what is a plane conic

Ted Shifrin

Both are largely number theory oriented, but great programs for motivated math talents like you.

Mike Miller

8:20 PM

What year are you, @Akiva?

Hiroto Takahashi

@TedShifrin maybe you can answer it

user 1618033

@AkivaWeinberger Working on multiple integrals with multiple polylogarithms of high order.

Akiva Weinberger

Going into 11th grade

Mike Miller

Kevin thinks mathcamp is better than Ross.

Akiva Weinberger

Kevin?

Mike Miller

8:21 PM

The guy next to me.

Ted Shifrin

Hiroto, I saw the question a few days ago and didn't answer it because I have no idea what level the question-asker is at. This would normally require some knowledge of algebraic curves or Riemann surfaces.

Akiva Weinberger

Hi, Kevin

Ted Shifrin

Ah, cool, @MikeM. I genuinely hadn't heard of it. I'm more familiar with PROMYS.

Akiva Weinberger

(Re: PROMYS) Ooh, six weeks. That's like 6/5 of MathCamp.

Ted Shifrin

And hi to Kevin from me, BTW.

MickLH

8:22 PM

@user1618033 Do share.

Mike Miller

I can give you his email if you want to talk to him about it.

Ted Shifrin

I know the guy that runs PROMYS and I had undergraduates/graduate students at UGA who went there as counselors, DogAteMy.

Mike Miller

He's taught at Ross and we're both familiar with mathcamp, given that we have friends that teach there and we'll apply to teach next year.

Hiroto Takahashi

@TedShifrin oh, I understand, somehow it seems hard

Ted Shifrin

Seems that none of robjohn, Pedro, DanielF, anon, or I are around here much anymore.

user 1618033

8:23 PM

@MickLH It's stuff new also for the math literature according to my investigations, but at some point I'll share more stuff.

Ted Shifrin

What subjects does mathcamp tend to concentrate on? (As I said, the other two are primarily number theory, not my particular interest.)

Akiva Weinberger

I could ask people at MathCamp what they think of PROMYS when I get there

Ted Shifrin

Where in Maine will it be, DogAteMy?

Akiva Weinberger

Ooh, lots of stuff. You can take up to four classes a day, with a lot of choices for classes. So, no, not primarily number theory.

Ted Shifrin

Is it problem-set based or lecture-based?

Akiva Weinberger

8:25 PM

Place called Waterville? It's at the campus of Colby college.

Lecture-based, mostly

Ted Shifrin

Oh, Colby, sure. Waterville, ME versus Boston hmm ... :D

PROMYS has fewer lectures ... more problem sets and collaborative learning.

Akiva Weinberger

Not that they encourage treating math as a spectator sport, of course.

Ted Shifrin

Well, I take that back. I guess there are some lectures and then lots of problems to wrestle with the stuff.

Akiva Weinberger

As I said, they move around; last year was Tacoma (by Seattle), year before Portland, Oregon.

Ted Shifrin

Well, I know you'll have fun, DogAteMy.

MickLH

8:27 PM

@user1618033 I've found myself stranded outside the literature in practical application before. I ended up finding a hypergeometric representation of one stretch of the function long enough to make up the rest from reflection identities. Then it was easy to integrate.

Danu

@Ted, long time no see!

@TedShifrin Oh, hi! :D

Ted Shifrin

I'm back from 2 1/2 weeks of travel, @Danu, for whatever it's worth :P

Danu

Where'd you travel to? :)

Ted Shifrin

Chicago and Atlanta

Contemplating a trip to Europe in the autumn.

Danu

DO ITTTTT

Meanwhile, I have a small question.

Ted Shifrin

8:29 PM

Your questions are rarely small.

Mike Miller

You could visit bananaman.

Danu

This one is, I hope

Mike Miller

Oh god, don't bring it up

Danu

Also, is that a good or a bad thing? :P

Ted Shifrin

And the three young'uns in Paris, @MikeM ... who've all disappeared.

Hiroto Takahashi

8:29 PM

Ted, maybe you can write as comment in the question what is a plane conic

Danu

Anyways, my question is about the cohomology ring of the 2-torus

MickLH

@Danu One small question... How do I prove Riemann?

Mike Miller

@Ted: The transition functions of the dual of a vector bundle are the transpose incerse, but not the conjugate transpose (or else the dual would not be holomorphic and many things would break). We don't understand why. I tried to help yesterday and got myself confused.

Oh, just that. Nevermind.

Danu

@MikeMiller Hah, I wasn't going to ask that.

But I also want to know that.

Ted Shifrin

OK, @Hiroto, I added a comment.

Ah, that's in my lecture notes you never bothered to read, @MikeM :) No difference in the real or holomorphic category.

Why would you expect conjugate if you're doing the dual of a complex vector space?

Hiroto Takahashi

8:32 PM

@TedShifrin That question is graduated level, right? Sorry to ask, I'm 14 and I study math most by myself

Ted Shifrin

Yes, @Hiroto, it is. So you shouldn't panic :)

Danu

We took some elements $\xi,\eta\in H^1(T^2)\cong \operatorname{Hom}(H_1(T^2),\Bbb Z)$ and found out that their cup product acts on $[c_2]$ (fundamental class, i.e. generator of $H_2(T^2)$) by:

Ted Shifrin

@Danu: Not to interrupt, but this sounds like it's totally down MikeM's line ...

Mike Miller

@Ted: The adjoint of a linear map is the conjugate transpose, is why. I expected to take the adjoint and invert it to get the transition functions.

user 1618033

@MickLH there is much magic in the area of hypergeometric representation like here $$\frac{\, _3F_2(1,1,n+1;n+2,n+2;1)}{(n+1)^2},$$ that arouse in my research at some point in the past.

Ted Shifrin

8:33 PM

no, no adjoint ('cuz that uses inner product)

Danu

$$ \xi\smile\eta ([c_2])=\det \begin{pmatrix} \xi(a) & \eta(a) \\ \xi(b) & \eta(b) \end{pmatrix}$$

Mike Miller

Aha.

Danu

Now, this is all fine. But then my lecturer mumbled something about "standard symplectic form" and I was too afraid to ask what he meant.

He repeated this mumbling when he generalized to all orientable surfaces, and I once again failed to ask.

Ted Shifrin

Represent cup product by wedge product of differential forms, @Danu.

Danu

So... Now I'm asking you guys!

Ted Shifrin

8:34 PM

Then you're doing the standard area 2-form.

Which is the symplectic form on a surface.

Danu

Okay, we haven't yet introduced de Rham cohomology in this course but I've seen it a year ago or so. I'll look it up

Ted Shifrin

Yeah, you need differential forms to talk about symplectic structures.

Mike Miller

Unfprtunately I spent a week working on a failed idea so I should spend today getting some non-failed work done before my meeting.

Ted Shifrin

Well, you don't absolutely need 'em, but it helps.

Mike Miller

@Ted That's not what they were referring to.

Ted Shifrin

8:36 PM

BTW, @Danu, you need to say what $a$ and $b$ are in that formula. They're of course the homology classes of the respective circles.

@MikeMiller Huh?

Danu

Sorry, yes.

MickLH

@MikeMiller Silly, "tested" isn't spelled with an 'f'

Mike Miller

Consider the cup product as an anti linear 2-form on the vector space $H^1(\Sigma_g)$. Then by picking the standard basis of curves, this is the same as the standard symplectic form on the bector space $\Bbb R^{2g}$.

They're talkinf about linear algebra, not differential geometey

Ted Shifrin

Oh, that was far from obvious from Danu's question.

The symplectic form coming from intersection pairing ... well, that's sort of tautological with Poincaré duality.

user 1618033

@MickLH have you ever written a proof on at least 30 pages?

Danu

8:37 PM

I'm doing this in the context of a course of algebraic topology, FYI @MikeMiller

Mike Miller

I know.

Ted Shifrin

@Danu: When your lecturer says symplectic form, to what is he referring, then?

MickLH

@user1618033 LOL that's a scary thought

Danu

@TedShifrin He often mentions things that are not part of what we're currently learning about.

user 1618033

:-)))

Ted Shifrin

8:38 PM

@MikeM, @Danu: Anyhow, regarding the dual bundle, just think about linear maps $T\colon V\to W$ and the dual $T^*\colon W^*\to V^*$.

MickLH

I'm not sure I could even read a proof with 30 pages of meat...

Danu

@TedShifrin Now, I thought I should define that using the inner product.

Ted Shifrin

@Danu: Probably best to ask him to which context he's referring.

Danu

Evidently, that's wrong.

Mike Miller

We did, just messed up repeatedly. I felt very foolish, though I suppose we both did.

Ted Shifrin

8:39 PM

I'm not making snide remarks this time, @MikeM, don't worry.

The dual map doesn't need any inner product, @Danu. The adjoint is a different creature. They just happen to agree for the standard inner product on $\Bbb R^n$. :P

Danu

So how do I define the dual?

Ted Shifrin

$T^*(w^*)(v) = w^*(T(v))$.

Mike Miller

@Danu: $(A^*\varphi)(v)=\varphi(Av)$, where

man

Ted Shifrin

Sorry.

Mike Miller

:P nothinf to be sorry about

Ted Shifrin

8:43 PM

When I'm at my desktop, I can type way faster than on my iPad.

Danu

Thank you both.

So, how I'm wondering what those bundles were good for. Do you know what they're used for, @MikeMiller?

Ted Shifrin

They're all over the place, @Danu.

Just like dual spaces.

Danu

You know which bundles I'm talking about? :P

Ted Shifrin

Oh, a different "those."

shuts up

Danu

:)

@TedShifrin :( no, don't!

Ted Shifrin

8:46 PM

@Danu: You do know that $Hom(E,F) \cong E^*\otimes F$, right? (Still persisting in blabbing.)

Danu

@TedShifrin Yes

Ted Shifrin

OK. Shutting up for real.

Danu

The bundles I was talking about were the line bundles $\gamma^k$ associated to the Hopf bundle $S^1\to S^{2n+1}\to \Bbb C\mathrm{P}^n$ via the representation $\rho_k:S^1\to U(1)$, $\rho_k(z)=z^k$.

My exercise was showing $\gamma^0$ is trivial, $\gamma^k=\gamma^1\otimes\dots\otimes \gamma^1$ and $\gamma^{-k}\cong (\gamma^k)^*$

Ted Shifrin

(You mean $z^k$.) Oh, those are the standard bundles that algebraic geometers write ${\scr O}_{\Bbb P^n}(k)$.

Danu

@TedShifrin blank stare

Mike Miller

8:48 PM

Those are all the line bundles over P^1, either topologically or holomorphic ally

Ted Shifrin

$\Bbb P^n$.

Mike Miller

Tho it's not an acceptable holomorphic def'n

Typo sorry

Ted Shifrin

Best to think about transition functions, probably, @Danu.

Or what local sections look like. Either way.

Danu

I'm done with the exercises now (this dual stuff was the last bit).

Mike Miller

I don't think he liked that suggestion when I made it

Danu

8:49 PM

But I'm trying to see why I'm asked to do this exercise.

Ted Shifrin

Well, we could beat him into submission, @MikeM.

Danu

@MikeMiller I like it in principle, it's just that my lecturer doesn't.

Ted Shifrin

He wants more algebra with representations.

Danu

and I wanna stay buddies with him ;D

He'll be teaching Seiberg-Witten theory next semester

Ted Shifrin

never bothers to please lecturers

Danu

8:50 PM

and I'm so excited

mathematics $\overset{?}{\Longleftrightarrow}$ physics

Eric Stucky

^ this may be the most accurate statement I have read all day.

Ted Shifrin

You like the ?, don't you?

Eric Stucky

yeah

Danu

Everybody likes the ?

MickLH

It's kindof amazing

Ted Shifrin

8:55 PM

oh, and hi, @EricS

Eric Stucky

hey

I gotta paste this thing before I forget and copy something else

Ted Shifrin

Wow, there's @robjohn!

O.O Hi!

Hi @Ted.

Hi @Balarka

8:58 PM

How are you? Back from the trip, I suppose?

Ted Shifrin

Yup, back.

Eric Stucky

Glad to see you again :)

Balarka Sen

How did it go?

Ted Shifrin

Tiring. But fun.

My penultimate cancer check-up, too.

Balarka Sen

Glad to hear that.

Mike Miller

9:00 PM

I just like it when Witten tells me what's true. It's very convenient.

I should really find a physicist who understands his Khovanov homology stuff at some point.

Ted Shifrin

Odd/annoying that people with relatively little background wander around looking at Harvard qualifying exam questions and expect to understand them, let alone to be able to solve them.

Balarka Sen

@Danu I see you learnt to compute the cup product structure in H_1(M_g).

Danu

@BalarkaSen If by $M_g$ you mean an orientable surface then yes

(but I learned it last week---I'm just now typing up the notes)

Ted Shifrin

Have you done any of my interesting diff top exercises yet, @Balarka? :D

Balarka Sen

That's what M_g means :)

Ted Shifrin

9:02 PM

@Danu: It was way easier to be a grad student before Knuth invented TeX.

Balarka Sen

@TedShifrin Oh, yeah, I had a question.

Danu

@TedShifrin One can dream!

@TedShifrin Hahaha... Except you'd forget what you wrote down.

God knows you don't actually remember that stuff.

Now, I have nice documents :)

Ted Shifrin

I had all my notes from grad school, @Danu. They were good. But they are no more.

My handwriting was pretty legible. :)

Danu

See, now they're gone. My pdf's will live forever.

Ted Shifrin

I did scan a few things I thought I might actually want at some point (like my own lecture notes from advanced courses).

Eric Stucky

9:04 PM

only if you are kind to your digital data, danu :)

Danu

@EricStucky The cloud is kind

Eric Stucky

kind of?

Ted Shifrin

Trump will build a wall and block the cloud.

Danu

@TedShifrin Close enough (?). I do this with the notes that are not worth TeXing (i.e. everything I'm taking except Leeb's courses)

Ted Shifrin

Ah, so you bestow upon him quite the hono(u)r ?

Balarka Sen

9:05 PM

@TedShifrin In problem set #2, I recall a (easy) exercise about the link of $x_1 x_6 - x_2 x_5 + x_3 x_4 = 0$ being diffeomorphic $S^2 \times S^2$. This is more or less easy to see by writing out the equation for $S^2 \times S^2$ in $\Bbb R^3 \times \Bbb R^3$ and the corresponding diffeomorphism.

Danu

@TedShifrin I guess I do. He doesn't appreciate it much though, I think.

Balarka Sen

But I think that equation actually cuts out the Grassmannian Gr(2, 4).

Ted Shifrin

Right, @Balarka: It's basically linear algebra. Well, the oriented Grassmannian is that.

Danu

The TA, who is a post-doc working with him, said I should probably not show him my script (which has a lot of TikZ illustrations that I'm very proud of).

Balarka Sen

Is there an easy way to see link of a point in the Grassmannian is S^2 x S^2?

I can't visualize Gr(2, 4) too well.

Ted Shifrin

9:06 PM

Why are you saying link?

The best way is to think about eigenspaces of the Hodge star operator. There are other ways. I have exercises on this in a graduate course or two. I didn't send you those.

Balarka Sen

Well, you're looking at intersection of a unit sphere at that point with that thing. Wasn't that called a link?

Danu

After I'm done with this course, though, I think I'll give him the script regardless if he likes it or not. I've worked too god damn hard for it to not let him know :P

Selfish motivators are the best ones.

Ted Shifrin

The oriented Grassmannian itself embeds in $S^5$ by Plücker.

The non-oriented ... in $\Bbb RP^5$.

Balarka Sen

Right.

I agree.

Ted Shifrin

So there's no link involved with discussing the Grassmannian. But that's how I originally came up with that question. Of course, it generalizes to other dimensions.

Balarka Sen

9:09 PM

@TedShifrin Hmm, ok.

Ted Shifrin

There are some cool advanced questions about blow-up and intersection numbers later on, too, @Balarka.

But good stuff in G&P, too.

At least you finally made it through that damn calculus stuff. :D

Balarka Sen

I have been progressing rather sluggishly through G&P for the past few days, mostly due to school.

Ted Shifrin

Well, passing school courses is advisable.

Balarka Sen

And today I fell a bit sick :(

Ted Shifrin

Not again!

Balarka Sen

9:10 PM

@TedShifrin Heh.

Ted Shifrin

(I had to get treated for a sinus infection in Chicago. Seems I get one every time I travel.)

OK, I have errands to run and then have to help young'uns (2nd to 7th grade) with homework. Good night, @Balarka. Get better.

Balarka Sen

Oh, sorry to hear that. Sinus infection can be horribly painful.

G'night, @Ted. Hope to see you around more.

Danu

Amen

Balarka Sen

@Danu Do you know the intersection interpretation of cup product already?

Danu

No, we introduced the cup product 2 lectures ago.

Balarka Sen

9:15 PM

Ah.

Danu

So far, we have definitions + examples.

Balarka Sen

Well, if you know the cup product structure of $H^*(M_g)$, then maybe it's worthwhile to compare the table made up of cup product of different generators of $H^1(M_g)$ with each other with the table made up of intersection (modulo perturbation/homotopy) of curves in $M_g$ representing generators in $H_1(M_g)$ with each other

Just sayin'.

Happy?

:P

user 1618033

This place becomes more and more boring, never the tough math I like (or exceptionally rare).

@MickLH aren't you bored without doing very hard math (I mean very hard for you)?

Balarka Sen

E.g., $\alpha \cup \beta$ is a generator of $H^2(T^2)$ where $\alpha, \beta$ are the two standard generators of $H^1(T^2)$. Correspondingly, the meridian and the longitude in $T^2$ intersect at a single point.

Danu

Yeah, I managed to do it for the torus, lol

Balarka Sen

9:21 PM

$\alpha \cup \alpha = 0$, and correspondingly a chosen meridian can be "perturbed" so that it becomes disjoint from itself.

hi again

Hello.

so what's good

@Danu It's a general fact. Embedded $k$-submanifolds of a (oriented closed compact) $n$-manifold $M$ always represent homology classes in $H_k(M)$ (triangulate it so that the inclusion map becomes a cycle), and those homology classes represent "dual classes" in $H^{n-k}(M)$. Cup product of two such fellows represent the "dual class" of the "intersection" of the two embedded submanifolds. It's just a fact worth keeping in mind, is all.

But I'll stop rambling if you don't want me to.

The slogan is that "cup product is dual to intersection"

@MikeMiller I worked my way through transverasilty theorem, but then I fell ill. So no good news. (I flipped through a complex analysis book, but that's not worth mentioning)

Danu

@BalarkaSen Keep going.

I'm reading it

Balarka Sen

9:33 PM

I think looking at more examples and comparing the tables like you did for the torus would be more enlightening than a rambling right now. But I don't know if you know nontrivial examples of cup product structure of any more manifolds.

Danu

That's all we did, so far.

Balarka Sen

Thought so.

Mike Miller

surfaces and flavors of projective space are probably the main insightful ones

products and connected sums too, I suppose

Danu

I'm guessing the FIRST thing we do next lecture is the connected sums of $\Bbb R\mathrm P^2$

Balarka Sen

first you have to compute H_*(RP^2)

Danu

9:34 PM

Done, already.

With a very neat trick :)

Balarka Sen

Oh, that's great. What's the ring?

I can tell you something interesting at this point then.

Danu

Don't know off-hand and I don't have the time to figure stuff out right now, sorry.

I gotta finish the top. notes

Balarka Sen

Ah, alright, then. I was going to tell you a proof of a version of Bezout's theorem using topology. But it's fine.

Danu

The 6 courses thing is really harsh on my free time :P

Mike Miller

i told you it was a bad idea

I used to say 3 things was my learning limit, but I think it's actually two

Danu

9:37 PM

Oh, I agree (for properly learning nontrivial math).

But the physics stuff is just a time-burner---everything comes reasonably easily so I don't have to focus on it.

The weekly problem sets just eat time.

I'm properly doing gauge theory + topology

half-assing Riemann surfaces

and then there's the 3 physics courses

Hiroto Takahashi

I have a question

I don'r quiet undetstand well this question

"Show that every absolute value is continuous in its own topology"

any hint?

Balarka Sen

on a completely different note, I watched Lost Highway earlier today.

Mike Miller

@HirotoTakahashi When you have an absolute value on a field, it gives the field a topology, with a basis for the topology being given by the sets $U_{x,a} = \{y \in K : |x-y| < a\}$. Agreed?

Hiroto Takahashi

@MikeMiller okay, I agree.

Danu

@BalarkaSen I watched La Strada recently.

Balarka Sen

9:42 PM

I liked it, but I felt somewhat sick watching it.

@Danu never heard of it

Danu

La Strada

La Strada (The Road) is a 1954 Italian drama film directed by Federico Fellini from his own screenplay co-written with Tullio Pinelli and Ennio Flaiano. The film portrays a brutish strongman (Anthony Quinn) and the naïve young woman (Giulietta Masina) whom he buys from her mother and takes with him on the road; encounters with his rival the Fool (Richard Basehart) end with their destruction. Fellini has called La Strada "a complete catalogue of my entire mythological world, a dangerous representation of my identity that was undertaken with no precedent whatsoever." As a result, the film demanded...

Shame on you ;D

Mike Miller

@HirotoTakahashi So this gives us a notion of continuous map. They're asking you if $K \to \Bbb R$, with this topology on $K$, is continuous.

You should write down precisely what continuous means and work it out.

Hiroto Takahashi

@MikeMiller thank you :)

Mike Miller

@BalarkaSen So what are your goals for the next 3 months? You should have some mildly long-term goals, perhaps a little aggressive, to keep you moving.

Balarka Sen

Erm, hrm, I want to learn manifold theory. But I also want to learn Riemann surfaces, hence complex analysis before that. But I am not sure if these two are consistent goals, so I am trying to stick to difftop.

Danu

9:57 PM

I think that what you're going to get from learning about Riem. surf. may depend to a large extent on what you're looking for: Geometry or Algebra.

But don't trust me, I don't know, really.

Mike Miller

Pick something concrete and ambitious and then do it.

Balarka Sen

What I don't want to have is unused pieces of what I learnt previously. I don't want to leave mult. calc. unused, so studying manifold theory for it. But I also don't want to leave the bit of alg. geo. I learnt unused :S

@MikeMiller I want to learn complex algebraic geometry. But I suppose that is too ambitious.

Danu

Maybe not concrete enough

Balarka Sen

As in, topology of complex varieties.

Semiclassical

@user1618033 worse, though the bifurcation point between "not great" and "bad" was between now and when you sent that message

namely, i just went to take the road test for my driver's license. based on my previous remark, you can probably guess the result.

Mike Miller

10:00 PM

You should know Riemann surfaces and, like, the literal basics of manifolds before you try to do that. G&H is not an easy book, and you're expected to be pretty alright at differential geometry. Ted could tell you about that. But it's fine to have general goals.

Do you want to do that because it combines thing you know, or out of some innate inherent interest? Go for the latter. The former is whatever.

Semiclassical

@balarka As an example of something concrete and ambitious, learn about elliptic functions and theta functions.

Balarka Sen

@Danu I don't know what classifies as concrete, but I was under the impression it was more concrete than schemey stuff one usually encounters in algebraic geometry.

Mike Miller

@Semiclassical Concrete to me meant like, some specific goal, like read a book etc

@BalarkaSen Concrete as in meaning explicit goals, is what Danu was saying, I think.

Semiclassical

ah. i was thinking some concrete application of it

Balarka Sen

@MikeMiller Both of them, I think.

Danu

10:02 PM

@Semiclassical The real question is why anyone would care about that (and no, string theory doesn't count ;D)

Semiclassical

my taste for complex curve stuff, though, is decidedly not in a modern/schemey sense

well, for one, it's pretty weird and cool @danu

for another, my research about two years ago revolved around figuring out how to apply Picard-Fuchs equations in the context of semiclassical analysis of QM

Mike Miller

@BalarkaSen OK. You've had a lot of big things you wanted to learn before. Think about 'em, again do not worry about whether you know much about it or whether it builds on things you know or how far away it is, and pick one.

Semiclassical

so that was actually pretty useful, albeit in a quite specialized way

Pick a mountain, and start walking towards it? @MikeMiller

Mike Miller

da

Forever Mozart

@Danu I liked Nights of Cabiria better than La Strada

Danu

10:06 PM

@ForeverMozart Still on my list.

Balarka Sen

@MikeMiller What do you suggest would be an appropriate goal for me? I have been off a specific goals after I finished the basics in Hatcher, and I don't know what I would like to do. It'd be nice if someone who knows the kind of taste I have could suggest a specific goal I'd want to look at. I mean, I don't know of too many options other than complex geometry.

Forever Mozart

Have you seen Satyricon?

Mike Miller

I find Fellini's Satyricon to be pretty tedious

Danu

No, I'm only breaking into the Italian directors now.

Balarka Sen

But if you don't have any specific suggestions, it's alright, I understand.

Mike Miller

10:07 PM

Like it's all about decadence and dreamlike imagery but I felt like I'd seen enough about half an hour in

Danu

I liked The Passenger by Antonioni a lot.

Forever Mozart

i liked it a lot, it stays with you

Satyricon that is

never seen an Antonioni film

Mike Miller

@BalarkaSen I mean, I want to get something predicated by your taste, not mine. Complex geometry is a nice goal, as are other flavors of geometry (symplectic, contact, foliations) which in my mind are a bit more topological in flavor. I also know them better. One could also learn some classical (and very important!) topology: characteristic classes, Morse theory. One could investigate the topology of 3- or 4-manifolds (again, obviously this is my taste, which is why I was hoping for input).

To do more or less all of these you need to learn more about manifolds. But I don't know if this is actually what you want, and not just what I want.

Danu

Morse theory is on my list---pretty much at the top.

Balarka Sen

I don't think I can choose anything specific out of the things you listed - I have been told bits about those (except symplectic/contact geometry - I have no idea what those are), and I thought all of them are pretty interesting.

user 1618033

10:15 PM

@Semiclassical no worry, you can try again I guess.

Semiclassical

true enough. just frustrating

Balarka Sen

So since studying all of them requires smooth manifolds, I think I should go on with smooth manifolds instead of choosing anything specific?

After I am done, I can choose what I want to. I think that'd be a good idea.

user 1618033

@Semiclassical The story with the driver license is one of the nicest one here.

Mike Miller

I guess, sure. One can get little tastes of everything after one has the basics they need. So sure.

G&P-style differential topology is less necessary but still often useful.

user 1618033

@Semiclassical I remember that from the group of our instructor I was the only one for which the instructor went to my parents and told them not to have expectations from him that I won't succeed to pass the theoretical exam (we were four or five there in the group).

@Semiclassical: The result: I got the only perfect score from all people there (maybe 200 in that session), and 3 or all from my group failed (not remember now).

Perfect score means 26/26.

Balarka Sen

10:20 PM

I am sorry if I am giving the impression that I am not liking what I am studying by being slow with G&P, in case you think that - I need some time to get used to the new schedule I have; I'd definitely want to study it.

Mike Miller

and of course this is all manifolds; I have said nothing of fascinating algebraic topology, of which there is a lot (a lot to digest even from Hatcher's webpage), or algebra, which I do not understand

user 1618033

@Semiclassical The thing is that my instructor taught us not proper stuff and I didn't want to attend his sessions, and hence his conclusion.

Driver license here is like an IQ test, really. For me it was pretty fun.

Balarka Sen

Honestly speaking, I am also a bit stuck at the 2nd chapter in G&P. I feel like I can get unstuck if I concentrate only a bit more, but the ideas in the proof of epsilon nbhd theorem, say, isn't coming to me naturally so I feel an obstacle in trying to go through it with the amount of attention I can afford to it, given the circumstances. But I do believe this will go away once I can manage my schedule and pay more attention to it.

Mike Miller

I don't remember it. Feel free to teach it to me.

user 1618033

@Semiclassical well, don't worry about the test, you simply weren't in a good shape/day, it happens.

Balarka Sen

10:24 PM

Thanks, I do feel like talking about it probably will help a lot, but I didn't want to bother you with non-genuine problems since you said you'd be busy.

Semiclassical

right.

Mike Miller

i'm stuck right now. go ahead

user 1618033

@Semiclassical hehe, at that time for me was more like a challenge and I wanted to give him a lesson. After the exam he never talked to me again, never greeted me.

@Semiclassical but I tell that despite the theoretical score, and the very good practical score, I consider myself poor at driving, unskilful.

Balarka Sen

@MikeMiller er, right now? I was hoping we could do this later, since I am a bit ill today.

Mike Miller

k

user 1618033

10:31 PM

@Semiclassical I very rarely drive a car. But it's OK for now.

Anyway, trying to get some sleep now. Tomorrow (well, today I mean) early in the morning I'll have some tutoring sessions.

Semiclassical

night @user1618033

Balarka Sen

I'll go to sleep, then. Thanks again for your time, @MikeMiller.

Kari

Any ideas on showing the Frechet differentiability of $f$ at the origin where $f$ is defined as below? $$ f(x,y) = (x^2 + y^2)sin \left( \dfrac{1}{\sqrt{x^2+y^2}} \right) $$

robjohn

11:37 PM

@TedShifrin so?

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$Mathematics$ Mathematics