Mathematics - 2016-12-04 (page 3 of 8)

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Abhishek Bhatia

5:00 PM

0

Q: Least-squares Projection

In least-squares for n-points, do we find n-dimensional hyperplane on which all these points lie? If yes, then say I want a line(2-D hyperplane) which is a least-squares approximation. Is that 2-D hyperplane a projection of the n-dimensional hyperplane we found earlier.

linear-algebra least-squares linear-regression projection-matrices

5:16 PM

@Ted Hi

Comment vas-tu ?

Hi @Astyx @robjohn and tern

Ça va, merci. Qu'est-ce qui se passe là?

Sur le chat ? Je sais pas exactement

Non, non, plus loin :)

@TedShifrin Hey there. Are you back home?

Comment ça ? Je ne saisi pas

5:18 PM

LOL, I've been back for 3 weeks or so, robjohn :P

@TedShifrin well, you seemed to be going places so much, I don't know where you are.

I wish my life were that exciting, @robjohn :P

But I am trying to plan a longer trip or two for the next year.

Are you still in touch with our series/integral master? Although I found her somewhat annoying, I sort of miss seeing the occasional interesting question.

Hi @Ted

Hi @Alessandro :)

Hi @Ted

why is only half my pic loading?

5:27 PM

I was wondering the same thing @meow

Hi @meow. I see your whole mugshot :P

I see half of it on the right side of the screen but all of it next to your messages

I just want to do a sanity check on my end, but given any connected, non-negatively weighted graph $G$, is the tree $T$ with the maximal weight from a maximum weight forest $F$ derived from $G$ going to always be spanning? (N.B. the graph in question is connected, and its edges non-negatively weighted.)

5:51 PM

1

Q: Surface integral (divergence theorem )

Evaluate: $\int \int \bar{F}d\bar S $, $ \bar{F}=(2-x^{2}yz+y^{3},xy^{2}z+ye^{z},y^{2}+z+e^{z})$ $\gamma : y^{2}+z^{2}=x^{2} $ between the planes $x=1$ and $x=2$ , The normal pointing away from the $x$-axis. Thanks in advance.

multivariable-calculus

Need help with this

hi

:)

0

Q: Set $\mathbb{K} = \mathbb{Q}((2)^{1/8},i)$. Prove that $Aut(\mathbb{K}/ \mathbb{Q}(\sqrt{-2})) = Q_8$.

I first proved that $G = Gal(\mathbb{K} / \mathbb{Q}) = D_8$. Since, if we consider the following maps $\sigma_1(2^{1/8}) = (2)^{1/8} \mapsto (2)^{1/8}\omega$ and fixig i. Also, we consider $\theta_1(i) = -i$ and fixing the other root. I proved that those maps generate the whole galois group. I ...

group-theory field-theory galois-theory

6:14 PM

@Ted im confused about the dual projective plane

Hey @TedShifrin

@TedShifrin o/

I'm (still) looking for some nice 'n' easy examples of bundles with nontrivial Euler class---are there any simple symplectic or complex manifolds with nonzero Euler class? Or perhaps some nice manifold with half-dimensional submanifolds of nonzero self-intersection (nontrivial Euler class for the normal bundle)? I've already got surfaces, $\Bbb CP^n$ and $\Bbb CP^1\subset \Bbb CP^2$ (for nonzero intersection number).

can someone give me irreducible polynomial with galois extension isomorphic to $Q_8$ and $Z_2 \times Z_4$ ?

@Hippa: Tu existes toujours!!??

6:21 PM

@TedShifrin On dirait :-) mais je ne fais pas de maths pour l'instant. Je recommence dans 4 mois environ

Interesting ... Somehow I missed all these pings.

@Hippa: dommage ... que fais tu alors?

@Danu: Think about adjunction for hypersurfaces of $\Bbb P^n$ ...

@TedShifrin Je suis en stage en gendarmerie. 7 mois au total, 3 de formation et 4 sur le terrain

Why are you confused, @meow? A point of the dual plane is a line in the original plane.

Ah... je ne savais pas ça, Hippa. Bonne chance!

@TedShifrin Merci :-)

@user51189 we are interested in some common subjects. If you need references about free lectures notes, for example about analytic number theory, call on me here. Good afternoon all users, bye.

6:26 PM

the idea is that the usual map from the plane to the dual preserve incidence and turns inclusions around @meow

There is no map, @Alessandro.

@TedShifrin I'm actually not too sure how to use that to get concrete examples. You mean the formula (fora hypersurface $Y\subset X$) $K_Y\cong K_X|_Y\otimes \det(\mathcal N_{Y/X})$, right?

I'm very annoyed. Apparently a new user doesn't know the etiquette: You don't just remove your question when you have understood the answer. How do we educate?

attack viciously

@Danu: So you can say precisely what the canonical bundle of a hypersurface of degree $d$ is.

6:28 PM

@TedShifrin I can't see deleted questions on math.se---can you still flag?

tern, but how? :D

If so, you should flag the post for moderators and explain the situation.

I voted to undelete, @Danu.

Maybe I should flag.

If not, go to a post by the same user and flag there, explaining that the flag is really for a different post.

@TedShifrin Also good.

@TedShifrin Okay... But I want the Euler class of its tangent or normal bundle (I don't know how Euler class behaves under linear algebra operations except direct sum)

@TedShifrin so the dual plane is the set of all lines of $\mathbb{P}^2$?

6:30 PM

yes, @meow

ah ok

In coordinates, homogeneous coordinates of a point in the dual plane gives you the normal vector to the plane in $\Bbb R^3$ which is a line in $\Bbb P^2$.

@TedShifrin I dunno how it behaves under tensor or exterior products, honestly.

I mean... I do know if the Euler class is just the top Chern class... But even if that's true (it is, isn't it?) I don't want to invoke that in my seminar talk.

so, if we consider their analogues in $\mathbb{R}^3$, the dual of a point in $\mathbb{P}^2$ (line in R^3) is the point in $\mathbb{P}^{2*}$ (a plane in R^3) which is perpendicular to that "line" in $\mathbb{R}^3$

Oh, wait, @Danu, I lied. What I said applies to $c_1$, not $c_n$ ... unless we're doing curves. But do curves in $\Bbb P^2$.

6:32 PM

@TedShifrin For curves I know the genus-degree formula... I guess.

Right. Or it's a special case of adjunction.

@meow: "dual of a point" doesn't make sense

In any case, I don't think I want to use examples that use non-zero amount of complex geometry. Any super easy symplectic examples? :\

Or submanifolds with nonzero self-intersection except $\Bbb P^1$ in $\Bbb P^2$? :P

I don't think about symplectic stuff unless it's complex.

@TedShifrin I don't mind Kaehler, if you know anything easy :P

All submanifolds have nonzero self-intersection

6:34 PM

Submanifolds of what?

of $\Bbb P^2$

Oh.

assuming you're talking complex submanifolds

They do?

Yes.

Intersection numbers are always nonnegative in complex land. We talked about this ages ago.

6:35 PM

Hmm... So there are no Lagrangian tori in $\Bbb P^2$?

A lagrangian torus would not be a complex submanifold, would it?

IDK :P

Couldn't it? :P

What's the definition of a Lagrangian submanifold?

@TedShifrin Well, nothing complex per se but it could also happen to be a complex submanifold, no?

Not if we just proved it couldn't. :)

6:37 PM

@TedShifrin By this, you mean strictly positive, I guess?

Sure.

Well, no.

Oh yeahhhh

Wait, no?

In the doubly ruled surface $\Bbb P^1\times\Bbb P^1\subset\Bbb P^3$, two rulings in the same family are disjoint, therefore have 0 intersection number.

In $\Bbb P^3$ okay

In $\Bbb P^2$ it's true though, right?

In a surface in $\Bbb P^3$.

Right, homology tells you it can't happen in $\Bbb P^2$.

6:39 PM

The symplectic geometry gives me a lot of more geometric ways of thinking about some characteristic classes stuff---he intuition the lecturer gave was the following

Fix one of the two charts: Either two lines intersect in this chart, or they are parallel---then they intersect "at infinity" (i.e. at the one point the chart misses)

Nothing symplectic so far.

@meow knows all about this :)

@TedShifrin I have no idea what rulings are (or a ruled surface).

lines

which field extension gives $Z_2 \times Z_4$ as galois group ?

This is the quadric surface $z=xy$ ...

Don't know, Karim.

6:41 PM

what about $Q_8$ ?

I really hate questions like this

I'm slightly annoyed by the lack of simple complex & symplectic manifolds.

Go look carefully at D&F.

how can you even solve it without brute forcing

Huh? @Danu

There are zillions of "simple" examples. You just haven't learned them.

@TedShifrin But then tell me some---really simple ones.

6:42 PM

I just did.

So algebraic subvarieties of $\Bbb P^n$

What more?

Curves in $\Bbb P^2$, surfaces in $\Bbb P^3$. In particular, smooth quadrics in $\Bbb P^3$ are projectively equivalent to the standard one I just said.

That's a ton. For non-algebraic, you have the Hopf surfaces.

The Hopf surfaces have zero Euler characteristic :(

Yes, they do.

I want nonzero Euler class :P

6:44 PM

Take the quadric surface in $\Bbb P^3$. It's got $\chi = 4$.

You can prove this at least two ways if you've listened to me. :)

Haha

at least

Well, I need to know $e = c_2$.

I think we did a proof in the course on four-manifolds last week---except for proving that the homology class of the zero set of a degree $d$ polynomial is really $d$ times the generator (of $H^{2n-2}(\Bbb CP^n)$). How do I see that?

But $S^2\times S^2$ ... I think you can compute that directly.

Poincaré duality, @Danu. Intersect with a line.

@TedShifrin Mhm... I kind of see that.

6:46 PM

Oh, and the fundamental theorem of algebra :P

Yeah, yeah

That basically runs like this right:

Say we have something like $\sum_{j=1}^4 z_j^4=0$ (this is for $\Bbb P^3$), then intersectin with a line (let's take the standard copy of $\Bbb P^1$) is like setting $(z_3=z_4=0)$. Then we have $z_1^4+z_2^4=0$ but since we're working in projective space we just solve $(z_1/z_2)^4=-1$, which has four solutions.

Right. You happened to pick a line that was transverse, so you got 4 distinct points. Yippee. :)

Generic somethingsomething

Sard/Bertini.

Bertini??

6:50 PM

That's the algebraic geometry version of Sard.

Generic element of a linear system is "good" (e.g., smooth).

I should learn that

I should learn complex algebraic geometry sometime.

not saying a word

Do you have recommendations for books? There is such a huge variety (lel)

@TedShifrin ?

danu my prof gave me many recommended books for algebraic geometry

I could email the list to you.

You will do better with Griffiths & Harris because it's more analytic/geometric. You don't want a super-algebraic treatment. Griffiths & Harris have some mistakes, but good intuition.

6:51 PM

I have a list...

135

Q: Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion per answer please. Also, please include an explanation of why you like the book, or what makes i...

ag.algebraic-geometry books big-list textbook-recommendation

@TedShifrin You think it's not essential to learn dat algebra at some point?

To get more familiar with examples, per se, look at Harris's Algebraic Geometry (except for its ugly typesetting).

@TedShifrin Good ol' G&H... Hmm.

@TedShifrin That's more basic, it seems, right?

@Danu: You don't need to learn scheme-theoretic derived functor stuff at this juncture. If it proves necessary later, then do it.

Harris's book is less about machinery and more about examples.

@TedShifrin Hmm

It's hard to prioritize things :P

It seems that now, I won't be needing much complex geometry at all for my thesis..

Perhaps not. No big deal.

6:55 PM

I've also heard that next semester there will be a seminar on Morse theory, and maybe the one char. classes will continue towards Atiyah-Singer.

Although maybe you'll get examples from hypersurfaces, etc.

Oh, those are both great.

Complex geometry is already paying off majorly in helping me understand 4-manifolds and symplectic geometry (the courses).

Everything is like a more intuitive version of Huybrechts haha

Learning usually pays off :)

I don't think Huybrechts was really my taste in terms of style

What is (that I know)?

6:57 PM

I like G&P a lot... But that was perhaps because it was a light reading after learning (co)homology already.

That's truly an undergraduate text, and a bit rambling. But of course I like it a lot.

I guess I haven't read many books, to be honest.

You like Forster better?

I think it's very clearly written.

hi chat

Forster was pretty good.

6:58 PM

Hi @Semiclassic.

But I didn't like the part about coverings

At the graduate level, you cannot put in all details or the book becomes thousands of pages.

It was very unclear to me, unnecessarily so. Talking to you helped a lot.

Oh, I never taught/read that part.

The bit about going around branching points etc

What Semi talks about ;)

6:59 PM

weeee

LOL

That was not clear in the book---too few examples.

Graduate texts are written so that in general the reader has to do a lot more for himself than ...

I write a lot of stuff myself... Notes n such. I like my own style hahaha

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