Mathematics

$n$ children sit in a line. They can change places, but each child can only move by one place at most. How many possible configurations can they reach?

fact: $f(n) = F_{n+1}$. This was sort of unexpected when I found it.

Balarka Sen

Fine, I guess. You're thinking about anything interesting, @AlexWertheim?

Alex Wertheim

Not to be a bother, but you should probably ping me as AlexW. I keep getting Alex Clark's pings, and I don't imagine he wants mine anymore than I do his =P

user147690

@BalarkaSen I wondered why you phrased it like that haha

Balarka Sen

yikes, yes.

Kaj Hansen

@SohamChowdhury, a post of mine regarding involutions: http://math.stackexchange.com/questions/795541/involutions-of-s-n/795544#795544

Not sure if that's what you're looking for.

Alex Wertheim

5:08 AM

Not at the moment though, @Balarka. I'm moving tomorrow/Monday, so there hasn't been much time for math in the last few days. :(

What have you been working on?

How goes it @Kaj? Have you started your REU yet?

Soham Chowdhury

@KajHansen I knew another derivation of the same formula. Yours is sort of "mature" and cool.

Balarka Sen

Nothing much, done some linear algebra -- now I'm gonna do a bit of multivariable calculus. Trying to answer algebraic topology question in MSE to keep myself from forgetting the little algebraic topology I have learnt.

Soham Chowdhury

Multivariable calc from?

Kaj Hansen

@AlexWertheim, I start in 9 days!

Alex Wertheim

That sounds pretty good. It doesn't seem like you know "a little" algebraic topology, quite a bit really, but that's nice. :)

Soham Chowdhury

5:11 AM

What's an REU?

@AlexWertheim He's like that.

Balarka Sen

@SohamChowdhury Ted's book.

Clarinetist

I did something wrong in Gran-Schmidt... could someone check my answer?

1

Q: Find an orthogonal basis for the space spanned by the columns of the given matrix.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ It is immediately clear to me that the columns of $X$ form a basis for the space spanned by the columns of $X$. How does one generate an orthogonal basis for the space spann...

My attempt:

To apply Gran-Schmidt, I used
$$u_k = v_k - \sum\limits_{j=1}^{k-1}\operatorname{proj}_{u_j}(v_k)$$
where $$\operatorname{proj}_{u_j}(v_k) = \dfrac{v_k \cdot u_j}{u_j \cdot u_j}u_j\text{.}$$
For this problem, $k = 3$. We have $$u_1 = v_1 = \begin{pmatrix}
1 \\
1 \\
1 \\
1 \\
1 \\
1\end{pmatrix}\text{.}$$
Now $$\operatorname{proj}_{u_1}(v_3) = \dfrac{v_3 \cdot u_1}{u_1 \cdot u_1}u_1 = \dfrac{10}{6}u_1\text{.}$$
Thus, $$u_2 = v_2 - \dfrac{10}{6}u_1 = \begin{pmatrix}
-2/3 \\
1/3 \\
4/3 \\
7/3 \\
10/3 \\

Alex Wertheim

That's exciting @Kaj, have you been doing much reading for it?

Clarinetist

Clearly $u_1 \cdot u_2 \neq 0$.

Kaj Hansen

Some, but not as much as I'd like. I'm going to hit the literature hard over the next week @AlexWertheim

Alex Wertheim

5:13 AM

Very neat @Kaj. Have you given thought to where you will be applying to grad school in the fall? :)

Kaj Hansen

Fortunately, we're going to get a crash course in everything in the first week and a half, but I want to come in prepared and not seeing things for the first time.

Balarka Sen

@AlexWertheim nah, all the cool stuffs are the ones I don't know about.

I haven't done (singular) cohomology for one. But I think I will get started on it at some point of time.

Soham Chowdhury

@BalarkaSen Can't you just study singular homology and reverse all the arrows? :P

Balarka Sen

There's quite a bit to it than being dual of homology, really.

Alex Wertheim

Well, at least you know what you want to cover @Balarka, so that sounds good.

Balarka Sen

5:15 AM

A graded ring structure comes naturally, for example.

It's the contravriance of $H^\bullet$ that makes all this happen.

Kaj Hansen

@AlexWertheim, not really to be honest. My grades are not that good; my GPA is a 3.4 right now, and maybe a bit lower if we restrict to just my math courses. That's rather discouraging to me as far as applications go. I will need to talk to my advisor and see where I might have a reasonable shot, also considering that I think I want to focus on algebra.

If I'm not mistaken, I will have taken >15 major-level math courses though by the time I graduate. Mediocre grades, but lots of exposure. I hope that counts for something.

Balarka Sen

@AlexWertheim Yeah, well, I was actually thinking about doing de Rham stuff before singular cohomology, as the former would be more geometric.

I will do both, but not sure which one I'd like to do first.

Alex Wertheim

Good stuff, @Balarka. The de Rham stuff is really fascinating, though I don't know much about it. I've heard about it more in the context of differential geometry.

Clarinetist

NVM I found what I did wrong

Alex Wertheim

When I finally get around to studying algebraic topology, I think my first big goal will be understanding the idea behind the etale cohomology.

Balarka Sen

5:18 AM

Yeah, I will do differential topology after I do a bit of mult. calc.

@AlexWertheim etale cohomology is not really algebraic topology :)

coincidentally, it's on my list of to-reads too.

Alex Wertheim

I see, @Kaj. I hope you're not too discouraged. Background really is important, and it's good to take hard/meaningful classes over just GPA preservation. You seem like quite a talented student, so I hope you do apply some top places. :)

@Balarka: really? I know of it as a major tool in arithmetic geometry, but I figured one must know some algebraic topology to understand what's happening.

Balarka Sen

it's something picked up from algebraic topology, but as far as I have heard, the flavour's completely different.

Kaj Hansen

Thanks @AlexWertheim ! Fortunately I got an A in both of my math courses this past semester, so hopefully they interpret that as some degree of improvement over my freshman and sophomore years.

The Substitute

off topic - is "show 7^(120)-1 is divisible by 143" a problem in groups? I can solve it, but now I wonder if there's a way to solve it using group theory.

Alex Wertheim

I'm sure it will, @Kaj. Not to mention, rec letters play a huge influence. If you have well-known mathematicians who can vouch for your skills, that can make a major impact on things.

@Balarka: interesting. I'm not sure whether that's a disappointing revelation or a relief, lol.

Kaj Hansen

5:23 AM

Indeed. I'm thinking about asking Pete Clark to write one of my letters, and also someone to speak to my algebra skills.

Balarka Sen

but you should study algebraic topology nevertheless. the ideas are from there.

@AlexWertheim any plans on what book you'll use?

EGA/SGA? (/kidding)

Alex Wertheim

For algebraic topology @Balarka, or the etale cohomology?

Soham Chowdhury

@TheSubstitute You might "work in $\Bbb{Z}/143\Bbb{Z}$". But it's really NT.

Balarka Sen

etale stuff.

The Substitute

@SohamChowdhury that's true i was using algebra without even knowing it ;)

Mike Miller

5:26 AM

@AlexWertheim: You should talk to John Yu. He's organizing an etale cohomology learning course this summer (before you get here). He'll be able to tell you what they're using.

I think you need to know Hartshorne beforehand, in practice.

Alex Wertheim

Ah, thank you, @MikeM. I'll definitely talk to him. It's a bit early, to be sure, but I do like to have some guides on my reading as I'm going along.

Balarka Sen

What are the prereqs for Hartshorne? Classical algebraic geometry/commutative algebra?

I only know a bit of commutative algebra and that's about it.

Mike Miller

Lots of commutative algebra. First chapter talks about a chunk of classical stuff. Eisenbud's book or the first halfish of Matsumura's should suffice.

Balarka Sen

I asked prof, and he said I should read the first chapter of Hartshorne once I have enough background, and then do Szamuely. Not sure if it's a good idea, though.

yikes.

@AlexWertheim is ahead of me on that :P

Alex Wertheim

Haha, uhh, I'm not so sure about that @Balarka. Hopefully I'll be progressing pretty soon though.

Balarka Sen

5:31 AM

well, you're doing heaps of commutative algebra from A-M for one.

Alex Wertheim

Planning to do lots, anyway. I've got to put my studying where my mouth is, first. :)

Balarka Sen

I did a few classes, did a few exercises, and threw away the book.

It's maddening to work out exercises from a book with no pictures.

(mostly because it reminds me of the horrific experience with the first few chapters of Rudin)

Alex Wertheim

LOL. I don't think I have the same geometric abilities or hangups. I've always had a very hard time visualizing things.

user147690

If $\tau=\{\emptyset,X\}$, then every sequence in $(X,\tau)$ converges and every $l\in X$ is a limit of a sequence. Is this because a sequence converges to a limit $a\in X$ if every open set containing $a$ contains all but a finite number of points of the sequence, and the only open set we can take is $X$?

Balarka Sen

@AlexWertheim The first chapter of A-M was ok. I got to draw my own pictures, a lot of topological stuff about $\text{Spec}$ to prove, etc. It got steadily non-geometric afterwards.

The only chapter I've had fun with afterwards was the one that talks bout Noetherian rings, but probably that's just because I have done an algebraic number theory class concerning those and I've got the motivation.

Alex Wertheim

5:35 AM

Well, this may be putting the cart before the horse, but maybe it'd be good to learn a little algebraic geometry alongside AM, @Balarka? That way you'd get the nice geometric picture which comes out of all that algebra.

Balarka Sen

I have no idea, I don't know any algebraic geometry.

Most of my motivation for com. alg. comes from algebraic number theory.

Clarinetist

Let $$X = \begin{pmatrix}
1 & 1 & 4 \\
1 & 2 & 1 \\
1 & 3 & 0 \\
1 & 4 & 0 \\
1 & 5 & 1 \\
1 & 6 & 4 \end{pmatrix}$$
Is there an easy way to calculate the form of a vector in $C(X)^{\perp}$, $C(X)$ being the column space of $X$?

Because what is unfortunate is that $C(X) \neq \mathbf{R}^6$ (of course that would be too easy!)

Alex Wertheim

Huh, interesting. Algebraic geometry would give you lots of motivation, especially geometric, to learn commutative algebra. :)

Balarka Sen

I guess I should really do commutative algebra from Dummit-Foote. It has a lot of pictures, and talks about algebraic geometry alongside.

Mike Miller

@BalarkaSen: Just a fair warning that that's not very much commutative algebra.

Clarinetist

5:39 AM

I do have an orthogonal basis for $C(X)$, if that's helpful at all...

Balarka Sen

Well, thanks. Anything you can recommend that'd suit my tastes, @Mike?

Mike Miller

Eisenbud.

Kaj Hansen

Indeed @AlexClark

Alex Wertheim

I've always been intimidated by the thickness of that book, @MikeMiller.

user147690

@KajHansen Thanks

Balarka Sen

5:40 AM

OK, never tried it, despite Ted's suggestions. I'll have a look, thanks.

Gotta run now.

Mike Miller

@AlexWertheim: There's a lot of commutative algebra.

Alex Wertheim

See ya, @Balarka.

Kaj Hansen

IIRC, uniqueness of limit points comes guaranteed only with the Hausdorff axiom.

@AlexClark

Mike Miller

Yes, that's correct.

user147690

@KajHansen Yep that is what I was about to learn

Alex Wertheim

5:41 AM

Sure @MikeMiller, just all in one place? :)

Obviously, one need not cover the entire book.

Mike Miller

No, only a little bit of it in one place, @AlexWertheim. :D

user147690

@KajHansen So we want all points to be in their own neighborhoods?

Kaj Hansen

Can you elaborate @AlexClark ?

user147690

What does neighborhood mean outside of a metric space though?

Alex Wertheim

Hahaha, fair enough, @MikeM. I suppose I'm only measuring using my already infinitesimal knowledge =P

(Which is never a good idea, really...)

Kaj Hansen

5:44 AM

@AlexClark, a set $U$ is a neighborhood of a point $x$ if $x$ is in the interior of $U$

user147690

Okay I think I want for all $x_i\ne x_j$ that $x_i\in I$ and $x_j\in J$ and $I\cap J = \emptyset$ then the space with this property for all points is Hausdorff

Kaj Hansen

Or in other words, there is an open set $V$ with $x \in V$ and $V \subset U$.

Oh yes, if there does exist open sets $I$ and $J$, then that does correctly describe a Hausdorff space @AlexClark

user147690

"Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be "housed off" from each other by open sets."

Mike Miller

@AlexWertheim: None of us know anything. Every year, we learn more at an exponential rate, and we learn how little we know at a superexponential rate.

3

Kaj Hansen

Indeed!

Clarinetist

5:47 AM

0

Q: For a given matrix $X$, find two linearly independent vectors in $C(X)^{\perp}$.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ Is there an easy way to calculate the form of a vector in $C(X)^{\perp}$, $C(X)$ being the column space of $X$? What is unfortunate is that $C(X) \neq \mathbf{R}^6$ (of cou...

linear-algebra vector-spaces orthogonality

Alex Wertheim

Haha, reminds me of that Gatsby quote, @MikeMiller: "So we beat on, boats against the current, borne back ceaselessly into the past."

Kaj Hansen

@Clarinetist, do you know how to find the null space of a matrix?

Alex Wertheim

Blurgh, long day tomorrow. Better get to bed now. Goodnight, friends.

user147690

Goodnight @Alex

user147690

(I'm allowed to do that :P the short way)

Soham Chowdhury