Mathematics

8:00 PM

A node like that could split into a pair of branch points under a perturbation e.g. $x^2-y^2=a^2$. I presume that to be the generic case, and the node surviving said perturbation to be nongeneric.

But I don't know what the terminology for the node to 'survive' a given perturbation is.

Astyx

Because this doesn't work in any field, does it ? @Tobias

Tobias Kildetoft

@Astyx Sure it does, you just need the right topology

Ted Shifrin

Right, @Semiclassic. I'm just saying that to me saying "the singular point $(0,0)$ is stable under the perturbation" means that the point itself stays singular, not that it moves.

Semiclassical

Yeah, I understand.

Question is whether there's a proper terminology for that.

Ted Shifrin

I would say something like "perturbation among curves with nodes" ...

Consider a family of curves with ordinary nodes ...

Mike Miller

8:01 PM

You would say that.

Ted Shifrin

@MikeM: Do you disagree (seriously)?

If you say a singular point is stable, what does that mean?

Mike Miller

Nah.

Astyx

@Tobias What topology would that be in, say, $\Bbb Z/p\Bbb Z$ ($p$ prime) ? I'm intrigued

Ted Shifrin

The usual differential topology language is that a property is stable/generic.

Mike Miller

I'm just agreeing with you.

Tobias Kildetoft

8:02 PM

@Astyx The Zariski topology

Ted Shifrin

Oh, ok ...

Astyx

@Ted I don't get what you mean

Ted Shifrin

Forget about it, Astyx.

Astyx

Let me look that up :) @Tobias

Ok

Mike Miller

Trying to get to my hotel via london transit before my phone dies.

Ted Shifrin

8:03 PM

Good luck!

Stop chatting.

Mike Miller

indeed

Semiclassical

The specific case I've got consists of a family of curves with parameters $g,\gamma,E,x$.

Ted Shifrin

$x$ is a parameter? seriously?

Semiclassical

ack

Balarka Sen

by physics standards that's perfectly fine

Semiclassical

8:04 PM

Hrm, yes it is in this context.

Ted Shifrin

LOL

Semiclassical

Which is rather silly

Ted Shifrin

Not when I'm writing equations of plane curves it's not.

Mike Miller

What do you expect me to do for the next hour?

Semiclassical

In the defense of the preprint I'm reading (It's their notation)

The variables on the curve would be $(t,\epsilon)$

Ted Shifrin

8:05 PM

Oh good grief

Balarka Sen

i wouldn't be surprised if $\slashed{\phi}_{ijk}^{lm}$ was a parameter instead

Semiclassical

And they're not doing any plane curves, I'm bringing that in myself.

Ted Shifrin

@Balarka: Go finish Gauss-Bonnet.

Semiclassical

But yeah, for the purposes of this conversation it's rather silly.

Balarka Sen

meh, I won't fix this

Semiclassical

8:06 PM

Hence, I'll instead say I've got parameters $a,b,c,d$. :)

(They also have $e$ as a parameter. Even I can't abide that.)

Ted Shifrin

It seems like you have too many parameters for a hyperelliptic cubic.

Astyx

I'm going, good day to everyone

Ted Shifrin

Bubye, @Astyx

Semiclassical

Eh, it's more symmetric than you'd think.

Ted Shifrin

Anyhow, I'm leaving in a few minutes for lunch. So get to the punchline.

Balarka Sen

8:08 PM

@TedShifrin Alright, thinking about it.

Semiclassical

It's the characteristic polynomial in $x$ of the matrix $$\begin{pmatrix} a-y & 0 & b c & d \\ 0 & a+y & cd & b \\ bc & cd & 1 & 0 \\ d & b & 0 &-1 \end{pmatrix}$$

Akiva Weinberger

Hielo

Ted Shifrin

@Semiclassic: Did you mean for it to be almost symmetric but not?

Hi, DogAteMy.

Semiclassical

Fixed.

Ted Shifrin

OK.

Semiclassical

8:12 PM

When $c^2=\frac{1-a}{1+a}$, the resulting curve $f(x,y)=0$ has a singular point.

Ted Shifrin

Now what?

So you want to perturb along that hypersurface in the parameter space?

Semiclassical

Yeah.

Ted Shifrin

OK, no problem.

You could explicitly substitute $a=a(c)$ and get rid of one parameter.

Semiclassical

And I want to know how one refers to the fact that the curve remains singular along that line.

...which, maybe one just says that.

Ted Shifrin

Not line. It's a hypersurface.

Yup, you just say that.

Semiclassical

8:13 PM

Right, I've got b,d still.

Ted Shifrin

This is analogous to the discriminant subvariety which parametrizes multiple roots.

Semiclassical

Right.

Ted Shifrin

So we're done for now? :)

Semiclassical

I'm interested, then, in perturbations which preserve singularities of the curve.

Ted Shifrin

No, that suggests that the singular points don't move.

Which preserve the singular nature of the curve.

Semiclassical

8:15 PM

Hmm, okay.

Ted Shifrin

Language is subtle.

I'm out of here for now. Back later.

Semiclassical

I'd say "perturbations which preserve the singular-ity of the curve" except that one already has "singularity" as a noun not as an adverb.

later, @ted

Socrates

does induction work for statements about matrices?

Krijn

@Socrates You could do induction on the dimension $n$ of a matrix, yeah

Semiclassical

(not adverb. I should've said: "singularity refers the points on a curve, not to the curve itself.")

Alessandro Codenotti

8:24 PM

@Krijn for a practical example that's how you usually prove that symmetric real matrices are diagonalizable

Daminark

Hello chat

Krijn

@AlessandroCodenotti Linear algebra seems ages ago, it was so much fun tho

Semiclassical

I've said it before, but I always find it a bit funny how 'linear algebra' means different things depending on what level you're talking about.

Balarka Sen

everything is linear algebra

2

(which explains why i am bad at everything)

Semiclassical

To a high-school student, linear algebra would be "simultaneous systems of equations." To a college student taking an intro course (not a full one), it'd probably mean "matrices." And to someone taking a serious course in linear algebra, it'd be stuff like vector spaces.

Balarka Sen

8:28 PM

on a more serious note, i really need to get some of my linear algebra straight: i don't really use a lot of linear algebra.

Semiclassical

I use certain parts of linear algebra a lot.

Eigenvalues-eigenvectors, diagonalization, etc.

Krijn

I don't really use it because every time it comes up the prof will wave his hands and say "this is just linear algebra"

s.harp

with linear algebra do you mean finite dimensional linear algebra?

Semiclassical

And there's stuff which I don't say a lot but which I know is there, e.g. null space. (An eigenvector, for instance, is nothing but an element in the null space of a particular matrix.)

Balarka Sen

@Semiclassical yeah, that's the kind of thing i don't use

Socrates

8:32 PM

@KajHansen hi, how is it?

Semiclassical

Can't do quantum mechanics without eigenvalues/eigenvectors.

Mike Miller

does anyone actively use that?

like that theorem

s.harp

btw if I'm looking a space $C$ of maps $X\to Y$, whats the topology called that makes all the evaluation maps $ev_x: C\to Y$ $f\mapsto f(x)$ continuous?

Semiclassical

Which theorem?

Mike Miller

or rather fact... doesn't really deserve "theorem"

@s.harp pointwise convergence?

s.harp

8:34 PM

@MikeMiller huh, right that makes sense as a name

Balarka Sen

the new neighbors are making hella racket.

Daminark

Functional programmers, huh?

Balarka Sen

i dunno what they are. having a party it seems

Mike Miller

back to advance wars til i get there i guess

Daminark

It was a joke if they're making "Hella Racket"

Mike Miller

8:36 PM

can't much work

Balarka Sen

i wish they'd just conk out after drinking to their fullest

Semiclassical

I finally broke down and downloaded VBA-M so that I could play advance wars.

Balarka Sen

cool

Semiclassical

I then got a bit annoyed with it and decided to download Castlevania: Aria of Sorrow instead :)

Balarka Sen

@Daminark ah i see

@Semiclassical traitor

Semiclassical

8:39 PM

lol

Mike Miller

I'm going through the war room in aw2

Liad

Hi, i got a question im not sure of:
Let F be a field. i need to prove that in the ring $F[x,y] $ $f \in (x,y) $ iff $f(0,0) = 0 $

Assuming $f \in (x,y) $ then $f(0,0) = a_{0,0} $ why does $a_{0,0} $ must be zero?

Sie

8:54 PM

So in the context of sets what is the appropriate way to say inbetween? The way I've been doing it is B = { x|x ϵ 1-100 }. Is that acceptable or is there a proper way to do it?

Balarka Sen

@MikeMiller Have you ever saved both the T- and B-copter in the Green Earth mission? I can't seem to be able to do it

(was replaying that one)

Socrates

@Sie $B=\{x\in\mathbb{R}|1\leq x\leq 100\}$ or with N, if you want integers

Balarka Sen

nvm, did it

Semiclassical

lol

JamalS

In the context of blowing up singularities, does anyone know what is meant by $\mathcal{O}(-2)$?

Balarka Sen

9:06 PM

It's the cotangent bundle over $\Bbb P^1$

Equivalently tensor product of the tautological line bundle with itself

JamalS

@BalarkaSen Thank you :) Can I ask why the notation is as it is?

Oh I see

Balarka Sen

If you take a generic section of it as a real 2-plane bundle, it intersects the zero section at two points, with opposite orientation each

aka the zero section of the underlying 2-plane bundle has self intersection number -2

or it's a bundle with Euler class -2, whatever you want to call it

JamalS

I'll go with the Euler class then, I'm more differential geometry minded :)

Balarka Sen

Me too :)

JamalS

@BalarkaSen On an unrelated note: Would you by any chance know of any resources you could point me to that discuss how to compute the period matrix of an algebraic variety $X$ defined as the zero set of a polynomial, which is a smooth manifold?

Balarka Sen

9:12 PM

I don't really know what a period matrix is. But you can wait for @TedShifrin, who's the resident complex algebraic geometer/differential geometer here.

Also I misread your previous "differential geometry" as "differential topology". I don't really know much about the former :P

Semiclassical

@JamalS You might look at Carlson's 'Period mappings and period domains' book. The first chapter, at least, is pretty readable.

JamalS

@Semiclassical Thanks, I'll have a look.

Semiclassical

@BalarkaSen The period matrix of a smooth genus-g curve has matrix elements $\int_{\gamma_i}\omega_j$ where $\{\gamma_i\},\{\omega_j\}$ are bases of cycles and co-cycles respectively.

JamalS

@Semiclassical Yes, I'm aware of the textbook definition so to speak, it's just that evaluating it by that means is not always possible, or at least not the easiest way.

Semiclassical

Don't I know it.

JamalS

9:18 PM

Hehe

Balarka Sen

@Semiclassical Ah

Semiclassical

The main tool I know is to find the Picard-Fuchs equation for the given elliptic curve and solve the resulting ODE.

That only works in cases where you've got a family of Riemann surfaces parametrized by a parameter, though, and even there it can be pretty hard to actually find the Picard-Fuchs equation.

Joe Slater

Hi

Can someone help me understand how to go from the first equation to the second - imgur.com/a/tH7Gh ?

JamalS

@JoeSlater You can work backwards, using the formula for the sine of a sum.

The phase shift is physically meaningful, and perhaps it's better if you understand this in the context of RLC circuits, as in your problem; Young and Freedman have a good section it.

Semiclassical

It may also be useful to compute $e^{i\phi}=\cos \phi+i\sin \phi$ given that definition of $\phi$.

Joe Slater

9:30 PM

Ok thanks. I will try to read more about it. But I am not able to see how using the eulers formula or formula for the sine of sum is useful in this case.

Simple Art

Say, should I delete this? :

-12

Q: Interesting patterns to the algebraic solutions of polynomials

In yet another attempt to find the solution to the quintic polynomial, I started looking backwards at the solutions to the quartic, cubic, quadratic, and linear polynomials to see if I could pick up any patterns. $\left\{ \begin{aligned} 0&=ax+b \\ 0&=ax^2+bx+c \\ 0&=ax^3+bx^2+cx+d \\ 0&=ax^4+...