mme

general recent conversations

$Mathematics$ Mathematics

Associated with Math.SE; for both general discussion & math qu...

mme

Nov 13, 2022 23:51

OK, see you all in two years unless you show up in my email inbox.

mme

Nov 13, 2022 23:48

I've complained about this before.

mme

Nov 13, 2022 23:48

@TedShifrin This subtlety is rarely discussed in the standard texts. They also love to tell you that if you're minimizing some function with respect to a constraint g(x,y) = k, you can use the constraints to do "substitutions" and minimize a 1-variable function. This is wrong: you may well miss all critical points corresponding to a spot where grad g projects to 0 on the corresponding coordinate axis.

mme

Nov 13, 2022 23:44

One has as a lovely corollary of the work in that chapter that any closed complex manifold has finite-dimensional biholomorphism group. Exercise: show that the additive group of holomorphic functions C -> C embeds into Biholo(C^2), so closed-ness is truly essential here.

mme

Nov 13, 2022 23:44

@LukasHeger This book will be good for your soul, especially the chapter on Hodge theory which --- to my eye --- is the best introduction for someone willing to get their hands dirty and who can handle the abstraction of differential operators on closed manifolds.

mme

Aug 16, 2021 00:21

i've had my fun for the night. enjoy

mme

Aug 16, 2021 00:18

that's nice

mme

Aug 16, 2021 00:13

too bad Ted left as I popped in. Hi, @Ted. I'm off aagain soon.

mme

Aug 16, 2021 00:12

that reminds me that there's a great miyakazi interview somewhere in which he poo-poos abe for wanting to remove the pacifist bit from the constitution (as you might expect, this was a really quite unpopular comment in Japan). they asked him why and he says, more or less, "we are very bad at war. no further comment."

mme

Aug 16, 2021 00:09

eg stalker is beautiful but you do have to be willing to sit there and watch the grass grow for 5m at a time

mme

Aug 16, 2021 00:09

it's a fine line. it can't be an action flick but it can't be boring. but so many good movies are very boring

mme

Aug 16, 2021 00:08

i haven't seen it but i'll add it to my list. my wife doesn't usually go for things which feel actiony so it'll be a hard sell

mme

Aug 16, 2021 00:07

it does really well perhaps because it's not just a kurosawa script with updated visuals

mme

Aug 16, 2021 00:07

which isn't to say it feels like you're watching yojimbo

mme

Aug 16, 2021 00:06

13 assassins is one of the better samurai movies since kurosawa

mme

Aug 16, 2021 00:02

catholics famously don't read the bible as much as protestants, which in an era with decreased churchgoing probably doesn't breed religious fervor

mme

Aug 16, 2021 00:00

Feels a bit Pascal's Wagery except you remove the conscious cynicism

mme

Aug 15, 2021 23:59

I should get her to watch some Hulot

mme

Aug 15, 2021 23:58

@leslietownes Wife says: "it didn’t really matter what was in your heart as long as you did the right observances, and people were pretty focused on the quid pro quo. also they “believed in” basically any god worshipped by a significant group. their concept of religion was very different than basically any modern religion, so i think it kind of depends on what it means to take them seriously"

mme

Jun 25, 2021 15:59

Secondary characteristic classes are more complicated and not seen as often. You probably do not actually want to look into those

mme

Jun 25, 2021 15:59

@SayanChattopadhyay This is the wrong name. If you want to understand the relationship between Chern classes and curvature forms, this is called Chern-Weil theory. It is explained in many references but I am fond of Morita's "Geometry of differential forms".

Secondary characteristic classes (like secondary cohomology operations) are defined in a setting when the characteristic classes you already know vanish. The standard example is the Chern-Simons invariant of a flat connection. The simplest case is really quite simple, so I'll explain it. Suppose one has a *flat complex line bundle*, so…

Mike Miller Eismeier

Apr 10, 2021 19:36

that's why i said transport

Mike Miller Eismeier

Apr 10, 2021 19:36

$$\int_{b = -\pi/2}^{a = \pi/2} e^{-\tan^2 u} \sec^2 u du = \sqrt{\pi},$$ now good luck computing this for $a \in (0, \pi/2)$

Mike Miller

Apr 10, 2021 19:34

@leslietownes Just transport one of your examples

Mike Miller

Apr 10, 2021 19:21

I hope she enjoys her time here. That's a very different part of the city than I am in. Busier.

Mike Miller

Apr 10, 2021 19:13

I do

Mike Miller

Apr 10, 2021 18:48

Though this one is more definitional.

Mike Miller

Apr 10, 2021 18:48

Sure, I agree it is the point, just as if [x]_beta means the coordinate expression with respect to a basis beta, the point of the matrix A expressed w/r/t basis beta is A_beta [x]_beta = [Ax]_beta.

Mike Miller

Apr 10, 2021 18:42

I'm glad the proof you gave of (AB)^T = B^T A^T is not circular, since I like that proof.

Mike Miller

Apr 10, 2021 18:42

This probably takes 10 minutes longer than I'd like it to and IMO at most three people would like a definition like this in this class, but I do like it as a mathematician.

Mike Miller

Apr 10, 2021 18:41

Then linearity follows by manipulating the LHS, and then the formula follows by considering v = e_i, w = e_j. Great.

Mike Miller

Apr 10, 2021 18:41

But let's take your approach seriously. First, one needs to justify that for fixed v, v * Aw is a linear function of w. This is clear enough from linearity of A and bilinearity of the dot product. Then you need to justify that every linear function of w is given by dot with a unique vector u. Straightforward. Then define A^T v by your formula above: it is the unique vector so that v * Aw = A^T v * w. OK. T

Mike Miller

Apr 10, 2021 18:38

That's what I did too yeah

Mike Miller

Apr 10, 2021 18:38

Or rather asked them to show that Av * w = v * Aw implies A is symmetric, which amounts to the same observation.

Mike Miller

Apr 10, 2021 18:38

I presented it the way I outlined above but emphasized that the point is that formula, use that formula repeatedly, and asked them to prove that formula characterizes the transpose on a HW.

Mike Miller

Apr 10, 2021 18:37

OK, I agree.

Mike Miller

Apr 10, 2021 18:37

Hi!

Mike Miller

Apr 10, 2021 18:37

Let me try to understand your approach.

Mike Miller

Apr 10, 2021 18:36

@TedShifrin I was just thinking about this. To me, the transpose is defined by (A^T)_{ij} = A_{ji}, so your formula v * Aw = A^T v * w requires proof. The most straightforward proof I know starting from that definition is to prove the formula (AB)^T = B^T A^T --- which comes down to symmetry of the dot product, as you can compute the ij'th entry to be (i'th column of B) * (j'th row of A) on one side and (j'th row of A) * (i'th column of B) on the other. Then your formula follows.

Mike Miller

Mar 15, 2021 12:02

OK, that's fair, but I think you should explain the informal point as well :)

Mike Miller

Mar 15, 2021 11:58

I think the picture I just outlined is the entire content of the picture here. Yes, yes, you need a directed system to formalize this.

Mike Miller

Mar 15, 2021 11:57

However, it should still be visually clear what is meant.

Mike Miller

Mar 15, 2021 11:57

I agree that one cannot take a literal union of $\{1, \cdots, n\}$ if your model for that is a subset of $\{n\} \times \Bbb R \subset \Bbb Z \times \Bbb R$.

Mike Miller

Mar 15, 2021 11:56

"At least once" should not be the first pass.

Mike Miller

Mar 15, 2021 11:33

Nevermind, you're going to be more satisfied with Thorgott's point. I'll step out.

Mike Miller

Mar 15, 2021 11:31

OK, then step back. Did you ever learn about the set-theoretic construction of the natural numbers?

Homotopy Theory

A room for anyone interested in homotopy theory, or any nearby...

mme

Sep 10, 2021 00:17

that is his Theorem 1. write the cup-1 product x cup_1 y as xy. his Theorem 2 is (xy)z + x(yz) + (xz)y + x(zy) = 0 mod 2; it's plausible that there's a version of this with appropriate signs over Z, but i don't want to compute the signs. his Theorem 3 is the observation on symmetric Massey products mod 2 (later reproduced by others): if x cup y = 0, then <x,y,x> = y cup Sq_{n-1}(x) mod the ideal (x).

mme

Sep 10, 2021 00:17

yes: https://gallica.bnf.fr/ark:/12148/bpt6k31937/f926
the entire back catalogue of the comptes rendus is available similarly

mme

Sep 9, 2021 21:02

does anyone know where to find a copy of Hirsch, "quelques propriétés des produits de steenrod"? This is the original reference for the Hirsch formula (ab) cup_1 c = a(b cup_1 c) + (-1)^{|b||c|+|b|} (a cup_1 c) b in my preferred sign conventions. I'm curious what else is in this paper.

Mike Miller

Mar 26, 2021 12:35

@ThomasRot If you email me what you're thinking about I'm glad to say what I can, which is probably not much