Mathematics - 2021-01-14 (page 3 of 3)

Thorgott

6:00 PM

choose the ones not containing the words "dynamic" or "system"

user2103480

Dynamical Systems III would be a bit ambitious anyways since I did not do dynamical systems I or II

Thorgott

lol

reminds me of when I wanted to take Algebraic Geometry this semester, but then I realized it's Algebraic Geometry III and I haven't heard I or II

user2103480

Although I have a tendency of taking lectures where I did not formally learn the prereqs

Needless to say, it's made lectures much harder

@Thorgott how did that work out

Thorgott

well, I didn't take it, that's how it worked out

BigSocks

ok yeah, if you literally write it down $1 -y = 0$ so $y = 1$, but if you were to do $(\Bbb Z / 2 \Bbb Z)[y] / (y)$, you'd write down $y = 0$ so $1 - y = 1 - 0 = 1$

user2103480

6:03 PM

@Thorgott nice

Thorgott

@BigSocks yup

user2103480

@Thorgott random dynamical systems might actually be interesting though

BigSocks

awesome, thanks again @Thorgott

user2103480

and hopefully not exactly like usual dynamical systems

Thorgott

inb4 it's just a regular dynamical systems lecture, but they assemble the topics in an order dictated by dice rolls

user2103480

6:05 PM

lmao

Thorgott

"And todays random dynamical system iiis....."

user2103480

There's lottery where we'll talk about some random dynamical system

@Thorgott hahaha great idea

Thorgott

right, they should hire me

user2103480

... you'd actually have to talk about the dynamical systems though

Thorgott

*rolls totally unbiased dice*
"...and todays topic is modelling dynamical systems in higher categories."

BigSocks

6:11 PM

is that... possible?

I think it's more of a meme... gotta be

user2103480

uhhh the category people have some seminar on applied category theory that tries to do exactly that

and it is quite a meme

BigSocks

tfw "both" is the answer

Alessandro Codenotti

@user2103480 JUST DO IT

user2103480

@AlessandroCodenotti lmao

Thorgott

yeah, so let me briefly explain why smooth topoi are the right setting for studying differential equations.... furiously waves hands around and throws chalk through the hall

BigSocks

6:14 PM

lel

user2103480

ncatlab.org/nlab/show/cohomology+of+dynamical+systems

BigSocks

tud.qucosa.de/api/qucosa%3A27340/attachment/ATT-0/?L=1

user2103480

nlab at its best

first two articles are in russian

BigSocks

jfc it's real

user2103480

I really hope they will get some useful results out of this in the next ten years, else I'll be mad that all this received funding

Thorgott

6:16 PM

lmao

ive been outdone by reality

user2103480

@Thorgott applied-compositional-thinking.engineering

Alessandro Codenotti

@user2103480 Dynamical systems are nice

user2103480

Session 10 (Jan 29th, 14:00 UTC): Parallelism: Parallel composition, symmetric monoidal categories.
Office Hours (Jan 30th)
Session 11 (Feb 1st, 18:00 UTC): Feedback: Feedback, trace, duals.
Session 12 (Feb 3rd, 14:00 UTC): Duals, compact closed categories.

.... yes engineers will definitely be able to appreciate symmetric monoidal categories....

poor blokes

maths researcher

If $\pi: E\rightarrow M$ is a vector bundle with connection $\nabla$, $p\in M$ and $X_p\in T_pM$ and $s,t\in \Gamma(E)$ and $s$ and $t$ agree on some smooth curve through $p$ with initial vector $X_p$, how do I show that $\nabla_{X_p}s=\nabla_{Y_p}t$?

Thorgott

lol

user2103480

6:20 PM

Abstract
Effective interoperation between multiple scientific disciplines is crucial to systems engineering.
Can the study of interoperabilitythe working negotiations and hand-offs between theories and
modelsitself be made into a hard science? Hard sciences are based on mathematics, so this would
require a mathematics of interoperability, a mathematics whose subject consists of the bridges
and analogies that make data- and model-integration actually work. I propose that category
theory serves this purpose exceptionally well.

I think "outdone by reality" is an understatement

Thorgott

glorious

user2103480

@Thorgott act2020.mit.edu

Cannot top this

"David Jaz Myers. Double Categories of Open Dynamical Systems"

Thorgott

"Richard Statman. Products in a category with
only one object
"

maths researcher

any have any idea?

BigSocks

oh dang that's twitter category people

Thorgott

6:24 PM

"Callum Reader. Measures and Enriched
Categories"
that one sounds good

user2103480

@BigSocks one of my weirdest observation in the last few months was that there is a twitter transgender category theorist bubble

BigSocks

yep, I have definitely noticed that

Thorgott

I'm surprised that the intersection of these groups of people numbers in more than one digit

BigSocks

idk how the intersection of those things is so large but yes-

wow there you go

user2103480

@Thorgott that's why it's so weird

I've found a random dynamical systems lecture in frankfurt and it looks nothing like a normal dynamical systems lecture

... perfect

Thorgott

6:32 PM

who's lecturing

user2103480

Hans Crauel

Thorgott

ah, I don't know him

user2103480

You took one probability course right

Thorgott

ye

user2103480

probably chose the right point to stop

Thorgott

6:43 PM

i also TAd probability for compsci students once

that sure was an experience

user2103480

oh boy they always wonder why they need to learn the stuff

I TA'd for "mathematics for computer scientists and mathematics educators" once

... even the computer scientists were better than the teachers

@AlessandroCodenotti yeah I plan to take that RDS course

maths researcher

@TedShifrin could I have a hint? without using connection matrix

BigSocks

So I get why $\Bbb Z[\sqrt 5] / (2) \cong \Bbb F_2[y]/(y - 1)^2$, but what am I supposed to think of what $\Bbb Z[ \sqrt 5] / (2) \Bbb Z[ \sqrt 5]$ is ?

user2103480

but the other lectures are interesting as well. Tbh nonlinear functional analysis looks like it's "just fixed point theorems", which is of course an understatement, but probably more something that one learns when one needs to

Thorgott

@user2103480 you know, there's a competence hierarchy

user2103480

6:49 PM

the lecture on functional analysis applied to modelling of molecular systems has a past website userpage.fu-berlin.de/dellesite/famolmod.html

@Thorgott yeh and math education students seem to be far far down

Thorgott

at the bottom, there are the Wirtschaftspädagogen

user2103480

lmao

but the problems seem to be fun

check this out, from a master's lecture for elementary school teachers

Thorgott

lecture I was once in: "Ich glaube, wir haben auch Wirtschaftspädagogen unter uns. Wer hier studiert Wirtschaftspädagogik? Ok, sie müssen sich auch nicht outen."

user2103480

lool

this one had me cracking up

there is exactly ONE obvious extension of the preceding figure, which has symmetry group Z_2

Thorgott

hahahaahahaha

user2103480

6:54 PM

and as a lecturer at a german university this is just ridiculous

Thorgott

this is extremely funny, because I thought about this recently as well

user2103480

a friend asked me if I could help a a friend of theirs with this sheet

The teaching student didnt react to the symbol at all, sad

But it's the "wrong" direction anyways (or, rather, it's the right direction if you don't want to be fired)

Alessandro Codenotti

@user2103480 nice

user2103480

@Thorgott what lecture was that?

Thorgott

some elementary algebra lecture, I forgot which

@user873110 did you see my recommendation?

user2103480

7:01 PM

anybody here have a clue what invariant measures for PDE do, or stochastic homogenisation

User 873110

yeah

user2103480

both things seem to come from the deterministic theory of PDE, rather than from stochastic ODE/PDE

Alessandro Codenotti

@user2103480 wait the picture I have in mind which would be inappropriate for a German university in particular can be rotated by 90 degrees too

user2103480

you saw exercise b) right

Oh wait

Alessandro Codenotti

lol

user2103480

7:09 PM

you mean... another Z right next to the Z?

fucking hell

ah wrong direction again

smart

Thorgott

huh, I don't follow?

user2103480

smart symbol

Alessandro Codenotti

I'm thinking about an ancient religious symbol

user2103480

ok now I'm lost too, do you just mean the svastika or another appropriated ancient religious symbol

Alessandro Codenotti

the swastika

user2103480

7:11 PM

that was the joke all along tho

Alessandro Codenotti

@user2103480 but the symmetry group is not Z_2

It can be rotated 90 degrees

Thorgott

it's supposed to be Z_4

Alessandro Codenotti

That's my point

user2103480

yeah yeah but exercise b) asks for symmetry group Z_4

Thorgott

siegrunen don't work, they have symmetry group Z_2xZ_2, no?

user2103480

7:15 PM

true, I was just perplexed that his is also not far away

Thorgott

thought I reached my low point that time I started talking about the automorphism group of an among us map, but good to know one can always do worse

user2103480

hahahaha

Rithaniel

Do you guys have any mathematical expressions that you consider particularly "beautiful?" Someone close to me has been asking about that sort of stuff

Karl Kroningfeld

oh god

maths researcher

@Rithaniel $K_p=\langle R_p(u,v,v),u\rangle$

gaussian curvature

Rithaniel

7:25 PM

Oh, that's a good one

BigSocks

x(t) = ((-46/31 sin(170/113 - 11 t) - 230/29 sin(67/43 - 10 t) - 39/29 sin(47/30 - 8 t) - 364/23 sin(14/9 - 6 t) - 5503/128 sin(25/16 - 4 t) - 135/26 sin(29/19 - 3 t) - 4563/28 sin(47/30 - 2 t) + 7685/27 sin(t + 107/68) + 34/31 sin(5 t + 52/31) + 187/26 sin(7 t + 107/68) + 17/7 sin(9 t + 13/8) + 27/17 sin(12 t + 136/29) - 19995/26) θ(63 π - t) θ(t - 59 π) + (-2/9 sin(124/83 - 24 t) - 24/23 sin(61/40 - 20 t) - 16/31 sin(70/51 - 19 t) - 31/17 sin(20/13 - 18 t) - 7/6 sin(25/16 - 14 t) - 100/13 sin(67/43 - 6 t) - 9259/35 sin(58/37 - t) + 4638/37 sin(2 t + 41/26) + 1327/30 sin(3 t + 19/12) + 377…

Rithaniel

Maybe a little long

BigSocks

still very beautiful

bit.ly/3oQOSG8

here's an easier visualization

Thorgott

beautiful

BigSocks

thank you :-)

user2103480

7:39 PM

🅱️eter

desmos.com/calculator/qp58oleeh4?lang=de

SPLENDID

BigSocks

I AM THE GLOBGLO-GRAPH-GALAB

Well look who we have here... @TedShifrin

user2103480

@BigSocks a particularly beautiful mathematical expression

BigSocks

indeed!

NoName

Free online grad intro course to elliptic curves taught by Álvaro Lozano-Robledo, following Silverman's book, and only assuming a year of grad abstract algebra.

BigSocks

ooo

thank you, looks nice

Ted Shifrin

7:49 PM

@mathsresearcher You have a typo. You mean $Y_p=X_p$, of course. So it suffices to take a section that's zero along the curve and see that its covariant derivative is $0$ along the curve. Presumably you want to multiply by an arbitrary function. Have you tried working this out?

It's merely an illusion, @BigSocks.

@mathsresearcher That's "particularly beautiful"?

BigSocks

hmm it seems to be fooling me... wanna see another cool illusion? check out my new picture

Ted Shifrin

No, it gives me nightmares.

BigSocks

hahahaha understandable

Ted Shifrin

Are you in fact local?

NoName

@BigSocks There's preliminary talk posted today on YouTube. xD

BigSocks

7:52 PM

no, no- I'm closer to the other coast. But you can still think of me as a giant pair of socks you see floating over the sunset

currently in Puerto Rico if you'd believe a pair of socks

Thorgott

should've said you're global

BigSocks

agh, what a missed opportunity lmao

@NoName I will likely listen to it once I take a break from these funny ideals

thank you

Ted Shifrin

@Thor: He's global, but he vanishes when you restrict him to local.

BigSocks

8:06 PM

ok wait then does that mean that I am a $C^\infty$ hypersurface?

1

Q: Local description of smooth functions that are zero on a hypersurface.

We say that $\Sigma \subset \mathbb{R}^n$ is a $C^\infty$ hypersurface if and only if, for all $p \in \Sigma$, there exists an open neighborhood $U_p \subset \mathbb{R}^n$ containing $p$ and a smooth function $f_p : U\to \mathbb{R}$ with $\nabla f_p \neq 0$ on $U_p$ such that $$ U_p \cap \Sigma...

jcora

hey @TedShifrin, I ran into a situation that you hinted at yesterday. same problem of finding the jordan basis but for a different matrix, and the system $(A - \lambda I)q_i = q_{i-1}$ didn't have a solution. but again I am having difficulty so I'd like to run this past you to see if there are any obvious problems with my reasoning

again there is only one eigenvalue $\lambda=3$, but with two blocks of sizes $3$ and $1$. the kernel of $A - 3 I$ is two-dimensional, so I have two eigenvectors immediately. one is for the 3-sized block, and the other is for the 1-sized block, so I called them $…

BigSocks

tried some google-fu, given my algebraicGeometry-fu needs some work

Ted Shifrin

8:47 PM

@jcora You did not take the time to understand what I was saying yesterday. You didn't do the exercise of understanding why $\text{im}(A-\lambda I) = N(A-\lambda I)$ in the case of yesterday. Today, you have to look at $\text{im}(A-3I)^2$. You cannot pick eigenvectors at random.

maths researcher

@TedShifrin if its not DG its not beautiful

jcora

9:00 PM

@TedShifrin hm I thought I understood what was happening yesterday, but I guess not since I don't remember that I was supposed to see those two were equal:/

I thought what you were saying was that I might run into a situation where I can't generate generalized eigenvectors easily with $(A-\lambda I)q_i = q_{i-1}$ because that system might not have a non-trivial solution

jcora

9:29 PM

ah I think I got something, I didn't ensure my generalized eigenvectors, the ones I picked from the difference of the kernels, actually correspond to the eigenvector in the sense that $A-\lambda I$ relates them. so I'm guessing I ought to start from $q_3$ and get to $q_1$, and it's not guaranteed that I'll get one of the basis vectors for $\ker(A-\lambda I)$ that I found originally

Semiclassical

9:47 PM

there are times when i look at someone's question and I just want to be like "...what?"

user2103480

10:03 PM

@Semiclassical me looking at the new problem sheet

BigSocks

10:28 PM

what are the categories that have bijective homomorphisms as isomorphisms?

I think Ab and RMod are examples of this, and Top and the category of varieties and (dominant) regular maps are nonexamples

Ted Shifrin

@jcora Right. So you need to find an eigenvector that is in the image of $(A-\lambda I)^2$. This is the logic I suggested yesterday.

Thorgott

you're asking for concrete categories where the forgetful functor reflects isos

an interesting example is the category of Banach spaces and continuous linear maps

the inverse of a continuous linear bijection is automatically linear for trivial reasons, but it is also automatically continuous - that's the Bounded Inverse Theorem

BigSocks

eyyy concrete, that's the word. that means that isos are actually on the underlying sets right? reflecting forgetful functor I mean

Ted Shifrin

sees functors doing evil things and flees

BigSocks

hahaha nooo

Thorgott

10:44 PM

a concrete category is just a category together with a faithful functor to Set, an iso in Set is just a bijection and for the functor to reflect isos means that if the image of a morphism is an iso, then that morphism is an iso, i.e. bijective (on the underlying set) morphisms are isos

BigSocks

ok this makes a lot of sense, I worded it like crap. bad to talk to people irl and phrase math. Thank you

Thorgott

any algebraic category (in a sense that can be made precise) has the property you're looking for

it usually fails in categories of topological flavor, as you say

Karim Mansour

I am interested to learn more of this stuff what does it mean that image of morphism is an isomorphism in precise terms

BigSocks

algebraic ones are my favorite. they have free objects and free $\dashv$ forgetful functor is a monad

Ted Shifrin

\turnstile?

BigSocks

10:47 PM

mb

user2103480

adjoint?

BigSocks

apparently it is dashv yeah

logic-poisoned-brain

Ted Shifrin

no comments

Karim Mansour

oh so whenever you have some clearly defined adjoint that turns out to be isomorphism

BigSocks

hahaha T e D

Thorgott

10:48 PM

also, though this is just juggling terminology, any concrete balanced category has your property

Karim Mansour

then the morphism is isomorphism in what is called reflexive category ?

BigSocks

no, I love that, I want more of these words

user2103480

@BigSocks shouldn't $\vdash$ be the logic-poisoned brain

BigSocks

yes, but calling the right one "turnstile" no matter what I think also counts lol

Thorgott

compact Hausdorff spaces are one of the rare categories coming from topological spaces where this holds too

also the category of fiber bundles over a fixed base has this property

BigSocks

10:50 PM

yeah but "compact Hausdorff" is like a god property and that gives you everything forever

Karim Mansour

yeah

BigSocks

also I am writing all these down for later rabbit holes for procrastinating

Thorgott

I guess you can also combine the two to get the category of fiber bundles over compact Hausdorff spaces, which also has this proeprty

Karim Mansour

Hi @TedShifrin

I will look at this stuff later for procrastinating as well.

Ted Shifrin

hi Karim

Thorgott

10:52 PM

well, in the non-algebraic examples given, these properties are slightly less trivial and actually very important in practice, so they're worth thinking about

Karim Mansour

Yeah I know most of non-algebraic ones and examples mentioned though not in the language of category theory.

I am gonna get this book should have everything I guess

amazon.ca/Category-Theory-Context-Emily-Riehl/dp/048680903X

brb

Thorgott

11:07 PM

oh, there's a classic example that I forgot to mention

the category of first-countable Hausdorff spaces whose morphisms are continuous proper maps also has this property

BigSocks

yeah gunna have to brush up on topology for that one

Thorgott