Mathematics - 2019-12-23 (page 2 of 2)

Balarka Sen

17:00

Oh wow try a two-parameter perturbation

$z^3 + tx + sy$

The leminscate at $(t, s) = (1, 0)$ desingularizes in two ways if you move $s$ to be positive or negative respectively

This is incredibly rich

Ted Shifrin

Looks like Thorgott's having lots of fun with Mathematica :)

Balarka Sen

Poor folks like us use desmos instead

Ted Shifrin

Yeah, but I confess that now that I retired and no longer had access to Mathematica from the university, I gave in and paid money to own my own.

Ah, English.

Thorgott

lol

that is desmos

Ted Shifrin

Aha.

Balarka Sen

17:05

Next semester I'll get Matlab access hopefully

Ted Shifrin

I hate Matlab. I found it so clunky and never got close to mastering it.

Balarka Sen

Have it in my course

Huh

Thorgott

I'm not even sure what Balarka's endgoal is, but this stuff looks cool

Gotta go shopping now, but I'll check in again later

Balarka Sen

I'm a man with no mission

Just demonstrating the fact that fold singularities are incredibly generic whereas holomorphic singularities are incredibly unstable

It's an old theorem that maps which are nonsingular everywhere except a 1-dimensional submanifold along which fold singularities occur form a stable generic subset of $C^\infty(\Bbb R^2, \Bbb R^2)$

Semiclassical

this is one of the weirder series expansions I've seen:

$$h(x) = x^{1/2} - 1 + \frac12 x^{-1/4} - x^{-1}
\sum_{k=2}^\infty 2^{-k}(x^{-2^{-k}} - x^{-\frac32 2^{-k}})$$

Mike Miller

17:17

I don't use software

Semiclassical

first time I've seen a lacunary series show up as part of an actual answer

well

i guess it's not really lacunary. smaller and smaller exponents

Mike Miller

@BalarkaSen How is that stable

Stable means open right

But why can't I just take the codimwnsion 1 subset, and instead of folding along it, first collapse a tubular Nbhd then good

Then fold

Balarka Sen

Is that a smooth homotopy?

Mike Miller

Feel like I can make it smooth using a bump function trick.

Balarka Sen

Yeah, seems like it should work

Mike Miller

17:22

Maybe that would start from something with an e^(-1/x^2) singularity along the codim 1 guy?

Which wouldn't be a fold

Balarka Sen

Ah, possibly

Mike Miller

@BalarkaSen Consider the graph of the map f, let's say it's a self map. Then the fold singularities are presumably transverse intersections of the graph with the diagonal?

That would explain why the folding submanifolds still make sense after perturbation

Oh, if they were transverse they'd be isolated f.p. lol.

But there should be something you're using here

I sort of buy it now

Balarka Sen

It's transverse in the sense that if you look at the space of jets they're the ones with rank 1 singularities

Rank 1 is the top stratum of the singular substratified space of M_{2x2}(R)

So yes, you can prove it by transversality

Mike Miller

Ok.

So you get stability of fold sets and this is enough to convince me stability in general

Balarka Sen

Just to write it down: A map $f : \Bbb R^2 \to \Bbb R^2$ gives rise to a holonomic section of the jet bundle $J^1(\Bbb R^2, \Bbb R^2) \to \Bbb R^2$, with fibers $M_{2 \times 2}(\Bbb R)$, and the singular matrices fiberwise forms a stratified subset $S$ of $J^1(\Bbb R^2, \Bbb R^2)$. By Thom transversality you can homotope the section $j^1 f : \Bbb R^2 \to J^1(\Bbb R^2, \Bbb R^2)$ a tiny bit so that it's 1) transverse to $S$ and 2) still holonomic

Balarka Sen

17:41

$J^1(\Bbb R^2 \times \Bbb R^2) = \Bbb R^2 \times \Bbb R^2 \times M_{2 \times 2}(\Bbb R)$ is an $8$-dimensional space, and $S$ is cut out by $\det = 0$ fiberwise so that it's $7$-dimensional. The section has dimension $2$

So they intersect in a codimension $\codim j^1 f + \codim S = 6 + 1 = 7$ dimensional subset, i.e., a dimension $1$ stratified subset.

So maps $f : \Bbb R^2 \to \Bbb R^2$ with folds and cusps as singularities (the singularity set is Whitney stratified with 1-stratum of folds and 0-stratum of cusps) forms a stable generic family by transversality.

The cusps are somehow not a problem if everything is oriented, but I dunno how to do this bit.

Thorgott

google doesn't give me a satisfying answer: what's a fold singularity?

Mike Miller

@Thorgott Fold a piece of paper

Balarka Sen

Couldn't have put it better myself. It's essentially any singularity which locally looks like folding a piece of paper along a line, like $f(x, y) = (x, y^2)$

Mike Miller

Aka locally your map is modeled on $M \times I \to M \times I$, given by $(m, t) \mapsto (m, t^2)$. I'd use $|t|$ if it were smooth

Balarka Sen

They also call Morse-type singularities fold in higher dimensions, $f : \Bbb R^m \times \Bbb R^n \to \Bbb R^m \times \Bbb R$, $f(x, y) = (x, y_1^2 + \cdots + y_k^2 - y_{k+1}^2 - \cdots - y_n^2)$

Morse along one factor

The cusps are like the Cerf-type singularities, appearing where two fold-lines meet tangentially

$f : \Bbb R^m \times \Bbb R^n \times \Bbb R \to \Bbb R^m \times \Bbb R$, $f(x, y, z) = (x, z^3 - 3yz + y_1^2 + \cdots + y_k^2 - y_{k+1}^2 - \cdots - y_n^2)$

I should eventually learn to see how handle-cancellation happens geometrically

Thorgott

18:16

I can see how $(x,y)\mapsto(x,|y|)$ would look like folding a piece of paper (along the $x$-axis), but $(x,y)\mapsto(x,y^2)$ does involve some stretching too.

Semiclassical

hmm. trying to find a function $f(x)$ which behaves like $x$ for small $x$ and like $\sqrt{x}-1$ for large $x$

Balarka Sen

@Thorgott Yeah it's just doing it smoothly.

Semiclassical

the best so far is $\sqrt{1+2x}-1$ which works for small $x$ but behaves like $\sqrt{2x}-1$ for large $x$

Balarka Sen

I agree there's some stretch. You cannot fold a piece of paper in real life smoothly

If it was made of rubber you could, by the $y^2$ trick

Hm

Semiclassical

also, it should be a smooth monotonic function

Balarka Sen

18:21

If you project it parallely to your plane of vision, the set of singular points looks like this:

Semiclassical

so $1/(1/x-1/(\sqrt{x}-1))$ doesn't work despite having the right asymptotics

Balarka Sen

I wonder how it'd modify the immersion if I tried to cancel a pair of cusps on the diagram

Thorgott

Hmm, ok. And how is the notion of "locally looks like" captured here? Composition with a diffeomorphism?

Semiclassical

that looks trefoil-y

Balarka Sen

@Thorgott Yeah, I say two maps $f, g : \Bbb R^2 \to \Bbb R^2$ looks the same if there are diffeomorphisms $\phi, \psi : \Bbb R^2 \to \Bbb R^2$ such that $\phi \circ f = g \circ \psi$

Basically "after reparamatrization on both domain and codomain", $f$ and $g$ match

@Semiclassical It's really a planar diagram though

Semiclassical

18:27

so equivalent up to composition with diffs?

@BalarkaSen ah

Balarka Sen

I wouldn't say that, I'd just say equivalent upto change of coordinates

Semiclassical

fair enough

Balarka Sen

"there exists $\phi$ such that $f = g \circ \phi$" or "there exists $\phi$ such that $f = \phi \circ g \circ \phi^{-1}$" are all different notions

The latter is known as smooth conjugacy

"or equivalent as dynamical systems"

Semiclassical

oh, nice. I've seen that concept before

Balarka Sen

the former, I dunno if it has a name, but you could study stuff like that. You're only allowing reparametrizations on the domain basically, so it captures topology of the map

Semiclassical

18:30

vis a vis logistic map versus tent-map

or whichever conjugacy it is

Balarka Sen

yeah

Hmmmm

Thorgott

Ok, makes sense. Can maps be classified up to locally looking like one another by the behavior of their rank? In the case of constant rank, I know this to be true.

Balarka Sen

This looks suspiciously like the swallowtail singularity

demonstrations.wolfram.com/TheSwallowtailSingularity

That's like cancelling two cusps right

I never actually understood the swallowtail catastrophe, but I guess I see the point now

@Thorgott For maps between 2-manifolds, yes. But in higher dimension this gets much, much harder

You can have different singularities of the same rank, interacting with each other. The whole setup is known as the Thom-Boardman hierarchy.

I don't understand it well

Thorgott

Huh, sounds cool

Ultradark

18:48

$\frac{\ln(x)\ln(y)}{\ln(1-x)\ln(1-y)}=1 $

is it possible to solve for $y$?

Mike Miller

@BalarkaSen the answer being yes there's a classification no it's much worse than rank

Balarka Sen

hahah

yeah

Damn this stuff is so beautiful why is this branch dead

I hate mathematicians

Mike Miller

19:11

@BalarkaSen Sociology

Not enough problems to build a community of grad students around

If you can't do that a field dissipates

Balarka Sen

Makes sense

Shine On You Crazy Diamond

19:32

Define $f : \Bbb{Z}/(q_1\cdots q_r)^* \to \Bbb{Z}/(q_1 \cdots q_{r-1})^*$ by $f(x) = x^2$ is surjective right?

where $q_i$ are the first odd primes

or any odd primes

@Ultradark

@JinLong

Ultradark

19:48

@shi I have no idea

Akiva Weinberger

20:12

What are the unknown knowns

Balarka Sen

Who knows

Mike Miller

goodbit

Ultradark

the answer to that question I asked above is $y=1-x$ lol

$\frac{\ln(x)\ln(y)}{\ln(1-x)\ln(1-y)}=1$

$\frac{\ln(x)\ln(y)}{\ln(1-x)\ln(1-y)}=2$

not sure how to solve for this though

It's almost as if the 2 wreaks havoc on the distribution of prime numbers

Ted Shifrin

20:34

a @Balarka: With McCrory and Varley I wrote several papers largely on singularity theory. I don't remember if I sent you copies of all. I know I sent one or two.

Balarka Sen

@TedShifrin Yeah you sent me one about singularities of Gauss maps on hypersurfaces.

Ted Shifrin

OK, you can find all sorts of versal unfolding stuff in there :P

There were ultimately 4 papers, I think, plus one monster one that never got written.

Leaky Nun

@BalarkaSen play?

Balarka Sen

Not now @Leaky

@TedShifrin I first learnt about Nash blowups from your paper, I think

Wow I am staring at your paper right now, this looks so good

loch

here's a recent talk by grant sanderson (3blue1brown) at mit if anyone's interested

Leaky Nun

20:45

@loch next time you should delete the fbclid parameter :P

Ted Shifrin

@Balarka: The Gauss map for surfaces in $\Bbb P^3$ was first; the big one you have, I think, was for threefolds in $\Bbb P^4$. Then there are some on theta divisors.

loch

oh

let me try again

Balarka Sen

Yeah it's about threefolds in $\Bbb P^4$

Leaky Nun

@loch halp i'm jetlagged

Ted Shifrin

The one for $\Bbb P^3$ is a good warmup.

loch

20:47

@LeakyNun you're in hk?

Ted Shifrin

@Leaky, are you back home?

Leaky Nun

yeah

loch

i see
im arriving hk on 26

Leaky Nun

nice

@loch what's the main point of the video?

loch

video.odl.mit.edu/videos/851813b02bd74da2995254daf20253b0

ok i dont think i'll be doxxed with this link

Leaky Nun

20:49

nice

Mike Miller

I'll doxx u

loch

it's about story telling

pls

Balarka Sen

boo

ask 3b1b to make a video on catastrophe theory

Leaky Nun

go be a patreon

Jin Long

What the hell is that?

Leaky Nun

20:54

@JinLong catastrophe theory

Balarka Sen

its singularity theory taken over by cranks

Jin Long

What does this image represent?

Balarka Sen

this was popular back in the day

Leaky Nun

catastrophe theory

Mike Miller

Good poster

Ted Shifrin

20:54

René Thom had all sorts of essays to popularize this stuff.

Jin Long

Interesting.

Balarka Sen

Thom said he switched to catastrophe theory because he couldn't understand what people were doing with Thom spectra

its in some interview

he was like I give up

cant keep up with these homotopy theorists

Ted Shifrin

I didn't read that, but I believe it.

Mike Miller

Here is a puzzle for you

You live on an island in which everyone wears a hat, either blue or red.

Balarka Sen

@TedShifrin I borrowed a book by Arnold which discusses applications of singularity theory in contemporary science. It's a very thin book called "Catastrophe Theory"

It's very good

Ted Shifrin

20:58

If you read Arnold to get the ideas, and not the details, he's wonderful.

In McCrory's and my paper on surfaces in $\Bbb P^3$, we spent five pages proving something Arnold totally handwaved in an earlier paper.

Balarka Sen

Hahah

amazing

Ted Shifrin

He just asserted "generically," and never justified it remotely.

Mike Miller

You've misplaced your hat. Everyone is staring at you. It's clear you're not welcome.

The horde of hatted people begin to approach you, chanting "no hat. No hat. No hat".

You walk backwards as they chase you towards the edge of the island. The water does not stop them.

Ted Shifrin

wonders if he stepped into a Trump rally with hats

Jin Long

This sounds like the humanity test in Blade Runner.

Mike Miller

20:59

As you begin to drown you ask: "what did I do to deserve this?"

That's it.

Leaky Nun

nice puzzle

very puzzling indeed

Ted Shifrin

Oh, I missed the introduction.

Balarka Sen

@MikeMiller Clearly, the answer is he misplaced his hat

Mike Miller

@JinLong I wasn't trying to channel Voight Kampff but I see your point

Ted Shifrin

This doesn't seem quite like the colored dot puzzle.

Jin Long

21:01

@MikeMiller what were you trying to do?

Mike Miller

Someone mentioned 3b1b so I was just trying to suggest a math puzzle

But everyone knows the usual hat game so I figured it's better to try something new

Balarka Sen

I forgot "Here is a puzzle for you" is what he starts with

Jin Long

You left out the "puzzle" part tho

Mike Miller

Nah check 2 minutes before the start

I just had to write the puzzle bro

Everyone's ganging up on me for providing content

Balarka Sen

It's very funny if I imagine it said in 3B1B voice

Jin Long

21:04

Maybe don't introduce it with "here's a puzzle for you"

Leaky Nun

@BalarkaSen 2-day correspondence?

Balarka Sen

Nah mane

That's too long

I'll play eventually

Jin Long

But instead, more accurately, describe it as "here's a bit of nonsensical content that is not at all a puzzle"

Leaky Nun

what is it with you

Jin Long

@LeakyNun is that chess?

Balarka Sen

21:05

he's triggered because he wanted a genuine puzzle

Leaky Nun

yeah

Ted Shifrin

I'm leaving before the riot/fight breaks out. BBIAB.

Jin Long

I have this weird notion that when someone presents a puzzle, I expect it to be an actual puzzle.

Bizarre I know.

nbro

If $\mathbb{P}$ is a probability measure, $X$ a random variable and $x$ an arbitrary realization of $X$, does it ever make sense to write $\mathbb{P}(X)$ or $\mathbb{P}(x)$? If yes, what would these notations more precisely mean (in terms of measure theory)? Note that I know that $\mathbb{P}(X=x)$ is an abbreviation for something else (which I don't want to write here)!

loch

@MikeMiller that's what happens when we're out of proofs for twin primes

Leaky Nun

21:06

lol

@nbro $\mathbb P(X=x)$

Mike Miller

@loch We should move on to triplet primes

Prove that there are infinitely many triples (p, p+2, p+4) with all three terms prime

Balarka Sen

as a wise man once said

Sep 2 '14 at 18:34, by Karl Kronenfeld

The conjecture relating to the number of twin primes is too difficult though, so I stick to prime twins

Leaky Nun

according to Dirichlet's theorem there are arbitrarily long arithmetic sequences in the set of primes

Mike Miller

Of course you get TWO copies of the twin prime conjecture as a corollary

nbro

@LeakyNun Yes, I've edited my comment to include that I know I can do that. However, in statistics, we often see $P(X)$ or $P(x), so I am wondering what those expressions would more precisely mean.

Balarka Sen

21:08

OH lmao

Mike Miller

One from (p, p+2) and one from (p+2,p+4)

Balarka Sen

Sep 2 '14 at 18:37, by Karl Kronenfeld

@PedroTamaroff I am working with EnjoysMath

@ShineOnYouCrazyDiamond You have been doing this for a long time aren't you

Leaky Nun

what are prime twins?

nbro

For example, in this blog post https://blog.tensorflow.org/2019/03/regression-with-probabilistic-layers-in.html, the authors wrote that $Q(w)$ is a probability distribution. Is $Q(w)$ a probability measure, cdf, or what? If it's a probability measure, what does $Q(w)$ exactly mean?

Mike Miller

Read a book on probability dude

5

Leaky Nun

21:10

@nbro can you quote?

nbro

I would like to note that I am very familiar with machine learning and probability concepts. I just would like to have a more rigorous (measure-theoretic) explanation of this notation and terminology.

Leaky Nun

where did they say that Q(w) is a probability distribution?

nbro

> layer uses a variational posterior Q(w) over the weights to represent the uncertainty in their values. This layer regularizes Q(w) to be close to the prior distribution P(w)

P(w) is thus a distribution, so Q(w) must also be a distribution.

Are they referring to probability measures, cdfs, or what?

Leaky Nun

it's a distribution

nbro

lol

Is this your explanation? It's a distribution? This is written in the article.

Leaky Nun

21:13

it's a random variable

nbro

lol

A distribution or random variable? And, when you say "distribution", are you referring to a probability measure?

Mike Miller

You're acting very high and mighty for someone who refuses to read a book

9

nbro

@MikeMiller I will read maybe in the future, but definitely not now (I don't have time for that now). I've already read books on probability in the past. And what? You forgot stuff after 1 week. Everyone forgets. That's part of the plasticity of our brain!

Why does this chat exist? If you're not willing to answer, just ignore my messages or questions.

Leaky Nun

is the distribution the random variable? they feel kinda different for me

nbro

You're the mathematician (not me) :) I studied computer science and AI.

Leaky Nun

21:17

a distribution is characterized by a measurable function from a probability space to R, how about this

but is it the function?

@MikeMiller I don't know about this

it feels like a philosophical question

nbro

The random variable induces a probability measure. The random variable is the measurable function.

Balarka Sen

@LeakyNun there is no loss of information in saying a distribution is a probability measure. dont respond if you dont know definitively

a random variable is said to follow a distribution if the induced measure is that distribution

Leaky Nun

'aight

loch

@nbro i am not a statistician** , but my guess is that for the formulas in the link they're referring to the corresponding density functions

nbro

@loch However, in that article, they are talking about "variational inference". All variational inference is about approximating a usually called "distribution" (the posterior) with another "distribution"

loch

21:31

yes but when densities exist they're "the same thing"! (meaning from a pdf you can get your cdf / your measure etc.)

nbro

What if they don't exist? Or when wouldn't they exist?

Semiclassical

Singular distributions are the issue

loch

i dont think people care about such high generality so it's not something one has to worry about (KL divergence makes sense in general though - according to my googling skillz)

last time thorgott gave you examples where they don't exist already!

Semiclassical

Eg cantor disturbution

Also cases where it’s a mixture of discrete and continuous, tho thorgott already covered that

Balarka Sen

its a theorem that every distribution can be written as a sum of an absolutely continuous part (has a density) and a singular distribution

this is known as the Lebesgue decomposition theorem

Semiclassical

21:42

If you’ve got a continuous cdf, then I think it has a density? I could be making too strong a statement tho

Balarka Sen

Nah that's not true

Semiclassical

Hmm

loch

@Semiclassical i think that's your counterexample :0)

Balarka Sen

yeah

Leaky Nun

the issue is that they're continuous but not absolutely continuous

Semiclassical

21:43

Wait, Cantor is continuous?

Leaky Nun

Radon--Nikodym requires absolute continuity

Semiclassical

Huuuh.

Balarka Sen

Cantor function is continuous yeah

Leaky Nun

continuity occurs iff singletons have measure zero

absolutely continuity occurs iff Lebesgue-null sets have measure zero

Semiclassical

How much smoothness do you need, then? Or is that the wrong question

Leaky Nun

21:44

Cantor set has Lebesgue measure 0

Balarka Sen

@Semiclassical if the cdf is of bounded variation then it has a pdf

Leaky Nun

wiki says:

for a compact interval,

continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere

Balarka Sen

ah so what i said is not true

something more is needed

Semiclassical

Huh

Balarka Sen

i forget what

Semiclassical

21:46

On the other hand, being continuously differentiable suffices

Balarka Sen

the "Luzin N property"

Leaky Nun

wiki says $x \sin(1/x)$ is also a counter-example

so there goes your smoothness a.e. assumptions

Semiclassical

Oof

Balarka Sen

well the cantor function is smooth a.e. isnt it

Leaky Nun

oh well

but this is even smooth except on a singleton lol

feels stronger

Balarka Sen

21:50

@Semiclassical yes because then you can differentiate the cdf and the derivative is integrable with integral the cdf :p

which then is your pdf

Semiclassical

Yeah

Balarka Sen

the key point here is that you only want the derivative to be integrable not continuous so continuous a.e. is sufficient so Lipschitz continuity is also sufficient because that's equivalent to C^1 a.e.

Semiclassical

And tbh that covers most of the practical cases

Balarka Sen

In most practical cases the cdf will be C^1 minus a few points, so yeah

Leaky Nun

except if you for some reason decided to take the sum of $X_i$ where $X_i \sim U(0,2^{-i})$

its CDF has a name that escapes my memory

Semiclassical

21:54

Ugh

Balarka Sen

the word "practical" has a meaning!

Leaky Nun

it starts with F

Mar 29 at 16:37, by Semiclassical

@akiva lots of values of the Fabius function here: https://arxiv.org/pdf/1609.07999.pdf

there you go

Balarka Sen

anyway nobody actually cares about these wild stuff. regularity is only an important phenomenon in probability when you start thinking about brownian motions

Leaky Nun

what happens in brownian motions?

Balarka Sen

with probability 1 the paths are nowhere differentiable

but they are holder continuous i think

i am not sure

Semiclassical

21:59

Yeah, this all falls under the heading of “pathological” for me

And I have a hard time getting too interested in it, simply because it’s typically so remote from practice

Leaky Nun

wait why is brownian motion not practical

it's literally physics

Semiclassical

all falls under is too strong, yeah

brownian motion, and stochastic systems in general, are a big exception

Balarka Sen

i have to learn something about them, someday

Semiclassical

same

I know it at the physics level, to an extent

but when I start seeing Ito calculus I know I'm out of my depth

Balarka Sen

Hahaha

probabilists love Ito

this stuff is too hardcore for me

Leaky Nun

22:05

who is Ito

Itô calculus

Itô calculus, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: Y t = ∫ 0 t...

Semiclassical

technically, I've worked on stochastic calculus stuff

but the work I did was predicated on an approximation method which immediately got rid of explicit stochasticity

("weak noise approximation")

Thorgott

22:20

oh yeah, I remember when a friend told me about one of his assignments last semester and said he has to solve a "stochastic differential equation". I replied "you what?" and I think that's where the conversation ended.

nbro

22:36

@Semiclassical Yes, that's right, he had provided an example, even though I hadn't fully understood it

Leaky Nun

22:47

How to play chess properly

►

Thorgott

lmao

Leaky Nun

@BalarkaSen ^

Ultradark

23:11

Why is there no implicit integration

say I have an expression I can't put in explicit form. Then there's no way to integrate it?

Thorgott

If you're talking about indefinite integration, then that exists and it's called solving a differential equation

Ultradark

I have an expression that I can't put in explicit form

I'm trying to figure out how to integrate it definitely

I wonder if it's impossible to represent the function explicitly

Transcript for

$Mathematics$ Mathematics