Mathematics - 2021-07-01 (page 1 of 2)

Ladies and gentlemen

12:00 AM

My friend said that he wanted to keep copyrights of his thesis or something

and his advisor found out like 1 month later

and now he's going to have to take that back

because the advisor got mad at him

there was a checkbox when he uploaded the thesis

and he thought it was cool to check it

copper.hat

bummer!

Ladies and gentlemen

and now he's gonna have to spend at least 5 hours doing paperwork to take it back ( my estimates).

leslie townes

i granted proquest a nonexclusive license to provide my thesis for eternity. paying them to offer it 'open access' was something like $60, no thanks. plus you can download it on researchgate and a million other places.

copper.hat

oh wow. i declined to have mine copyrighted.

everything in it was published anyways.

leslie townes

that was the main issue with my dissertation too.

copyright litigation is fun. i need to get on another copyright case.

copper.hat

12:03 AM

it took me a few years to realise that i was not cut out for academia.

leslie townes

there was a weird case out of new mexico where a guy had a draft of his thesis, and the university archived it and had it available in the library without his permission, and he sued them. i don't think there was evidence that anybody ever accessed the draft thesis, but that doesn't matter. making publicly available is enough to trigger liability

copper.hat

regardless of my marginal potential

wow.

leslie townes

i don't think he was cut out for academia either, although it made for an interesting case.

copper.hat

law is a funny mix of testosterone and mathematical precision.

Ladies and gentlemen

did he get money out of it?

like enough to buy a car?

copper.hat

12:05 AM

have a few acquaintances who are defense attorneys. all typically greet with a hard punch in the shoulder.

Ladies and gentlemen

or house money?

leslie townes

i doubt it. i didn't follow what happened after the appeal but my guess would be he didn't register in time. he probably got them to remove it from the library.

Ladies and gentlemen

oh

leslie townes

nobody ever registers in time.

except for like, disney, and DC comics, or whoever. they're good about that.

Ladies and gentlemen

I would never sue a library to get my draft removed

I hate paperwork way too much

leslie townes

12:07 AM

i think it was just one prong in some kind of long standing dispute with his department.

Ladies and gentlemen

oh ok

leslie townes

there was lots of weird stuff going on in that case. academia really sucks for a lot of people and they deal with it in different ways, sometimes by internalizing the toxicity, and sometimes by lashing out. anyway, it led to a nice appellate decision which is all that matters.

there is not a lot of copyright law in the tenth circuit.

in fact he did get them to remove it for the library. i remember requesting the thesis via interlibrary loan. the record was still in the system and i got back 'no can do.'

why did i request the thesis? my mind just works that way

Ladies and gentlemen

there was probably the proof of a homicide in there

I love how in the us detective movies everyone always comes up with a super complicated homicide and gets figured out in 4 days

but irl people just commit homicides in completely straightforward ways and they can't find out who does it

well, at least in Mexico

leslie townes

it's the same here. they often don't clear those cases unless there is a connection to something else they are interested in.

Ladies and gentlemen

wait: I don't love it

I should have said: it's interesting

leslie townes

12:14 AM

it's just not interesting to dramatize a routine killing. you need all that extra plot stuff so there is something to follow.

there was a homicide in my hometown that went unsolved for 40 years and then some DNA or something showed up and they got the guy who did it. he was like 80 at that point but they got him.

Ladies and gentlemen

although in many of the shows the murderer acts like a complete jerk

leslie townes

if there's something that murderers have in common, 'complete jerk' might be it.

Ladies and gentlemen

"there's no way I could have killed her I was winning the soccer world cup at the time"

while smiling smugly

leslie townes

columbo brilliantly repeated this premise over and over.

there was a funny piece in some magazine that was a list of things the author would not say if they were investigated for a murder. it ran down everything that every villain in columbo did. i've lost track of it.

copper this might be a convexity question. i don't recognize the form of the matrix formula. math.stackexchange.com/questions/4186355/…

gavin newsom is emailing and asking for $3 by midnight. this is distressing. i thought he came from money.

leslie townes

1:37 AM

i just paid my registration renewal fees on my car. it caused me to notice that my wife's registration expired a year ago. she should do something about that.

EM4

tell her asap.

leslie townes

she says she has the receipt of paying for her registration but that it never came. i said, maybe contact the department of motor vehicles about that. a cop doesn't want to see a receipt when he asks for your registration.

i don't think cops even look at minivans, however. it's just noise to them

there was a period where she kept getting pulled over for broken turn signals and a missing license plate (it kept falling off). a level of chaos that i could not put up with.

Ted Shifrin

I worry too much to do things like that.

leslie townes

one time a collection agency was going after her for some fines on overdue books at some library at a place we lived at like 10 years ago. she didn't tell me. i overheard it on a phone conversation.

i said, please let me clear this up, i can't live my life as a fugitive and this is probably f-ing up both of our credit for no good reason.

Ted Shifrin

Collection agency for a library. Now I've heard everything.

leslie townes

1:45 AM

it was several hundred dollars. i don't know what she was doing although i do sometimes find library books around the house.

Ted Shifrin

rolls uncountably many eyes

EM4

Hahahah "I can't live my life as a fugitive" this made my night.

you sir @leslietownes are legend.

leslie townes

:)

one time we were moving a microwave (our apartment had an in-built one, so we didn't need the one that we owned), and the door of the microwave popped open and two library books came out.

i don't know why. i may have married an insane person.

Ted Shifrin

She says the same.

EM4

HAHAHHAHA!

I hope the books weren't cooking books.

Ted Shifrin

1:48 AM

They were cooked books.

leslie townes

she'd borrowed the only copy of someone's thesis from a university library and never given it back, i think that pissed them off.

Hawk

this covid vaccine is slaying me right now...

Ted Shifrin

Quite irresponsible.

leslie townes

hawk, good luck with that. i had about 48 hours of misery.

EM4

you took pfizer?

Hawk

1:50 AM

Moderna

leslie townes

i had pfizer and it was a doozy.

EM4

same, Pfizer put me to sleep in my final exams.

Hawk

it feels like someone punched my left arm and bruised it

Ted Shifrin

I had Moderna. I keep commenting that it’s supposed to be harder on younger folks.

Oh. Site pain is usual.

leslie townes

i spent about a day in bed, my wife says i was incoherent at times. and then a day of feeling miserable. the second shot was just a really bad punch in the arm.

EM4

1:51 AM

Moderna my friend took it...he was sweating like crazy. I told him ahhh Olympics came early.

Hawk

i slept like 3 hours after the vaccine because it made me feel drowsy

Ted Shifrin

You’re almost always incoherent.

leslie townes

you aren't wrong. hahaha

Ted Shifrin

How could she tell the difference?

leslie townes

it's all relative. our daughter will interrupt dinner conversation to tell us about an opossum who teamed up with some ducks to climb a mountain. the boundary between reality and fantasy is negotiable in this house.

Ted Shifrin

1:53 AM

Well, yes, a 3 yr old ….

leslie townes

i have no excuse. i was raised in a broken home by a family of circus performers.

Ted Shifrin

So you repeatedly reiterate.

leslie townes

that's somewhat pleonastic.

Ted Shifrin

Intendedly so.

leslie townes

i just wanted to say 'pleonastic.' life is scrabble and i want the triple word score.

Ted Shifrin

1:57 AM

I know it from French syntax.

Hawk

2:30 AM

this might sound a bit ambiguous, but if I have a plane, then the Riemann integrable is not necessarily 0, but the Lesbegue one is right? since a plane is a subspace of R^n?

*and the plane has measure zero.

leslie townes

are you integrating something? if so, what?

Ted Shifrin

My question.

Hawk

yes but literally anything

so not the zero funciton

Ted Shifrin

No. Make sense.

Hawk

literally any f, lets say it is continuous so no funny business

Ted Shifrin

2:32 AM

Do it in $\Bbb R^2$.

Hawk

an might as well make it positive

leslie townes

i'll give an example. it is common in multivariable calculus to compute 'double integrals' of two-variable functions f(x,y) defined on a subset of the plane. this is meaningful and can be nonzero.

Hawk

yes the Riemann one is, but not necessarily the lesbegue one right?

leslie townes

the 3d lebesgue measure of the domain of f does not really enter into that kind of computation.

Ted Shifrin

He wants to integrate over a thin set, but he’s saying nonsense so far.

leslie townes

2:35 AM

my daughter is singing happy birthday to her stuffed cat toy. every day is a birthday at this house.

Hawk

don't we have inequality always \int_E f \leq K\mu(E)?

and the RHS is 0 in this case.

yeah that's why i say it will be ambiguous from the getgo

leslie townes

mu(E) in that example might be computed at a dimension lower than the dimension where you look at the graph of the function.

for example the line from (0,0) to (1,0) as a subset of the plane has planar lebesgue measure zero. this doesn't mean that the integral from 0 to 1 of f(x) dx is going to be zero.

the mu(E) there is a measure on the domain of f, which might not be where the graph of f lives.

Hawk

ok so my claim is false.

thanks.

Ladies and gentlemen

0

Q: Three PhD courses on the same semester

I will begin my PhD in Mathematics next month, with no scholarship, but depending on my scores I might get one later in the course of the PhD. My question is: it is possible for me to take three PhD courses at the same time? The three available are: Smooth Manifolds, Functional Analysis and Rings...

Can a course in functional analysis only cover up to Schauder basis and Banach algebras?

Also, what is Banach algebras as a topic in a course.

Ted Shifrin

You should ask your faculty for advice, not us. We don't know the level of the courses, how much work is assigned, whether there are exams, etc.

Ladies and gentlemen

2:42 AM

Oh nevermind the question got edited haha

Ted Shifrin

In the US certainly strong students will take 3 Ph.D.-level courses at a time.

Oh, it wasn't your question ...

Ladies and gentlemen

No I usually do really well in school

Maybe I've just been to easy classes

well there was this one class that I took that was hard, but the hard part was the oral presentation

but the teacher was my friend because I helped her in some teaching projects for kids

So I didn't get pressed super hard

In other news I'm using stylus to get a dark mode

this is a very preliminary version

Oh I just realized I'm ignoring the graph-isomophism tag for some reason

leslie townes

3:12 AM

nobody asked but that sounds like a standard course in functional analysis. fairly common to take in combination with two or three other classes but of course the difficulty and expectations vary tremendously by instructor and whatever else is on the syllabus. and other course offerings.

my grad school did not have a class called functional analysis. the stuff was spread across a second-semester general analysis class and classes specific to certain kinds of functional analysis. the 'intro' class syllabus would have looked very elementary because there were whole courses devoted to the other stuff.

if there aren't i could see a tendency to pack more subject matter into one or two classes.

for more answers to questions that nobody asked, visit my patreon and subscribe to my newsletter.

Ladies and gentlemen

Can I subscribe to your pateron with LeslieCoin ?

leslie townes

a course on banach algebras could be expected to do a lot of stuff in B(H) and abstract C* algebras and also fourier analysis in L^1(G).

G hopefully a locally compact abelian group, maybe more general

yes, investments in lesliecoin are welcome.

all of the world's leading investors are placing enormous amounts of fiat dollars (which are imaginary) into lesliecoin (which is real and durable and indelible on a blockchain).

i will keep hawking this here until the SEC shuts me down.

Ladies and gentlemen

Hopefully you don't get whacked like McKafee

leslie townes

most of lesliecoin is served on an offshore oil rig that is not subject to any law, so i don't see that the SEC would even have jurisdiction to whack me.

Ladies and gentlemen

Well, he got whacked in spain

leslie townes

3:22 AM

if they do, my attorneys will file jurisdictional objections, you can count on it.

Ladies and gentlemen

great news!

troll bot

3:41 AM

Ladies and gentlemen

dropping the dupe hammer is clearly the chadest move ever

leslie townes

i see how that could be annoying on other SE sites but duplicates on math.SE tend to be verbatim duplicates. you don't even need to change the numbers (even if doing so would not change the problem). it's exercise 5 on page 46 of popular textbook X over and over again.

Ladies and gentlemen

not everything that shines is gold

not everything that gets closed as dupe is a dupe

But I don't care

leslie townes

i do think dupes ought to be reserved for verbatim dupes. i don't vote to close if it's conceptually similar.

Ladies and gentlemen

I have all of my questions and answers saved in my drive

so when people delete mys tuff I don't really care anymore

besides my nooby database is actually searchable

leslie townes

3:51 AM

there are PDE where you change + to - and it's a completely different universe, so i can't say "blah blah just change the numbers" in good faith.

even if that's what it is.

Ted Shifrin

You mean hyperbolas and ellipses aren’t the same?

leslie townes

ugh. geometers.

i guess that would be another example of that, yes.

if you want to come up with completely convoluted things outside of the realm of PDE

Ted Shifrin

I was referring to PDEs.

leslie townes

i withdraw my hostility to geometers if hyperbolas and ellipses were meant in the PDE context.

Ted Shifrin

I spoke merely symbolically.

Ladies and gentlemen

4:03 AM

0

Q: books on curves

I've read a few introductory texts on differential geometry, focusing on my specific interest in plane and space curves in Euclidean space. The breadth of coverage seems more or less the same (Frenet, Hopf, Fary-Milnor, four-vertex, isoperimetric, etc.). I have an impression that some advanced or...

differential-geometry book-recommendation curves

leslie townes

wonderful.

i've never heard of aberrancy. i think if you've digested the other stuff you are probably going to be OK with some other notion. i did read a very elementary book on the algebraic geometry of real planar curves but i forget who wrote it.

once you know that a curve is determined up to isometry by its curvature i don't know what else to do with it.

Ted Shifrin

There’s affine and projective geometry to do with curves, but I have no idea what aberrancy is.

Ladies and gentlemen

4:26 AM

the dark theme of the site is pretty cool ngl

copper.hat

i'm still festering here

Ladies and gentlemen

I don't know what that means :;(

leslie townes

copper is pustulent

Ladies and gentlemen

I wonder what happened with the seagull

leslie townes

it flew away.

copper.hat

4:28 AM

pus is a great word for many reasons

Ladies and gentlemen

poetic

copper.hat

poetic pustulence

leslie townes

one unexplained bird phenomenon i have observed is that seagulls and crows almost always seem to have french fries. if you see them flying with food, it's going to at least look like a french fry. i want to know where they get them.

copper.hat

the great chipper in the sky

Ladies and gentlemen

sounds like a 'Murica thing if I'm being honest

copper.hat

4:30 AM

ever since my suspension i have lost all interest in answering tedious detailed questions.

leslie townes

they ought to give badges for suspensions.

Ladies and gentlemen

they should

it means you're a man of the people

copper.hat

i have noticed two blue herons with rodents in their mouths recently

leslie townes

blue herons are f'ed up. i love to watch them at the bolsa chica preserve in OC.

Ladies and gentlemen

well, suspensions used to be bad back in the day

leslie townes

4:31 AM

they are very much about murder.

copper.hat

i am working on my second suspension, i have two lines going presently

Ladies and gentlemen

The day I got suspended I answered a totally non-psq

leslie townes

blue herons use their beaks as shivs and it's exactly what you'd expect on a prison yard when they get to using them.

Ladies and gentlemen

But it was a dupe

so after someone pointed it out I gold badge hammered it and copied my question to the dupe target

and then the dupe target got closed for being a psq

copper.hat

qoute from joe: Oh no, wait, I forgot this is the entitlement chapter

Ladies and gentlemen

4:34 AM

then I got suspended

and for the final twist it turned out the dupe wasnt a dupe

it got reopened and my solution worked for the "dupe" but not the "dupe target"

tragic shit if you ask me

leslie townes

i hope you're sorry

copper.hat

a lot of peculiar stuff

Ladies and gentlemen

I have repented for my sins thoroughly

I will no longer close as dupe

copper.hat

i haven't really

Ladies and gentlemen

I decided that I'd try to answer more bounties since those were more "legit questions"

but two of the bountied questions I've answered were featured in CURED after I answered them

copper.hat

4:37 AM

i only answer convex psqs now

Ladies and gentlemen

so maybe there's no hope

copper.hat

there never has been. it was a myth

2

Ladies and Gents

:(

Martin Sleziak

5:30 AM

@copper.hat If that's the case, have you considered changing your parent user? (So that you're still able to chat, in case of your suspension.)

I should maybe think about something like that, too.

copper.hat

@MartinSleziak Hi Martin, I was joking above; what do you mean by parent user? (I was only suspended for a few hours from chat.)

Martin Sleziak

By parent user I mean parent user: chat.stackexchange.com/users/35622

The one that you can see in your chat profile.

In my case, my parent site is Mathematics: chat.stackexchange.com/users/11287/martin-sleziak

Which means: 1. If I am suspended on Mathematics, I will not be able to talk in chat for the duration on suspension on this site. 2. If I am suspended on [mathoveflow.se], I will still be able to chat.

I guess it is better explained in this feature request by Mad Scientist: Don't extend suspensions from the parent site to chat.

huzaifa abedeen

6:40 AM

-5

A: i need "intuition" about fraction exponents like 4^(1.2) what it exactly means to do with number 4?

Please help me sir, my personal account sir, I am a swiggy delivery boy, my ID is hidayay4130@gmail.com, please help me sir, not working for 5 days, my contact number, personal account sir, please help me sir.

-2

A: i need "intuition" about fraction exponents like 4^(1.2) what it exactly means to do with number 4?

Please help me sir personal account sir swiggy delivery boy my ID hidayay4130@gmail.com please help me sir 5 days to not working my contact number personal account sir please help me sir https://www.facebook.com/ads/library/?id=432994437529521

copper.hat

@MartinSleziak Thanks Martin! My noise about suspension is mostly joking.

huzaifa abedeen

The swiggy delivery boy needs help

Balarka Sen

7:42 AM

@Thorgott The minimal geodesic goes from x to y, the vector is at y. The vector should point in the general direction of the geodesic ray obtained by extending [x, y] beyond y, ie at the tangent vector of [x, y] at y. So it should make acute angle with this tangent vector at y, hence obtuse angle with the geodesic [x, y] at y ie with the negative of the tangent vector

Mad Spaces

8:30 AM

let $ a, b ,c , d > 0 $ if $a-b < c +d $ how can we mathematically transform the equation, or conclude somehow that $ b < c +d +a $ i see this in a lecture nd i am not sure how one reaches this, however i think it is true.

robjohn

8:44 AM

Hey, Ted! Up late?

@MadSpaces you can show $b$ is greater than $a-c-d$, but that's about it

Mad Spaces

what then rob?

Sorry if this is easy, but i am not seeing it

user3733558

say $\{a,b,c,d\} = \{1,10,1,1\}$, then $1-10<1+1$, but $10>1+1+1$, or am I missing something?

Mad Spaces

hmm well, i might have not told the whole story, its basically distances and i just used letters for them, let me write it correctly out

$d(x_n, y_n) - d(x,y) \leq d(x_n,x) + d(y,y_n) $ he then concludes that $ d(x,y) \leq d(x,x_n) + d(y_n,y) + d(x_n, y_n) $

And obviously between is written "trivially one sees" xD

wierdly enough he changes the order for the points, but that should not make a difference right?

i guess i have to mention that $(x_n, y_n) \rightarrow (x,y) $

Which then would make sense for n goes to infinity, but seriously nothing is mentioned of that, so i am a bit confused

robjohn

9:04 AM

@MadSpaces $d(x,y) \leq d(x,x_n) + d(y_n,y) + d(x_n, y_n)$ That is true by (two applications of) the triangle inequality.

Mad Spaces

Oh......

user3733558

yeah, @robjohn beat me to it, I was about to say the same thing

robjohn

application 1: $d(x,y)\le d(x,x_n)+d(x_n,y)$

application 2: $d(x_n,y)\le d(x_n,y_n)+d(y_n,y)$

use the estimate for $d(x_n,y)$ from the second in the first: $d(x,y)\le d(x,x_n)+\color{#C00}{d(x_n,y)}\le d(x,x_n)+\color{#C00}{d(x_n,y_n)+d(y_n,y)}$

@MadSpaces: does that make sense?

Mad Spaces

yes... i see why he did this, the formulation of his proof is quite misleading

But now i see what he means.

thank you

he wanted to reach this equation, simplified $ -a < b < a \Leftrightarrow | b | < a $ i did not knwo this relationship is something used, but now i see its true

yes sure

10:19 AM

@TedShifrin I just took the exam, I did all integrals correctly!! The hard things were the theory questions, they were really subtle... Only 3 people passed the exam lol, i was one of the lucky guys. The highest grade was 20/30 lmao

yes sure

11:17 AM

hmmm my message just duplicated lol. sorry about that

troll bot

11:45 AM

I frickin like this song

epsilon-emperor

12:38 PM

Is anyone here a fan of Convex Geometry?

If yes (or even if you like Analysis), please check my latest question on Dvoretzky's theorem.

It needs hardly any background to understand, and I'd appreciate any help.

Alessandro Codenotti

Your question is extremely long and is actually many questions, often people don’t like this

Anyway I don´t have time to go through the whole thing, but I can offer some intuition for question 1, that might also help with question 6

Namely suppose that $T\colon X\to Y$ is an injective linear operator, what does it mean that $\|T\|\|T^{-1}\|<1+e$, geometrically?

epsilon-emperor

@AlessandroCodenotti Everything is pretty much related, and I had difficulty separating this into several questions :(

Alessandro Codenotti

It means that if you take a sphere in $X$ then its image through $T$ is still kind of spherical (up to $e$)

epsilon-emperor

@AlessandroCodenotti Aha, I see. I guess that makes sense

Alessandro Codenotti

Think about what happens in the easiest case: finite dimension. $\|T\|$ is the biggest eigenvalue of $T$, while the norm of the inverse is the reciprocal of the smallest eigenvalue of $T$

So you´re taking a ratio telling you ¨how much the direction that is stretched the most compares to the one that is stretched the least¨

If this ratio is close to $1$ then all directions are stretched by pretty much the same, so sphere remain kind of spherical

epsilon-emperor

12:48 PM

Yes, agreed. Thanks!

Alessandro Codenotti

While if this ratio is very big it means that some direction is stretched way more than another, so you get long ellipses instead

So it makes sense to use the Banach-Mazur distance as a measure of how close two spaces look to each other. In this sense sections that are within $1+e$ of euclidean ones are almost spherical

Vitalie Ghelbert

Conclusions so far: $ \forall x, \frac{x}{0} = \nexists, \ \therefore \ 0^0 = \frac{0}{0} = \nexists $

Alessandro Codenotti

For question 5 what´s the difference between a $k$-dim subspace and a linear embedding of $\Bbb R^k$? Exactly, there is none

For the randomness part it´s explained on page 50 how to use the probability measure on the orthogonal group

you could also look at the Stiefel manifold of $k$-dim subspaces which comes with a canonical invariant measure as well, that should be the same

epsilon-emperor

@AlessandroCodenotti Agreed, I had trouble understanding the "random embedding" though

It's on Pg. 50, yeah

For any particular unit vector $\psi\in\mathbb R^k$, why is it highly probable that $\|T\psi\| \approx M$?

anakhro

1:45 PM

happy canada day to our canadian math friendz

Balarka Sen

2:30 PM

not a proud day for canadians this time, @anakhro

at the very least it shouldn't be

anakhro

2:49 PM

It wasn't a proud day when those atrocities were done and hidden. That doesn't stop canadians from celebrating their shared and very diverse community while keeping in mind that they need to move towards reconciliation.

Balarka Sen

While the Canadian PM is still perusing legal battles with indigenous survivors in court, hmm...

Astyx

What's up with Canada?

leslie townes

if you don't know, don't google it. it's horrific.

anakhro

The usual: the government ignoring first nations people.

leslie townes

or maybe do google it but expect it to ruin your day.

Balarka Sen

2:53 PM

@leslietownes yeah

Astyx

They're also having record high temperatures right?

Edward Evans

leslie townes

the climate certainly isn't helping.

Thorgott

Ah ok, I got the directions mixed up. So, to see this formally, say the point is a regular point. $f\circ\gamma$ is a strictly increasing function, so $df\vert_y(\gamma^{\prime}(t))=(f\circ\gamma)^{\prime}(t)>0$ ($t$ the time s.t. $\gamma(t)=y$). $df\vert_y$ vanishes on a hyperplane and $\gamma^{\prime}(t)$ lies on the positive side of that hyperplane. In order to read positivity off of the angle with $\gamma^{\prime}$, we want that $\gamma^{\prime}$ is orthogonal to that hyperplane.
This is plausible geometrically, but I don't quite see it.

leslie townes

i've never been happier to live near the ocean. it can boil inland but stays around 80F here. and we're on a steep hill so probably won't lose the house to rising sea waters.

earthquake, maybe.

Balarka Sen

3:00 PM

@Thorgott I am very confused. $\gamma$ here is the geodesic joining $x$ to various points near $y \in U$, or a flowline of the gradient-like vector field?

Thorgott

gamma is the minimal geodesic from x to y

Balarka Sen

OK, and what exactly do you want to show?

Sorry for not following

Thorgott

you want to say that the angle a vector field forms with $\gamma^{\prime}$ (at $y$, the rest follows locally) determines whether $f$ decreases along the vector field or not

it seems to me that for this to be the case, $\gamma^{\prime}$ should be orthogonal to the kernel of $df$

Balarka Sen

Right, so I have a gradient-like vector field $X$ such that $\langle X, \gamma' \rangle < 0$ pointwise. I need to show $f$ strictly decreases along the flowlines of $-X$. I agree this requires a proof, but it should not be too hard. One moment.

In the Morse case, $\langle X, \gamma' \rangle = X \cdot f$, but we cannot quite say that here.

The issue is that $y$ can be part of the cut locus of the exponential map. If it was outside the cut locus, then kernel of $df$ would have been the tangent plane to the geodesic sphere passing through $y$

Which is obviously complementary to $\gamma'$.

Thorgott

ah yeah, if it is the tangent space to the geodesic sphere, this follows from the Gauss lemma

I don't see that though

you agree this holding is equivalent to the orthogonality of $\gamma^{\prime}$ with $\ker(df)$, right?

Balarka Sen

3:08 PM

Yeah so the crucial case is $y$ is actually a singular value of $\exp_x$. If it's a regular value it is that, because $\exp_x$ is a local diffeomorphism at $y$ (or it's preimage under $y$).

@Thorgott Yeah, I agree.

Well, I mean, not really. Because $f$ can be singular

$df$ need not make sense

Thorgott

aren't we assuming we're at a regular point

Balarka Sen

But $f$ as a function need not be differentiable. This notion of regularity has nothing to do with differential topology

Thorgott

wait it isn't smooth??

Balarka Sen

Of course not. It's Lipschitz.

Thorgott

I thought there was no issue away from x

Balarka Sen

3:14 PM

There is trouble near the cut locus.

For example, try it on a cylinder, or an ellipse. It'll be a mess.

Thorgott

yeah ok, I get your point

but I hate this

Balarka Sen

There may be points $y \in M$ such that there are two minimal geodesics $x \to y$ meeting at very awkward angles at $y$

Those will typically not be smooth points

Thorgott

I hate to admit that you're making sense

Astyx

what's the context?

Balarka Sen

It's actually very beautiful. But let me try to answer your question, it's a good one.

Thorgott

3:17 PM

3

A: Diameters of Lens spaces

The diameter of $L(p; q)$ is exactly $\pi/2$. There might be a more direct way to see this but it follows from Morse theory with the Riemannian distance function, see Gromov's "Curvature, diameter and Betti numbers". Given a Riemannian manifold $M$ and a point $x \in M$, the analogue of a Morse f...

Balarka Sen

OK, we want to show that for a flowline $\sigma$ of $-X$, $g(t) := d(\sigma(t), x)$ is strictly decreasing.

The point is we don't know if this thing is differentiable at $t$. We can nudge it off a little bit though

Consider $d(\sigma(t), x')$ where $x'$ is very close to $x$. That should now be differentiable at $t$

This is because $d : M \times M \to \Bbb R$ is Lipschitz, it's almost everywhere differentiable. If stuff is measure zero I can nudge

Thorgott

3:35 PM

oh right

that was a theorem

it had some name, which I forgot

Balarka Sen

Radamacher or something

Thorgott

rademacher, I think, right

Balarka Sen

So I guess given $t_0$, I can always pick $x'$ such that $h(t) := s + d(\sigma(t), x')$, where $x'$ lies along the minimal geodesic joining $x$ and $\sigma(t_0)$ and $s = d(x, x')$, is differentiable in a neighborhood of $t = t_0$. So $h(t_0) = g(t_0)$, and $h(t) \geq g(t)$ in a neighborhood of $t_0$. If $h'(t) < 0$ in a neighborhood of $t_0$, hence $h$ is strictly decreasing in a neighborhood of $t_0$, hence so is $g$

This is some comparison theorem. Sturm, probably

I mean no, this is dumb lol

So suffices to do the calculation for the nudged distance function which in fact is smooth

If $g(t)$ is smooth this is kind of clear. $g'(t) = \langle -X, \gamma'\rangle$ exactly, where $\gamma$ is a minimal geod. joining $\sigma(t)$ to $x$.

Oops, signs are all wrong.

But I hope this is convincing

I can add it to the answer in more detail if desired

Thorgott

3:58 PM

@BalarkaSen that last claim is clearly false in this generality, no?

Derivative

How do you figure out what the professor wants on a test? For example in a calc test we were asked to calculate $\lim_{x\to 0}\frac{\ln(x+1)}x$ and I used a change of variables to transform it into $\lim_{x\to 0} \frac{e^x-1}x$, which we proved in class to be 1 so I just stopped there, but turns out the professor wanted us to prove that too from the definition of $e$.

Astyx

When in doubt, prove that you know

Derivative

it didn't occur to me that I'd have to prove that, because we proved that in class. Plus the test was really short on time

leslie townes

professors are fallible. if it seems that somebody is being slightly inconsistent, that may be the case.

there isn't one script that everybody follows.

Balarka Sen

@Thorgott Hm? $g(t_0+\varepsilon) \leq h(t_0+\varepsilon) = h(t_0) + h'(\xi)\varepsilon = g(t_0) + h'(\xi)\varepsilon < g(t_0)$.

Derivative

4:07 PM

okay so given the fallibility of professors, how do I deal with this problem so that I can get the grades so I can get into grad school?

Astyx

Be as precise as possible

leslie townes

if you have time to go back to first principles or rules that you can indisputably use because they are established and of general application, do that, even if some creative way might be quicker.

Ladies and Gents

Well, now you know that professor is delicate you just write everything as much as possible

Thorgott

@BalarkaSen only for $\epsilon>0$

Balarka Sen

That's what decreasing means, right?

Thorgott

4:08 PM

I mean, it can also be smaller than $g(t_0)$ to the left of $t_0$

it's only decreasing on $[t_0,...)$

Balarka Sen

Huh?

Oh, but you have such a $h$ for every $t_0$

leslie townes

my real analysis instructor offered extra credit for a proof of something on a homework assignment. i submitted a proof and was not given extra credit because i had not submitted the solution they were expecting. it was a fine solution, it assumed nothing we hadn't proved. she just didn't know enough analysis to understand it.

Balarka Sen

If it's decreasing on $[t_0, \cdots)$ for every $t_0$ it's decreasing

Derivative

@LadiesandGents But now it's too late to get a good grade in calc 1, and no none of the homework I turned in has been graded yet

Thorgott

ok yeah, I guess it doesn't make a difference for what we want to ultimately conclude

leslie townes

4:11 PM

derivative, i have never been in the driver's seat on this issue but i think performance in calc 1 is regarded as less probative of success in grad school than performance in other classes. which is not to say to give up on that class, but it wouldn't be the end of the world if you weren't at the top of the class.

Ladies and Gents

maybe you need to investigate how harsh teachers grade beforehand

Balarka Sen

Yeah this seems to work. This is basically like the analogue of nudging a Morse function to be Morse-Smale, but you cannot do this globally anymore. Locally you can always dodge the cut locus

leslie townes

i got a C in linear algebra, which happened to be my worst grade in college, and later a phd in what was basically linear algebra.

Derivative

what does a C mean out 10?

Thorgott

@BalarkaSen is it immediately clear that this is the derivative when we're smooth?

leslie townes

4:13 PM

6 or 7, depending on how you round.

but grade inflation plays into this, it probably would have been a 4 if measured objectively.

Derivative

4 is a fail in my university

leslie townes

i probably would have failed without generous partial credit. i knew very little about linear algebra. the instructor was terrible, although i didn't know this at the time.

Thorgott

it is clear that $\gamma^{\prime}$ is orthogonal to $\ker(df)$ by the earlier argument, but does that already imply this

Balarka Sen

@Thorgott Oh yeah at a regular value $y$ of $\exp_x$, derivative of $d(x, -)$ is $\gamma'$, isn't it?

Just by chain rule and the fact that derivative of norm on $\Bbb R^n$

is $x/\|x\|$

Then use chain rule on $d(x, \sigma(t))$. $\langle \gamma', \sigma' \rangle = \langle \gamma', -X\rangle$

By the way, this is one half of the quarter-pinched sphere theorem, which says a simply connected manifold with $1/4 < K \leq 1$ is homeomorphic to a sphere

Thorgott

4:31 PM

what's the derivative of $\exp_x$ itself at points other than $0$, I actually don't know this

Balarka Sen

That doesn't matter as long as it's an isomorphism, we just need to know the sign. No point in thinking more about this particular detail

I mean just forget this junk, the derivative of $d(x, -)$ is the gradient, direction of steepest ascent, which must be the direction of the minimal geodesic joining $x$ and $y$, at $y$

Thorgott

yeah ok

but that's true at the cut locus as well, right

ah wait, the issue is that differentiability fails at the cut locus

Balarka Sen

Yes

Thorgott

but whenever we're differentiable, exp is regular and this works

Balarka Sen

Correct

Thorgott

4:43 PM