Mathematics - 2020-12-05 (page 2 of 2)

Balarka Sen

22:00

Yeah

I think every closed $3$-manifold is a boundary of a $4$-manifold, modulo orientability hypothesis I might be missing. Every closed $3$-manifold (adjective?) is a surgery on a link in $S^3$, $S^3 = \partial D^4$ so I think you can do clever handlebody stuff at the boundary.

Being nullcobordant is I think a surgery invariant

Ted Shifrin

Yes, every closed $3$-manifold is orientable anyhow :P

This is stuff I knew in grad school but never taught or used, so ... Hippa's meme.

Balarka Sen

What about $\Bbb{RP}^2 \times S^1$?

Ted Shifrin

Hmm, what am I missing?

Balarka Sen

Why did you say it's true? I think I get this wrong everytime as well

Ted Shifrin

Because it's a famous fact !! LOL ...

Obviously we don't need simply connected, because every simply connected guy in every dimension is orientable.

Balarka Sen

22:04

I get this wrong because I keep RP^n as canonical nonorientable guys in mind, but RP^3 is of course very orientable as you'd say.

Ted Shifrin

Yeah, only even $n$ for that one. WTH are we thinking of, darn it?

Maybe it's time for me to retire for good :P

Balarka Sen

We're probably thinking every orientable $3$-manifold is parallelizable, or something like this

Ted Shifrin

Yeah, that's it.

Whew

Balarka Sen

Haha yeah no this is confusing

Ted Shifrin

Not really, but glad you sorted it out :)

I'm not sure I ever knew that proof, either. We get a trivial line sub-bundle for free.

There's got to be some obstruction that vanishes because of dimension.

Balarka Sen

22:07

Yeah, I don't know a proof either! But maybe it follows from our discussion about cobordism?

Thorgott

every orientable 7-manifold is a boundary, apparently

Balarka Sen

If something is oriented nullcobordant it's parallelizable... by Thom's theorem haha

@TedShifrin I think this is the morally correct idea but I don't know enough

Ted Shifrin

@Thor: It's true in every dimension not divisible by $4$, I believe.

Assuming orientability.

Actually, Balarka, this proof I did know/understand at one point.

Thorgott

crazy

Ted Shifrin

There's an argument based on spin structure, but I thought I remembered something easier.

Balarka Sen

22:11

hmmm

Ted Shifrin

Ah, PVAL (I miss him!) gave an argument here.

Balarka Sen

every 3-fold is spinnable?

Ted Shifrin

yeah, orientable, still.

Balarka Sen

whoops right

oh ok

all SW classes vanish somehow

Ted Shifrin

Odd. I would have started with a trivial line bundle (because $\chi = 0$).

Balarka Sen

22:14

that definitely means you can take three lin indep sections by the zero locus interpretation

Ted Shifrin

And then I just want to frame the orthogonal complement.

Well, how do you get from mod 2 to $\Bbb Z$?

Balarka Sen

oh fair enough

i don't know

you're right, lifting the frame bit by bit must be the correct argument

Ted Shifrin

I think I must have read a proof in Steenrod years ago. I wish I'd kept that book.

On a more geometric note, @Balarka, what do you think of this discussion I had with love-sodam on his post?

His book seems to have a number of false questions. It's discouraging him; I think it's sort of funny. It makes me prouder of mine :D

Balarka Sen

Wow

I was thrown off by the question until I saw your answer saying it's false

Ted Shifrin

You probably did my exercises about its being true if the geodesics are at right angles (or any constant angle).

I'm giggling, because this reminds me of the angry emails I got from the guy who insisted that Fraleigh had a question saying that any number of degree 4/$\Bbb Q$ was constructible by ruler and compass, so stuff in my book had to be wrong.

Balarka Sen

22:22

@TedShifrin Yeah I know this one I think

Ted Shifrin

I think there's a tendency for students to trust all books. Mistaken tendency.

Balarka Sen

That's ridiculous!

Ted Shifrin

The pretty argument is with local Gauss Bonnet. Look at holonomy around a geodesic parallelogram. (Usual contradiction proof if $K\ne 0$ somewhere.)

Balarka Sen

Right

Ted Shifrin

Of course, I love the moving frames proof :P

I think love_sodam gets particularly riled up, though, based on what I've seen here.

Karim Mansour

22:23

@TedShifrin I found few typos in your book. Though perfect.

Balarka Sen

Wrong exercises are the worst.

Ted Shifrin

@Karim: I corrected a bunch in a corrected printing. And I do keep a page of errata. But there are no blatantly false exercises. I learned after my first book to write a solutions manual before sending the book to the publisher.

Joe Shmo

stumbled on this exercise - and I have an answer - but I wonder if there's a more analytic solution. Like idk something about the 20th derivative of $f' - Kf$ being zero or something; If $f$ is differentiable on $[0, 1],\ f(0) = 0,\ |f'(x)| \le K|f(x)|$, then $f = 0$ on $[0, 1]$.

Ted Shifrin

I was so upset when I found errata (and typesetting flubs) in my algebra book (first). But then I learned that we all make mistakes and I just accept it. But this new geometry book has some whoppers in it.

Karim Mansour

Yeah haven't found any false exercises. I have solved half of the book I might keep my solution manual when I teach this class at some point.

I won't post it online

Ted Shifrin

22:25

There actually is one for teachers available from the publisher.

No, please do NOT post solutions online anywhere!

I finally started putting my exam solutions behind a secret password for my students.

There are a few typos in some exercises. Check the errata sheet, @Karim.

Oh, Taylor's remainder formula, @JoeShmo.

Spivak has a few exercises of this flavor, too.

Joe Shmo

ah ha! I knew it

Thanks

Karim Mansour

yeah will check it I will definitely use your book when I get chance to teach this class

Joe Shmo

I did MVT and creeped to $0$

Ted Shifrin

Well, to be honest, @Karim, very few places have students that can handle this sort of course.

Karim Mansour

Yeah I agree

Ted Shifrin

22:28

And very few places have an integrated linear algebra/multivariable course, let alone a proof-oriented one.

Karim Mansour

My undergrad didn't have that

It is a shame

Ted Shifrin

Places like Harvard, Yale, Vanderbilt, UGA (for now) have it, but not very many places.

Thorgott

The condition says that the logarithmic derivative of $f$ is bounded, but $\log f$ tends to $-\infty$ at $0$, so that cannot be

Ted Shifrin

Chicago, of course. But they go off the deep end typically and do graduate functional analysis and no multivariable.

Thorgott

ok, that's not a rigorous argument, but hey

Ted Shifrin

22:30

Oh, @JoeShmo, I take it back. I thought your $f$ solved some ODE.

Thor is right. The right proof is based on a famous thing that shows up in differential equations. Gronwall's inequality.

It's a wonderful theorem (and I did add that to Spivak's 4th edition as an exercise).

Khallil

@TedShifrin I'm guessing it doesn't go any further than this? lol ($f \colon x \mapsto xx^{\top}$) $$ \begin{aligned} \text{d}(\text{tr} \circ f)(x) v & = \text{d} \text{tr}(f(x)) \circ \text{d} f(x) v \\ & = \text{d} \text{tr}(x x^{\top}) (xv^{\top} + vx^{\top}) \\ & = \text{tr} \left( (xv^{\top} + vx^{\top})(xv^{\top} + vx^{\top})^{\top} \right) \\ & = (xv^{\top} + vx^{\top})^{2} \end{aligned} $$

Ted Shifrin

Whoaaa.

It has to be linear. What's all this stuff?

You're misinterpreting basic stuff.

Khallil

oh no I have forgotten how to do maths >_<

Mike Miller

Did someone already tell Thorgott everything he needed to know

Ted Shifrin

Trace is linear, so you take trace of $df(x)v$.

Oh, magical @Mike is here. Yes, but then Balarka and I got stuck on things.

Mike Miller

22:34

Stuck on anything non-trivial, though?

;)

Balarka Sen

Magic Mike lol

Thorgott

Ah, here's how to fix my idea. If $f$ is not the constant $0$ zero function, there is some interval $(a,b)\subset[0,1]$ such that $f(a)=0$, but $f(x)\neq0$ for all $x\in(a,b)$. Then $\log f$ is well-defined and differentiable on $(a,b)$ and has bounded derivative. But bounded derivative on bounded interval implies bounded by FTC, yet $\log f(x)\rightarrow-\infty$ as $x\rightarrow a$.

Ted Shifrin

@Khallil: What does it mean to say $T$ is a linear map? What is $dT(b)$ for any vector $b$?

Mike Miller

I'm just playing

Ted Shifrin

Actually, @Thor, if you've never seen Gronwall, look it up. It's beautiful, and it's based exactly on your differential inequality thing.

@MikeM: The "best" proof that any orientable 3-manifold is parallelizable.

Joe Shmo

22:36

nice, I'll look it up

Mike Miller

There is no way to avoid Wu's formulas I don't think

Once you know that an orientable 3-manifold is spin you're basically finished

Balarka Sen

What if you use Thom's theorem instead

Thorgott

I've not seen it, this is me being ad hoc

Mike Miller

How are you getting that M is spin?

Thorgott

I'll add it to my reading list

Mike Miller

22:37

Let's clarify this in the next 2 minutes so I can get back to commenting on projects

Balarka Sen

oh I dunno spin, I was thinking you can prove by hand 3-folds are orientable nullcob, so parallelizable

Ted Shifrin

I thought there was a straight-forward obstruction theory argument (e.g., in Steenrod).

Coming from having a trivial line bundle and then being left with a $2$-plane bundle.

Joe Shmo

here's a proof based on the MVT - suppose WLOG $f(b) > 0$, then on $f(b) = f(b) - f(0) = f'(c) (b-0) = f'(c) \cdot b \le Kb \cdot f(c)$. You could iterate $c \rightarrow 0$, whence by continuity of $f$ you have $f(b) \le 0$

Mike Miller

Why does that 2-plane bundle have w_2 = 0? This is not clear to me.

@BalarkaSen What is your argument?

There are of course plenty of guys which are orientable null-cob but not parallelizable

Balarka Sen

oh er i guess

i dont have argument sorry

Mike Miller

22:41

I don't even know how to use that to show that w_2 = 0. All of the SW numbers have w_1 as a factor (or are w_3 = e mod 2 = 0)

Khallil

@TedShifrin $T(ax+by) = aT(x) + bT(y)$ for scalars $a,b$ and $\text{d}T(b)v = T(v)$

Balarka Sen

yeah good point

Mike Miller

IF someone can show that w_2 = 0 you get that the tangent bundle is classified by a map $M \to B\text{Spin}(3) = \Bbb{HP}^\infty$. Cellular approximation (AKA obstruction theory) implies this is null-homotopic.

Ted Shifrin

Oh, right $w_1(L)=0$, so no info.

Mike Miller

Showing that w_2 = 0 is necessary in any argument (obstruction theoretic or otherwise).

Ted Shifrin

22:42

@Khallil: Right, $dT(b) = T$. Look what you wrote above.

Mike Miller

Ted's approach (TM = R + P) gives us that w_2(TM) = w_2(P) but what then?

Maybe there is something but I don't see it :)

Time for me to go

Ted Shifrin

Bubye. Thanks.

Khallil

Oh my goodness

Never do maths after midnight >_<

Thank you, @TedShifrin

Balarka Sen

its like 4 AM and im about to start my assignment

Ted Shifrin

Balarka: You're a bad influence on everyone here.

You break everyone's sleeping.

Balarka Sen

22:45

i should put up a CONTENT WARNING

CONTENT WARNING: The author does not endorse severe insomnia and sleep deprivation as a lifestyle in any way, shape or form

6

Ted Shifrin

Nevertheless, some readers may find ...

Balarka Sen

lol

Shaun

This PhD situation is really getting me down, @TedShifrin. I know that I'm in a very privileged position & all, but, well, that kind of makes me feel worse, like I've wasted the opportunity.

Ted Shifrin

What makes you want to be an academic, and at what level of academe?

Khallil

I'm getting old

The 4am lifestyle is no longer viable

Alessandro Codenotti

22:51

@BalarkaSen I thought you were supposed to be the author of your assignments, but apparently not

Joe Shmo

Ted, is there a geometric "interpretation" to the logarithmic and exponential functions?

Ted Shifrin

LOL @Khallil. Define "old."

Balarka Sen

@AlessandroCodenotti Lmfao

skullpatrol

early to bed, early to rise?

Ted Shifrin

Hmm, area under curve @JoeShmo for log? I don't know what you mean. I can parametrize a hyperbola by $\cosh$ and $\sinh$, analogous to the circle.

Balarka Sen

22:52

@Alessandro A track

Alessandro Codenotti

Oh Porcupine Tree, noice

Back when Wilson didn't do pop

Balarka Sen

yeah man

wtf is his new material

Shaun

To be skilled enough to teach at a university level and get paid to do research seems to me a noble goal. I think I'd get a lot of satisfaction from inspiring students, hopefully, the way some lecturers have done with me. I get a lot of pleasure out of studying and I would be honoured to do it for a living.

Alessandro Codenotti

@BalarkaSen I haven't really been keeping up with his stuff after Hand. Cannot. Erase.

Balarka Sen

good lol

he's doing EDM now

wait let me find a meme

Ted Shifrin

22:54

What country are you in, Shaun?

Shaun

England.

Joe Shmo

he spelled programme earlier

Thorgott

@Ted I just had a look at the Grönwall inequality, very neat and natural result. It's also precisely what you get when you generalize my logarithmic derivative argument from above (though you'd have to pay a bit more attention to signs and zeros in general when taking that approach).

Ted Shifrin

So I don't know the variety of opportunities there. In the US there are different levels of colleges, and a number of teaching-oriented people end up in community (or 2-year) schools and find it fulfilling to make a difference to students there.

Sorry, @JoeShmo — I'm talking here and answering on main simultaneously.

Balarka Sen

@Alessandro

Alessandro Codenotti

22:56

Just noticed that Vildhjarta got 25 days left to deliver their promised 2020 album, why do I fall for it every time

lol

Ted Shifrin

Salut @Astyx.

Astyx

Hello

Balarka Sen

damn they keep delaying

skullpatrol

Hi

Ted Shifrin

@Thor: Yes, it is logarithmic derivative, with a limiting argument at $0$.

Very powerful tool, though.

Shaun

22:57

That sounds like the equivalent of a sixth form college, @TedShifrin, where students get their A Levels. I'm much more oriented towards conducting research than I am, teaching.

Alessandro Codenotti

@BalarkaSen their modus operandi seems to be "release a 90s long teaser and then wait for a few more years for the next one" (it sounds great I must say though)

Balarka Sen

thats hilarious

Ted Shifrin

Academic life is full of lots of responsibilities other than research, unless you are a research super-star. And let me just say, from my personal experience, that research is itself far more pressure-filled than teaching.

Alessandro Codenotti

woops I meant to link a specific one

Balarka Sen

the comments below those videos

yeah im listening to the one you linked

Thorgott

22:59

which book do I have to read to acquire the sacred knowledge that oriented 7-manifolds are null-cobordant

is this in Milnor-Stasheff

Balarka Sen

why do you want this exactly

Ted Shifrin

I think so, @Thor. I no longer have it on my shelf to check.

Balarka Sen

is this for the exotic sphere

Alessandro Codenotti

At least the album will be good when it comes out, Architects keep releasing new singles from the upcoming album and they are each worse than the previous one

Thorgott

yes

it's literally the second sentence

Mike Miller

23:01

You need to know the Wu formulas, identities for Steenrod operations, and the fact that a manifold is null-cobordant iff all of its Stiefel-Whitney numbers vanish

So yes that book

skullpatrol

@TedShifrin pressure from the pubiish or perish community?

Shaun

I was hoping, failing a career in academia, to work in the R&D department of some scientific company, so that I may have the opportunity to make a significant contribution knowledge.

Ted Shifrin

And, in the US certainly, from university administrations to bring in grant money and publish.

Thorgott

I feel like I'll have to read at least half that book as preliminary to this paper

I'm scared

EM4

what book?

Ted Shifrin

23:02

You don't need a PhD to work in industry, Shaun.

But you probably do need computation skills.

Balarka Sen

@Thorgott that paper is too hard

i dont have the patience to learn the algebra to understand it

characteristic classes are algebra at the end of the day

Mike Miller

It's also important understand that, say, 10% of PhDs at top schools ultimately land tenure-track jobs; this is not because 90% are poorly skilled but rather because there are comparatively few open positions (and none are opening much); plenty of incredibly intelligent and incredibly diligent people do not get tenure-track jobs, and not for lack of trying, and this causes a lot of distress

Thorgott

algebra isn't scary

but the topology is scary

Mike Miller

This is not to say you shouldn't pursue a PhD, which I'm not commenting on here; it's information one should be equipped with

EM4

I agree with you @Thorgott.

Ted Shifrin

23:05

The system is a bit different in England, though, @MikeM. However, now that Brexit has happened, they're separated from the rest of the EU and that has a definite impact on education and mobility of both faculty and students.

Balarka Sen

characteristic classes are the perfect topology an algebraist like you'd read thorgott

its just manipulating with laurent polynomials

Mike Miller

Fair enough, I'm commenting specifically on the US here --- sorry

Thorgott

hopefully

I'll have to figure all this stuff out within the next, uh, less than 3 months

Mike Miller

He's worried about proving that SW numbers 0 => (unoriented) boundary

Ted Shifrin

Well, I'm not sure how universal it actually is. I know that France has for years had a separate research environment that the best young people went into, and then they blended into teaching. I'm not sure what that status is now.

Shaun

23:07

I've often compared wanting to be an academic to wanting to be a professional football player, except it's less glamorous and more demanding. I'm aware the odds are against me. It looks like dropping down to the Master's is my first step towards an alternative career. But a PhD at least seemed attainable. My friends & family know I'm after a PhD. I don't know what to do. It's embarrassing.

Ted Shifrin

You giving a lecture on it, @Thor?

Astyx

What do people who don't get tenure track jobs do?

Thorgott

a seminar talk

Ted Shifrin

Ambitious for anyone, certainly an undergrad. Do the best you can, and black-box some of the worst.

Mike Miller

@Anixx I don't really want to do this conversation right now, probably should not have made my original comment (edited to clarify who this is directed to)

Yes you should black-box this fact

Ted Shifrin

23:08

@Shaun: They're not judging you (at least they shouldn't judge you), so you need to stop judging yourself.

Astyx

That's fine

Ted Shifrin

Honestly, when I was a Ph.D. student at Berkeley, more than 60 or 70% of the students ended up finishing with just a masters or less.

Joe Shmo

oh wow

that's a shocking statistic

Balarka Sen

it is impossible to do topology without algebra its so annoying

fuck it ill leave topology

Mike Miller

Become a banach space theorist

Ted Shifrin

23:09

I don't know the figure at UGA, but certainly over a third of our students finished with masters or less.

Balarka Sen

@MikeMiller probability on banach spaces

Joe Shmo

wow..

Ted Shifrin

So funny, a @Balarka, since Balarka 1.0 and 1.2 were all about algebra.

Joe Shmo

yeah I spoke to my cousin yesterday (did his doctorate at scripps, right by you)

Thorgott

diagram chasing is fun man

Astyx

23:10

What did 1.1 do?

Ted Shifrin

I think he skipped that version.

Balarka Sen

lol

Astyx

"We prefer not to talk about that version"

Joe Shmo

wanted to be an academic when he went in for his doctorate, quickly switched to wanting to go into a private lab, and eventually just wanted to get into industry

the maturation process of a doctoral student

Balarka Sen

what is this $\lambda$ invariant man

why would you define this

WTF milnor

Mike Miller

23:12

It's a relative characteristic class it makes sense

It's the same thing as the Chern-Simons functional

Ted Shifrin

Oh really?

Balarka Sen

where can i read the Chern-Simons approach

Thorgott

ah yes, of course

Ted Shifrin

Was Milnor around the same time as Chern-Simons (early 70s)?

Leaky Nun

Liquid tensor experiment, by Peter Scholze via Kevin Buzzard

Balarka Sen

23:17

My friend recently linked me an old post in AoPS by Peter Scholze where he gives a pure algebra proof that $\Bbb Z^n$ admits no nonconstant positive harmonic functions

Thorgott

profinite set lmao

Balarka Sen

He was like 14 then

Mike Miller

You have something discretized on an n-manifold with boundary (in the case of the Chern-Simons functional it is $\int_M tr(F_A^2)$ in $8\pi^2 \Bbb Z$ for A a connection on a bundle over a 4-manifold, in the case of Milnor it is the expression $45\sigma(M) + p_1^2(M) \in \Bbb 7Z$).

Then for (n-1)-manifolds, take an n-manifold bounding it and calculate that same expression. It will depend on the n-manifold, but only up to a discrete ambiguity: take two n-manifolds with the same boundary Y. Glue them together along their boundary. The expressions above are additive under gluing, and they lie in this discrete subspace.

Hence the ambiguity between the various things we are computing lies in this discrete subspace (8pi^2 Z in the first case, 7Z in the second case) while the general value can be anything (in R or Z). Thus there is a well-defined value: cs(A) for A a connection on a bundle over a 3-manifold (this lies in R/8pi^2 Z), or lambda(M) for M a closed 7-manifold (this likes in Z/7Z)

This is also the same as the proof of the Jordan-Brouwer separation theorem.

Ted Shifrin

Wow.

skullpatrol

@TedShifrin have you looked through Thomas' 14th edition calculus textbook?

Ted Shifrin

23:19

Hell no.

I hated Thomas from the outset.

It hasn't been Thomas in 4 decades.

Shaun

Well, it's not over yet. The thing is that I have an assessment on Wednesday and my supervisor & I are both pessimistic. By some miracle, I could scrape by with the minimum necessary for the continuation of my doctorate. It just doesn't seem likely, and this is largely because I've been unwell, not because anyone thinks I'm incapable. I have produced original research; I just have to write it up as a paper.

Ted Shifrin

The writing it up seems to be where lots of people get stuck (I don't mean this in a mean way).

user2103480

@Thorgott that's a nice one

Ted Shifrin

All I will say is that grad school is tough enough when one is mentally strong.

Thorgott

thanks

Mike Miller

23:23

I guess I should really say this is the same as the definition of mod-2 intersection numbers.

If you can find the ambiguity, when you mod out by it, you have something well-defined.

Balarka Sen

@MikeMiller Explicily, the integrand in CS is $A \wedge dA$ plus some terms?

Ted Shifrin

@Balarka: You should look at the original CS paper.

Mike Miller

something like that

Balarka Sen

Asking because I think $\int A \wedge dA$ for a magnetic potential $A$ computes linking number of field lines or something like this

@TedShifrin Hm ok not sure if I will understand though

oh but the connections here are on principal $SU(2)$-bundles, unlike $U(1)$ in EM?

Ted Shifrin

The original CS paper was written for physicists? Nah.

They actually do the set-up for connections in a $G$-bundle.

Balarka Sen

23:27

Oh OK

Ted Shifrin

At the end they apply to $3$-manifolds.

LOL, the last sentence I'd forgotten. "We do not see how this helps to settle the Poincaré conjecture."

Balarka Sen

@MikeMiller this is a nice analogy but is $45\sigma + p_1^2$ actually an understandable geometric quantity?

Mike Miller

@BalarkaSen Is the Hirzebruch signature formula geometrically comprehensible?

Balarka Sen

or does it pop out of calculations

i don't know the signature formula but ill look that up

Mike Miller

That's where I got it from

In dimension 8 it says $\sigma(M) = \frac{1}{45}(7p_2(M) - p_1^2(M))$

Balarka Sen

23:30

ohh i see

it kind of makes sense, $\sigma$ is a cobordism invariant

Mike Miller

Genus of a multiplicative sequence

In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e. up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties. == Definition == A genus φ {\displaystyle \varphi } assigns a number Φ (...

Shaun

I've been compiling the write-up from the beginning. My supervisor said it's a great habit of mine. All I need to do is reorganise then add a few tables summarising my results. Perhaps I'm asking too much of myself, then, @TedShifrin; I'm no Nash. 2019 was an awful year for me. I was hospitalised twice for my mental health, for about two months at a time. I could have died a couple of times too. (I won't go into it.)

Mike Miller

It's related to $\frac{\sqrt{z}}{\tanh(\sqrt{z})}$ sorry to say

Balarka Sen

lol

thats awful wtf

user2103480

@BalarkaSen apparently he read hartshorne while sitting in high school classes. pure beast

Ted Shifrin

23:32

@Shaun: I do not know you, of course. But of course I sympathize and I would urge you to think about how your health must come first. If the stress of graduate school is making it worse, then you should think about healthier options.

All these multiplicative sequence things come from crazy Laurent series, @Balarka. :P

Balarka Sen

@user2103480 i dont think thats such an unusual thing anymore

ive met lots of people who are/were like that

user2103480

uhhh what's your standard

that's like 0.1% of math students tops

skullpatrol

he's always had high standards :P

Alessandro Codenotti

@LeakyNun The title is amazing

Ted Shifrin

In my long career, I have taught a few brilliant students. But, even counting Berkeley and MIT, not just UGA, no one quite in that league. I taught a high school kid in AoPS who might be close to that league.

Balarka Sen

23:35

plenty of high school kids read unusual things for their age; a friend of mine in high school used to bring Schopenhauer in class

Ted Shifrin

In my days, the odd-ball kids read LOTR.

Leaky Nun

@AlessandroCodenotti so are condensed sets

Balarka Sen

avid literature fanatics are pretty up there as well

skullpatrol

true

Balarka Sen

its something to do. young people are bored and wile time away in obsessive things

Alessandro Codenotti

23:38

@TedShifrin I read LOTR in high school

Ted Shifrin

I have never been a fantasy sort.

Alessandro Codenotti

I wanted to see the movies, but my mother was always a big fan of reading the book first

Turns out that she was right though

user2103480

@BalarkaSen fair enough, but I was an outlier for doing some uni mathematics while still in high school, and it was far from anything on that level. I can't really weigh in on the difficulty of hartshorne, but people don't call it easy if I recall correctly

Shaun

That's a helpful perspective, @TedShifrin; thank you. I'll have some fun with some group theory for 'rest o' night. Oh, and I've read LOTR too.

user2103480

people like kripke, scholze, wolfram are immense outliers

Balarka Sen

23:40

i wasnt really particularly thinking about math

LOTR is great

Ted Shifrin

@Shaun: I truly wish you nothing but the best. :)

4

Alessandro Codenotti

All of Tolkien's work are tbf

skullpatrol

Take care @Shaun

2

Balarka Sen

i truly miss being in 11th grade, in which there's no serious exams you have to give unlike 10th (middle-to-high graduation) or 12th (uni admission exams); i remember reading so much (and not just mathwise) and listening to so much music and watching so many different movies

it was an obsessive thing, because there was so much time

Alessandro Codenotti

I bought a copy of Dune recently but haven't started it yet

Balarka Sen

23:44

there's no time anymore. these things wear off when you have no time

skullpatrol

ultimately, we are all destined to run out of time

Thorgott

Kreck told us a story a while ago about how he gave a lecture at Bonn aimed at postdocs where he summarized every single piece of work in topology that earned a Fields medal. Apparently most people quit the lecture cause it was too tough, but there was one person who not only followed it, but even corrected Kreck on every of his mistakes on the board.
One day, Kreck approached him after the lecture, asking if he was a topologist. He replied that no, he was an algebraic geometer. Of course you know who he was; I believe it was before he got his PhD even.

Ted Shifrin

I don't know who. Who?

Mike Miller

the now-fields medalist

Alessandro Codenotti

That boy? He was Albert Einstein

Ted Shifrin

23:52

Oh, Scholze, whom people were referencing earlier. I know none of these people, even with Wikipedia.

Alessandro Codenotti

@Thorgott that sounds like an interesting lecture, but it would be rather short if done for set theory/logic instead

user2103480

@AlessandroCodenotti lmao

Thorgott

I don't think I know any set theory fields medal result

what's an example?

user2103480

@Thorgott there's only one

Balarka Sen

CH

user2103480

23:53

cohen

Thorgott

oh, duh

ok, I do know that result

Alessandro Codenotti

@user2103480 and the fact that Shelah didn't get a fields medal is a strong indicator that there will never be another one

Thorgott

suck it, set theorists

your spaces=no extra structure=boring, our spaces=lots of structure=exciting+cool

Alessandro Codenotti

Yet as soon as set theorists do some topology you complain it's awful

Ted Shifrin

Highly technical point-set topology is awful.

If there's some interesting differential topology, let me know.

Alessandro Codenotti

23:57

Grrrrr

Balarka Sen

Puzzle: Given a configuration of spheres in $\Bbb R^3$, call it's nerve graph to be the one obtained by taking a vertex for each sphere, and an edge for every touching pair of spheres.

Thorgott

ok done, I called it

Balarka Sen

Are all graphs nerve graphs of some config of spheres?

Mike Miller

Discrete ricci flow probably proves it

Transcript for

$Mathematics$ Mathematics