7:00 PM
Why the cubic in $\Bbb P^2$ is a torus comes from your slicing and thinking about it. But that has nothing to do with understanding the embedding of the torus coming from $\wp$, as far as I can tell.
Put in $w=e^{it}$ and rewrite as a complex line integral, @Liad.

you want me to do the substitution $w = e \ ^ {it}$ ?

@TedShifrin I think we don't need to worry about combining eigenspaces, because if you look at $J$ (JNF of $A$), then we have that the eigenvectors correspond to the first column of each block. To compute $q(J)$, we just take polynomials in each block of $J$, and so the eigenvectors are preserved. That is, this sort of argument should show that $v_i$ is an eigenvector for $\lambda_i, A$ if and only if it is an eigenvector for $q(\lambda_i),q(A)$. I believe that works.

Ah OK
I see what you mean

so $dw = iw dt$ , right @TedShifrin ?

No, @Antonios. What you just wrote is wrong, I believe.
$i\,dt = dw/w$, yes, Liad.

7:01 PM
@TedShifrin It's possible I've mis-written (so to speak). What seems objectionable?

ok so putting $dt = dw/(iw)$ i get

@Antonios: What if $q(t)=t^2$? Then the $1$-eigenspace of $q(A)$ becomes the direct sum of the $\pm 1$-eigenspaces for $A$.

$\dfrac{1}{2 \pi i} \int \dfrac{w+z}{w-z} \dfrac{u(w)}{w} dw$ @TedShifrin

Let me write this example out in detail in my book real quick.

You left out the $1/w$, Liad?
OK
So now argue that that is a holomorphic function on the unit disk.

7:04 PM
hm.
i need to use Chauchy's formula?
because the $u(w)$ term make it difficult :P

No. It doesn't. You want a function of $z$ here.
As i said above, this is the kind of thing you've been doing since the beginning of the semester.

@TedShifrin, I think I see exactly what you mean.

I think I can account for it, though.

For a second I read that as "I don't think I see exactly what you mean" and Ted's smile was... confusing, to say the least

7:09 PM
loll

Good thing you're up in the hills, Demonark.

@Daminark :)

True, I do fear for the hill inhabitants though

@Balarka kek

7:10 PM
It's like the Flowey smile

pls no

You know it's twisted

I'm gonna be sans real quick so I can, you know...
(Make puns at Flowey)

starts noteblock version of Megalovania

@TedShifrin i can write it as $\dfrac{1}{2\pi i }\int \dfrac{u(w)}{w-z}dw$ + something similiar
and this is $u(z)$ maybe ?

7:14 PM
Don't even do that, @Liad. Why is your original formula giving a holomorphic function of $z$? As I said, we've done this numerous times.
Stop trying to make it look like the CIF.

@BalarkaSen :')

@TedShifrin Take a 4x4 matrix $J$ with 2x2 Jordan blocks $A,B$ –– $A$ for eigenvalue 1, $B$ for eigenvalue -1. Let $q(t)=t^2$. Indeed, when we square, we have that the resulting matrix has two 2x2 blocks for eigenvalue 1. This isn't a problem, though. I am having an immensely difficult time communicating what I want to say lol.
But this still fits into the formula I claimed above (unless I mis-typed).

No, as you said something with $\lambda_j$ iff $q(\lambda_j)$.
If $q(\lambda_j) = q(\lambda_k)$, then bets are off.

I should use a theorem ? @TedShifrin a hint would be great :P

I agree. But, we have that the eigenspaces for $\lambda_j,\lambda_k$ direct sum to the new eigenspace for $q(\lambda_j)=q(\lambda_k)$.

7:17 PM
I've given you the same hint four times, Liad!
Since the beginning of your course, you've been using differentiation under the integral sign.
@Antonios: Yes, that's my point. But that kind of thing appears nowhere in your original statement.

i thought about it!!!, but i dont see a function that this is its derivative :/

@Balarka hmm, I'm starting to hit a crossroads wrt the paper, I'll ask Peter about what to do here, but it turns out the Dold-Thom theorem itself is a statement that's technical and... takes a long time to prove, my book just shoved it in the appendix

You want to argue that that integral gives a complex-diff function of $z$. That's all.

So, given a pointed map $f:X\to Y$, it's pretty clear that there's an induced map on $SP(X)$

My original statement was decidedly wrong (depending on how far back we go). My revised statement I claim to have proved is that "If the characteristic polynomial of $A$ is $p(x)=\prod (x-\lambda_i)^{d_i}$, then the characteristic polynomial of $q(A)$ is $w(x)=\prod (x-q(\lambda_i))^{d_i}.$" In the case where $q(\lambda_i)=q(\lambda_j)$, we have that terms of the product combine to $(q-\lambda(i))^{d_i+d_j})$, which accounts for the direct summing behavior.

7:21 PM
This construction happens to be functorial, and homotopy equivalent maps are sent to homotopy equivalent maps

Huh, i thought about it the other way around.
Thanks ,i will write out the details . @TedShifrin

OK, Antonios. Somewhere you made an iff claim on $\lambda_j$ and $q(\lambda_j)$. That's what I was bitching about.

hehe, sorry I should have said sooner that I don't agree with that anymore.

@Liad: Make sure you use somewhere that $u$ is real-valued :)
LOL, OK, @Antonios.
I'm done, then ;)

I found my faulty reasoning for that one last night. It was a basic error, involving precisely that I assumed that $q$ was injective (or didn't account for the case where it wasn't).

7:23 PM
@Daminark I see.

I slept only like 3.5 hour last night. That's likely not helping me express myself. That's the part of my brain that goes first... lol

Now if $X$ is a Hasudorff space (pointed) and $A$ is a path-connected subspace with a neighborhood deformable to it. If $\rho: X\to X/A$ is the quotient map, then $\hat{\rho}:SP(X) \to SP(X/A)$ is a quasifibration, and the fibers are homotopy equivalent to $SP(A)$

Right, that makes sense

No sleep for final exams is a big error, Antonios :)

Anyways, thanks @TedShifrin. This was quite helpful.

7:23 PM
You're welcome.

@Antonios-AlexandrosRobotis love yourself and sleep my dude

Oh, I'm still a week out. My night before policy is 12 hours of sleep, believe it or not.
Actually, I was up talking to family in Greece about something haha.

12 is deadly. I'm a zombie if I sleep too much (unless I have cancer or anemia).

12 hours of sleep a week?

Now, I'm wondering (if I can't do both, which I'm starting to think might be the case) if it's better to focus on the proof of this theorem or on using this to talk about homology

7:24 PM
Noooo. Night before. I can sleep as long as I want to lol.

@Mathein right? That's so much!

On the flipside I can make due with ~5-6 if need be. But I've noticed that relaxing and sleeping before tests increases my performance more than studying the night before.

@TedShifrin Did you see my question about having visually obvious proof of $SP^2(S^2) \cong \Bbb{CP}^2$?
@Daminark Which, the fibration SP(A) --> SP(X/A) --> SP(X) theorem you just wrote?

No, but PVAL was talking about it.

Yup @Balarka

7:26 PM
My proof that $\text{Sym}^k(\Bbb P^1) = \Bbb P^k$ is definitely not visual.

I know the polynomial thingy
but im not very happy with it

LOL, OK. I guess I'm now more algebraic than you.

hahah

Differential Geometry: An Algebraic Approach ... agh.
9
I knew you'd star that.

7:27 PM
LOL

@TedShifrin Howdy. I did one of your undergraduate exams last night. How many points did people end up averaging on these?

Not very high on the final, Kevin. Midterm probably around 60 average.

maybe would give some intuition about dg

I'd have to look.
There's a lot of algebraic structure to tensors, bundles, etc., @Mathei.
And complex geometry, for sure ...

7:28 PM
I know but I have absolutely no intuition for stuff that involves differential equations

@Daminark Let me have a look at the proof of Dold Thom

which seems how a lot of difficult stuff is proved

Can you point me to where it is?

"Maybe an algebraic approach would give some intuition about dg" (Though actually kinda same tho, I struggled hard with it the first time around)
@Balarka look in Appendix A of Aguilar

No, not that much. But the existence of solutions of ODE can be taken as a black box ... to get geodesics, parallel translation, etc.
You didn't exactly give it serious effort, Demonark.

7:30 PM
I also like the Jacobi equation

my dude Jacobi is gold

truly

Cheeger is always talking about using solutions for Elliptic PDEs to prove stuff in DG.

yeah I don't understand Jacobi fields at all

@Eric helped me a lot with it

7:31 PM
Yeah, at the research level for sure.

when the picture was clear, it was very beautiful

depending on what part of DG elliptic PDE doesn't necessarily show up thaaat much
in some parts it's literally every all the time

With Cheeger everything is research level lol :P

sometimes you can black box it
or get what you need out of Evans

I did a very different flavor of diff geo, myself ...

7:31 PM
speaking of DE's, I'm learning about hyperbolic sets in dynamics now

@TedShifrin Ah ok. I did the 2011 4220 final. I think I got 1, 2(a,b) (we didn't cover c), 3, and 5(a). I completely whiffed on 4 and 6, so clearly I need to do some more intersection and degree problems.

Oh, I love that product of spheres problem, Kevin.
That was the year I got cancer and missed the last 2 weeks of the semester. Wasn't my best teaching semester. But the students survived.

@TedShifrin do you know anything about $L^2$ Cohomology?

@KevinDriscoll Find the self-intersection number of the diagonal S^2 in S^2 x S^2. Don't mess it up.

A little bit, @Antonios.

7:33 PM
@Ted at the beginning I did, since it was probably what I was most excited for going in, like I put Schlag-levels of time into the first pset and still barely got anywhere. Afterwards I definitely put less because I found that I was actually understanding probability, but... Eh, I dunno

That wasn't the exercise, Balarka.

@TedShifrin Oh I know, I just gave him a random exercise

@TedShifrin Might have some qns on that next weekend. Cheeger told me to read his research paper on it for our last HW problem of the semester LOL.

on intersection numbers

Cool, Antonios.

7:34 PM
He said "It's extra credit, even though everyone knows there's no extra credit in this course."

Wait now $L^2$ cohomology is a thing? Does every concept in math have an associated cohomology or something?

@Daminark tbf probability is more similar to stuff you did in analysis, diff geo is more it's own thing

yeah L^2 cohomology is interesting

so there's not as much of a through line with which to understand things

L^2 homology is like
the chains are not just finite linear combos of simplices anymore
they are allowed to be infinite

7:35 PM
I'm just going to start appending 'cohomolgy' to things and acting like it exists

but the coefficients form an L^2 sequence

I was actually the best in the class on the dg problem sets up to some point. But when there were exercises on solving geodesic equations and computing integral curves for vector fields I just dropped the course

@Kevin: #4 is generalizing the argument principle from cx analysis. @Balarka: #6 was to prove that any map $S^k\to X\times Y$ (where $X$ and $Y$ are positive-dimensional compact, orientable, dimensions adding to $k$) has degree $0$, using intersection #s. Then prove it using cohomology.

Okay, all. It's been fun. Time to get back to reading about Ramification of prime ideals...

Mathei: Those were likely easy calculus exercises. ... And Demonark struggled because he doesn't know calculus (especially multivariable).

7:36 PM
@TedShifrin Ah I see.

@Antonios-AlexandrosRobotis back to the good stuff, you mean!

@MatheinBoulomenos It's all a matter of perspective :P

Oh ive actually never heard of this argument principle

@Balarka: #4 was the exercise I gave you to do the prove that the winding number tells you how many times you take on a regular value inside.

Yep I remember

7:37 PM
@Ted the calculus ones were not the ones I struggled with, it was really the geometric stuff

It's an essential step in one of many proofs of Hopf degree theorem

@TedShifrin these were not easy at all. I solved one geodesic equation using isometries (I had to prove all the relations between isometries and geodesics myself, we didn't do that in class)

With surfaces there's more geometric intuition that's important, Demonark. But the first p-set was just curves?

but people who tried to solve it directly couldn't do it

7:38 PM
Well, obviously, I don't know what you covered, Mathei.

not even physicsts who had been solving a lot of differential equations since the first semester

Symmetry is a powerful ploy, though ...

@TedShifrin I was angry I didn't get 4a, but I wasnt sure if I was allowed to use the IVT or not.

OF course, @Kevin. But you need more — you have to analyze "rising" and "falling" behavior. This came up in a homework problem.

The curves problem that stumped me was the one about support lines of a convex plane curve

7:40 PM
Oh, that problem is totally just calculus, Demonark.
And normal vectors :P

I should work through the curve problems in your notes at some point

Some are quite interesting; some aren't.

I need problem solving creativity for my admission exams

Huh, when I was working on it I had difficulty figuring out interpreting the situation

It didn't really help that the prof was a mathematical physicist who just assumed everyone knows a lot of vector analysis and solving differential equations like the physics majors do

7:42 PM
Normal vector is $(\cos\theta,\sin\theta)$, so equation of the line is ...

Like I just kinda stared at the picture, tried to reproduce it, failed, repeat for a few hours, move on to a different problem

@Mathei: I always taught with way more differential forms than most books/profs. We all have our biases.

I'm familiar with differential forms
we did that in analysis 2

Geez, Demonark, you should have mumbled at me or Eric.
I think geometrically with differential forms way better than with vector fields, but most Riemannian geometers avoid forms like the plague.
Thinking about how orthonormal bases twist as you move along is very geometric to me.
@Balarka: I would imagine your math problem solving is way more developed than most high school kids (except the ones who do tricky competition problems, and I've never been good at those).

Well, not entirely true, but I have been trying to improve for some time. I think it's paying off
I am an @AkivaWeinberger wannabe

7:46 PM
i think being able to translate between vector fields and forms is really important

LOL, well, DogAteMy is particularly quick and intuitive with hard things. I'm not sure I've ever taught someone at the college level quite like him.

he's too good
@EricSilva I think in terms of vector fields

And I've taught a lot of talented kids who've gotten PhD's or ...

I find schemes easier to work with than vector fields, for example

I actually had to use embedded components in one paper. I was very proud.
I think I sent Balarka that paper.

7:49 PM
The Whitney umbrella paper?

yup

aha

different singular Chern classes

i think the dudes who do geometric flows really dont like doing the cartan form game

I've done enough algebro-geometric stuff that scheme structure has to be there.
Most Riemannian geometers avoid it totally, Eric.
I remember arguments with some big-named Riemannian types when they were postdocs and I was a grad student. But we're friends anyhow :P

7:51 PM
the situation for the geometric flow guys is that writing out the structure eqns and working out how the time dependence work is apparently a nigthmare

Mostly it's Chern students and Bryant students who use them a lot.

sometimes anyway

Yeah I guess the pinch point has schemey badness in it
k[x]/(x^2) things

Yeah, although I convinced you that variational stuff can play nicely with the structure equations and the Cartan formula for Lie derivative.
You get embedded components in the Nash blow-up, if I remember right, Balarka.

Ah yeah makes sense

7:52 PM
Why were there arguments? You'd expect that whether or not someone wants to work with forms would be a personal thing, no?

yeah but that's also way nicer than if you try to work with like Ricci flow or smth

Yeah, I think I concede on Ricci flow. But I haven't tried hard enough.
Demonark: It's a question of whether they even present things when they teach.
Sort of the same issues I have with the courses you've taken.

Ah

on one exercise the TA said something like "So this exercise turned out to be a lot more nontrivial than we thought and nobody got it right which is not surprising as you have to use nontrivial differential topology which presumably nobody of you knows."

It's not like I totally avoided the vector field viewpoint when I taught. That would be difficult and also incompetent.

7:54 PM
@TedShifrin Id like to run the brief version of my solutions by you to make sure I did what I think I did. But Im not sure what the economical way to do that is. Is chat ok or do you think something else would be easier?

I tried for a couple hours this summer to work out some stuff w the moving frame and ricci flow and i can say it's very nasty

@Kevin: If you wrote stuff down, feel free to email me.

I think the best living expositor of the world at this point is Misha Gromov

but Milnor is alive

I did but its a bit fo a mess. I think Ill just Latex something

7:55 PM
The Jacobi diff eq to prove Cartan-Hadamard can be done with the geodesic normal coordinates moving frames approach, Eric. I only have seen that in one book, actually — Helgason.

@Eric Gromov is still the best

Don't waste time, Kevin, unless you feel LaTeXing it will make you think more.

I should work out that one
@Balarka but Milnor is soooooo gooooooood

@EricSilva lol u don't get memes

I have had issues with some of Gromov.

7:56 PM
@MatheinBoulomenos We had a similar issue with a linear algebra problem. It was posed as an extra challenging problem, but it was not solvable by the TAs, even when we tried using everything we knew for it

Gromov is probably one of the baddest expositors out there

some of his writing is ok

who are also exceptionally good mathematicians

@Eric I'm pretty sure Soug knocks both of them out of the park

@Eric I can't into the Hyperbolic Groups paper

7:57 PM
his like high level overviews are pretty good

When I took grad real analysis from Irving Segal, he gave us a takehome final with several undoable problems on it. I spent 20+ hours on the final and got an A even though I couldn't solve them, but wrote my thoughts.

i tried to read that one and i couldnt get past the typewriter font
if im recalling correctly

@TobiasKildetoft I once solved an LA problem (which was really about ring theory) using the fact that a graded ring modulo a homogenous ideal has an induced grading. They knew nothing about graded rings and homogenous ideals ...

@Daminark 50 problems from Brezis after each exposition

@MatheinBoulomenos I still haven't found a way to do this problem.

7:58 PM
I actually am very fond of Griffiths's writing. He makes mistakes, but he gives great intuition and I like his viewpoints.

does Brezis even have 50 problems

@Balarka after? His exposition will be precisely the list of problems

@Eric I agree Milnor is A+

@Eric I thought it had like, tons

@Daminark aw yiss

7:59 PM
OK, time for me to do stuff. Bye.

cya

i havent read enough gromov to know how bad or good some of his writing is but ive read stuff that was good

I'm not sure since Smart's not even gonna use Brezis
@Ted see you!

@Ted bye!

@Daminark Brezis distinguishes problems vs exercises
tons of exercises
not tons of problems
the problems were usually things that people have written papers on

8:00 PM
I mean Soug's gonna assign them all anyway

@MatheinBoulomenos The lecturer thought it should not be too hard, but it turned out he did not actually have any good idea on how to solve it

problem, Brezis? [Souganidis trollface]

someone should tex gromov's hyperbolic groups paper so i can read it

I'm a bit sad the grad students didn't get to experience the full glory of Soug

i tried to penetrate it like a bunch of times
9

8:02 PM
@EricSilva um

@Daminark they probably told him to cool it
@BalarkaSen I know what I said

r/nocontext maybe?

Not really, from the beginning of the quarter he said that he would be assigning just 10 problems a week, while he was set to do the 50/week game in 207 before someone complained

no shame

And he was assigning the undergrad students triple the homework

8:03 PM
"triple the homeless" looool

maybe the people complaining finally broke his basketball spirit\

That was quite the autocorrect
"What if math actually is a spectator sport?" -Soug

I bet you learn a lot by solving 50 problems a week

you learn the spicy sougi memes

sleep and social life are overrated anyway

8:06 PM
not a lot necessarily but you learn how to do things

Honestly I don't disagree, 207 definitely taught me a good bit

im actually kind of in agreement with souganidis in that i dont think we do anywhere near enough problems
most classes at UC just bark results at you and you dont feel them in your bones

Like I think I wouldn't use the same book as he did, since Sally's problems weren't often that good
I'd rather he just used Rudin and did something like, do all the problems of chapter n on week n + the linear algebra psets

doing all the problems in rudin is too much during a school term

@Eric In any case if someone ends up TeXing Hyperbolic groups, I feel like it shouldnt be Gromov

8:10 PM
On many weeks Soug basically assigned nearly all the Rudin problems from the relevant chapters

His TeXing is getting oddball recently

It was definitely a lot, like I spent upwards of 40 hours on some weeks and didn't finish them

But like, the week has 168 hours, you could probably make it work if you try hard enough

Hello world

8:11 PM
@Astyx yo, how's it going?

hey @Astyx

Good n'you ?

Oddball how @Balarka

didnt you check out those papers

8:14 PM
check it out and you'll see
especially the last paper

@BalarkaSen $\heartsuit$

\heart broken?

It's \heartsuit apparently

Why is his contents page so big tho

scroll down, it gets weirder

8:16 PM
"oh god I'm so good"-Benson farb
Just overheard live

Classic Benson

lmao

Oh shit ur right balarka this is crazy
All these weirdo shapes

told you so
@AkivaWeinberger I know :)

Ah OK

8:18 PM
But the compliment was actually meant with bilious anger

GOOD N YOU ? @Daminark
Hum
Sorry

@BalarkaSen For some reason I don't believe you

Read "Notes from The Underground" and your belief will be falter. It's my bible.

@Astyx woah woah calm down, i'm ALRIGHT THANKS!!!

OKAY, GLAD TO HEAR IT !!!!!?!

8:20 PM
WHY ARE WE YELLING!!!!111
4

@Astyx I mean we're not talking out loud, so I'm not sure if you're hearing it but...

Point.

@Eric turns out I'm in the system for the wrong reading course
Peter was like "Yo I'm entering your grades in but the two of you aren't registered for my class" and we were like um frick.

Hello!!
Let $f:\mathbb{C}^n\rightarrow \mathbb{R}^m$ be a $\mathbb{R}$-linear map. I want to show that the map $F:\mathbb{C}^n\rightarrow \mathbb{C}^m$ $x\mapsto F(x):=f(x)-if(ix)$ is $\mathbb{C}$-linear.

Since $f$ is $\mathbb{R}$-linear we have that $f(z_1+z_2)=f(z_1)+f(z_2), \forall z_1, z_2\in \mathbb{C}$ and $f(\lambda z)=\lambda f(z), \forall \lambda \in \mathbb{R}, z\in \mathbb{C}$.

To show that $F$ is $\mathbb{C}$-linear, we have to show that $F(z_1+z_2)=F(z_1)+F(z_2), \forall z_1, z_2\in \mathbb{C}$ and $F(\lambda z)=\lambda F(z), \forall \lambda \in \mathbb{C}, z\in \mathbb{C}$, right?

8:42 PM
I am asked to prove that for a homogeneous polynomial P of order k on R^n the following expression defines a k-tensor on R^n. The only thing that really needs prove is the multi-linearity. I wanted to prove this by first working out the differential, but the only way to do so is by using the product rule over and over, which seems to be a terrible job. Is this the (only) way to go? imgur.com/i0L0RiL

Hey @Astyx !! Do you maybe have an idea about my question above?

@MaryStar I can give you the representation theoretic reason why it holds

@MaryStar The spaces have actions of the cyclic group of order $4$ by multiplying by $i$
this makes the space of $\mathbb{R}$-linear maps into a representation when the $\mathbb{C}$-linear maps being the fixed-points
and in general, if $V$ is a representation of the finite group $G$ and $v\in V$ then $\sum_{g\in G}gv$ is a fixed point
in this case, two of those summands cancel out to give the function you need to show is $\mathbb{C}$-linear