I think I'll be fine, I've done well on the homeworks and they're probably gonna be way harder than the exams

@Eric has now seen that I was industrious enough to write homework exercises (and grade the ones that people turned in).

Though Soug's final was kind of crazy too, I'd say harder on average than any of Schlag's tests or Marianna's midterm

You'll be fine, @Eric. Make sure you can state the main theorems, too.

plus your exercises are quite fun @Ted

@Ted Lol, Soug's approach to homework was probably to use random number generators and give us 50 problems from 3 books,

Thanks, Eric, I'm glad you find them interesting.

I mean not really, there was one time where he assigned us problems from Buck, and he only had a physical copy so he actually wrote down 23 exercises by hand, but yeah his problems were 100% book problems

lol @Daminark when I took souganidis's class the final was just Schlag's final from the previous year, when completely different material was covered in the course

I hate Buck, Demonark. My book is infinitely better :P

most people tanked it, I did well only because I had a bigger background than most in the course and had seen some of the differential equation stuff

Just do my sequence exercise. Nothing to do with uniform convergence, @Little. If $x_n\le 10$ for all $n$, can $x_n\to 11$? Why not?

Soug actually wrote our final but it was kind of... I'll say that there was one problem where part b was proving that if $|f(x)-f(y)| < |x-y|^2$, then $f$ is constant, and part e was proving that if you take Gaussian kernels and convolute them with a continuous function $u$, then this converges uniformly to the function

then the $x_n$ can be possibly $>10$, by using definition and letting $\epsilon = \frac{1}{2}$

@Ted I agree with you there, I heavily disliked Buck with his insistence on $\mathbb{R}^3$ at most and avoiding vectors

@Little: In fact, they would eventually have to be bigger than 10, which cannot happen.

Demonark: It bothered me that Buck did everything with coordinates and coordinate functions rather than vectors. Really old-fashioned and not conceptually satisfying.

Right, we agree for once :P

I see, its the same analogy on the sequence of functions

The stars seem to have aligned in a very specific way

:P

Right, @Little.

Thanks :) gonna head for exams soon >.<

Make us proud, @Little.

Honestly I wasn't too fond of the books we used. Rudin and HK were nice, Sally had some neat content but was kind of meh in writing style, and Buck was awful

One of my former students who's now TAing out of Hubbard/Hubbard says he now realizes how great my book is. :P

It made me feel good.

What book is it, Ted?

@Daminark noone agrees with me it seems, but I'm actually reeeaaaally not fond of Rudin

Integrated multivariable calc/analysis and linear algebra.

@Eric: That's because you and I have similar taste, it seems.

It. Has. No. Pictures.

Rudin is a great reference, very poor for pedagogy.

I dislike Rudin intensely, although I could teach out of it and give insight and draw lots of pictures, but I've only done reading courses with students out of it.

Hell, my algebra book is full of pictures, dammit :D

At least a few of us were using your book/lectures for differential forms and manifolds. In first quarter no one knew about your book, I used it once for implicit function theorem because I had no idea what Soug was doing.

Well I guess it is the pictures I am paying for when I shelf out >$200 for your book, Ted?

And lol I was kind of raised on Rudin in conjunction with Spivak so I guess I developed a soft spot. I have never looked at chapter 8 and was not fond of chapter 9, heard chapter 10 was garbage

I'm TAing a multivariable analysis class now that uses Rudin and honestly it feels like no one has the pictures in their head and it makes me very sad bc the subject is so beautiful and Rudin is like the worst book to learn it out of

I have no control over the pricing. I did my best to tell publishers to keep prices down. At this point, I give up.

@Eric: TOTALLY.

Would a company like Dover have taken on your book?

@anakhronizein: It's still in print (in fact, they're finally doing a corrected printing of it, although that has turned out to be a huge saga).

But my lectures are on YouTube for quite free.

One of these days I wish to write a book.

Last quarter I TAed the same class and it was much worse, the lecturer went straight from Rudin. This quarter they're being taught by a geometric analyst who infuses some classical diff geo and they seem to be improving over the quarter which is nice

Eric, you're making me convinced that virtually all pedagogical sense at Chicago is absent. :P

Honestly accelerated should probably use Pugh or something

@Daminark I like all of the books I've read from Spivak personally

I took courses from Pugh but I don't know his book. I'd imagine it's quite good and has good exercises.

Spivak is great.

Great exercises.

@Ted I personally believe that few instructors here do things in a way that's pedagogically sensible

Spivak's Calculus on Manifolds is tooo dense/terse. His 5-volume Diff Geo book is totally the opposite. It's too wordy. I think Calculus he gets the balance just right.

I'd say to use your book @Ted but they'd want topology in the general context of metric spaces

I took a course from Pugh in high school and it was reeeeaaally good I thought

Demonark: My book is meant to learn calculus, not abstract analysis.

Lol they use calc on manifolds in 163 actually

@Ted I actually love the first intro to diff geo

Munkres's Analysis on Manifolds is a slightly better rewrite of Spivak's book, but I don't like it that much, either.

Yeah, volume 1 is ok, @Eric, but the rest is way too drawn-out (and no exercises to speak of).

but I used it in conjunction with other books so I didn't spend a lot of time reading it, just doing a load of exercises

I love Spivak and I learned my grad diff geo from him, actually, but his book is not ideal.

I find Munkres too wordy and conceptual sometimes.

Only for half a quarter and they don't do much, but in first year people are using that book. My professor didn't want to do stuff like topology since we'd do it in analysis, so we used Stewart (not a fan...) and did a bit on stuff like quadric surfaces

I used it in my reading course with Farb when I was solidifying some basic diff top

Later we went back to Spivak for derivatives and integrals in $\Bbb R^n$

I actually like Boothby as an introductory manifolds/diff geo book. It's pretty concise and has the important stuff in it.

Is Lee's manifolds book any good?

This was one of the books I used as reference for that course actually^

which one @anakhronizein

Students love it. I don't like his Riemannian geometry book much, @anakhronizein.

But Lee writes clearly. He actually took complex manifolds from me when he was a grad student, so I've known him a long time :)

Actually around 207 I found this very... interesting book, Fleming

I took a course from Fleming. It's unique because it combines multivariable analysis and Lebesgue integration. It's not bad. I probably would never have used it to teach, myself.

I used Lee's smooth manifolds for the undergrad diff top book and felt frustrated with it at the time but have used it a lot since and grown to appreciate how detailed it is.

Kobayashi was what I learned diff. geo, with.

That's unreadable unless you know it all already, @anakhronizein.

I was recommended KN by a grad student and got freaked out when I picked it up

that book is terrifying

Yes the introduction is terrifying.

But it was well-needed rigour for a course on tensor analysis I took.

Kobayashi was very, very smart. I think he wrote that more or less like Lang — at the typewriter. :)

does it have exercises?

Not a one.

Honestly I kind of liked Lang's analysis book, either that or Pugh would be my pick for accelerated analysis

oop that sucks

That's why I'm proud of the exercises I write (for books and courses), @Eric :)

no exercises makes me a sad budding geometer

It's fun making your own exercises too!

Exercises that are more than "verify this computation" can be nontrivial to make up.

I was given a book I think by Chern, chen, lam by a professor and that didn't have exercises either :( sucks because the exposition from the first couple chapters had really good exposition i felt

NO exercises, either. When I TA'ed for Chern's 140 and topics course follow-up at Berkeley, I wrote the exercises (and lectured some).

There's still some good stuff in that book, Eric.

it does moving frames right?

Yeah, Chern does everything with moving frames.

Oh, public service announcement. For moving frames, check out Jeanne Clelland's brand new book (AMS) on moving frames and diff geo.

She is at Colorado, a student of Robert Bryant's.

I may read the rest of the book in conjunction with your notes then @Ted

Cool, @Eric.

oooh that book sounds exciting

It's not super advanced, but lots of concrete stuff and good exercises.

@Eric the counter influences of Neves and Ted

Eric can take good taste from each of us :P

We were asking Neves in office hours earlier whether he'd be doing forms and he was like

Neves is absolutely the research star.

Rehi DogAteMy

"Eh, the class is titled integration, so I should probably do that, but forms are all notation and no content. Might show it anyway but I want to do other things and time is tight"

That is bullshit ...

That kind of statement makes me livid.

retracts high praise

I'd like to see him say that sentence to Chern, Griffiths, or Bryant.

So Neves isn't particularly fond, he seems to be in the DoCarmo camp

Oh snap

@Daminark I think that while I'm still young I should be open to multiple perspectives on things I like

Even doCarmo would never have made such a statement.

But I complain about how algebraists teach by symbol pushing sometimes ... Perhaps that's equally bad (although my statement is true) :D

Does anyone know any references that connect the different spectral theories, say of functional analysis and algebraic geometry?

Something tells me you hate Bourbaki, Ted. ;)

I don't have any idea what you're referring to @anakhronizein re spectral theories & algebraic geometry.

I mean do Carmo did have a book on forms. But yeah that echoed Soug, who also felt like it was just formalism to push stuff to manifolds. I'll wait until I see more of the stuff like moving frames before saying anything

I'm not fond of formal mathematics, so I'm not so fond of Bourbaki. Correct. However, I rather like Dieudonné's 4- or 5-volume analysis treatise that has everything in it.

Spectrum in algebraic geometry versus spectrum in functional analysis.

And @Eric that's very fair

oh you mean Spec of a ring?

Neves did give some pretty high praise to Bryant when we were speaking about Willmore stuff

Bryant has done some brilliant mathematics ... a lot.

But a lot of geometers really are uncomfortable with forms, and so they make insulting remarks instead of just keeping their mouths shut.

Spectrum of a linear operator gets its name from physics, really — spectral decomposition is like doing spectrometry.

I hadn't thought about where the name for rings came from. It must be connected. I'll think about it.

There is a math overflow question, let me find it

I mean a ring is a connected set so there's that... @Ted

Um, no, Demonark.

No I was joking about a physical ring

I remember when I first saw diff forms in my analysis class first year I got really excited just because I thought the name sounded really cool

Then when I found out the moving frame was a thing I just thought "why doesn't everyone know about this, this is incredible"

It's actually so much more geometric than the del approach, @Eric, but people are stubborn.

I remember fighting with Chris Croke when I was a grad student and he was a postdoc at Berkeley.

Lol @Eric, we were kinda waiting for when Schlag presented them to us, at the time it felt like it unified things nicely

I think the aversion to it probably lies in the fact that you have to do a lot of algebraic work beforehand

at least this is what Neves seemed to indicate

the six favorites on this 21-minute old question make me wonder...

Not really that much. But all Riemannian geometers need to do tensors, regardless. So it's just a tiny bit.

yeah that's true

It's bullshit, @Eric. You have to define wedge product ... although in the grad course I short-circuited a lot of that.

@Semiclassical weird

it seems fishy.

Heya tern!! I hope you're healthy again! (I mostly am, but I had a dental implant done today, so that's pain.)

Huh, @Semiclassic? Did you link the wrong thing?

nope.

Oh, I see what you mean.

I hardly ever look at such things.

Yeah, fishy.

nor I, tbh.

Soug kinda told us that he didn't really want to do forms, he also felt like it was just a bunch of formalism (pun only partially intended) that you used to prove integration by parts. Soug seems to be less geometric, though, so I'm not surprised

Schlag's solution was to do things all in the plane

I think people who never developed any intuition by actually using forms are predisposed to find them a useless formality.

It's basically covering up an insecurity.

I'm not very intuitive with measure theory or representation theory, but I don't walk around calling them useless.

Didn't have to get into the full generality of the exterior algebra stuff (though I'm fond of that kind of thing)

I think my favorite thing about forms is how it unifies div-grad-curl

I also found exterior algebra really pretty when I first learned it

@Semi and all that? :P

It's the usual thing in life, people have certain likes and dislikes and make harsh statements about those things they dislike.

quite so.

(i've never actually read that book.)

@Semi when I first learned that forms did that my mind was blown

That's just scratching the surface, @Semiclassic. If you look at the moving frames proof of Gauss-Bonnet for surfaces versus the usual "classical" one, you will be convinced of more.

I buy that.

@Eric: You should seriously thinking about working with Robert Bryant (if he doesn't retire — but he hasn't mentioned that to me).

G-B is literally Stokes' theorem in the moving frames set-up.

Well, it's literally Green's theorem the classic way, @Balarka, but a whole lot of mess ...

One thing I'm personally fond of (though it's a very silly thing) is using 2-forms to get all the Maxwell relations in thermodynamics.

@anakhronizein that's a fair point

Right, for boundary terms.

@Ted he's at duke right?

Either way.

Yup, @Eric (he was the head of MSRI and at Berkeley, but returned to Duke).

@Semi My professor in EM actually did do this 4-vector/SR/tensor thing whatever it was to get Maxwell's equations down to one, along with just a mathematical identity

Namely, $dT\wedge dS=dp\wedge dV$.

It's not a good thing to take people too seriously when they call something in math entirely useless.

@Semiclassic: Well, all the thermodynamic stuff is basically equality of mixed partials everywhere, so $d^2=0$.

Of course.

Right.

Balarka, isn't it un-sleep time?

One thing that the 2-form approach makes really obvious, though, is the signs.

It is indeed

Those are otherwise a bit of a hassle to remember.

If he's not retiring any time soon I will strongly consider that @Ted

The reason it's silly, though, is that you'd never have a reason to go beyond 2-forms in thermodynamics.

That's part of what makes thermodynamics interesting at times, of course, but it also means that the full scope of forms isn't on display.

@Eric: He's really great with students and a great teacher, too.

I have to admit, 'moving frames' is something I am almost entirely ignorant about.

@Semi Are they inertial or not?

...lol

sends Demonark to study for his exam

thinking about grad school is terrifying

glares menacingly @Daminark

@Eric Same...

And lol I should probably do that

Fun fact, that remains true if you insert any of the following: entering, leaving, staying in...

I don't help matters if I say professional life gets more terrifying after grad school.

no, you don't :/

Cauchy sequence of gulps