sup nerds

Yeah. This is actually pretty old. link.springer.com/chapter/10.1007%2F978-3-322-84243-5_3

sup Eric

sup @Eric

@TedShifrin is it obvious that real analytic manifolds are real analytic spaces (I think the latter is defined in analogy to varieties where you allow analytic maps instead of polynomials)?

So here's something a little bit worrying

https://youtu.be/OCLaeBAkFAY

@AkivaWeinberger yeah that

Yes, @Mathein: A real analytic space has a stratification by real analytic manifolds.

Look at Whitney's book Complex Analytic Varieties. It has great stuff in it.

I will do that when the exams are over

LOL ... no rush for me :P

tfw you have a pset and an exam and a lab and an 10 page paper all on the same day

kill me

Memo to self: Plan ahead and minimize procrastination.

good memo

if only I could tell past me

When I taught at MIT, the freshmen in my multivariable calculus lecture typically had exams in physics, calculus, differential equations, and organic chemistry all the same day before spring break. I warned them several times of that same memo.

horrible

Whereas UGA students would whine bitterly if they had two tests on the same day.

i had 4 exams back to back once my first year

that was an experience

OK, so you knew about my memo :P

I think two exams on the same day are not optimal, but still okay, but more than that is really not optimal

it all worked out ok

Lol I've only ever had two finals 30 minutes apart once

Most my exams were kinda boring so it probably would've been okay I guess

Which was not fun, especially since both were morning. Beyond that there was usually a few hours

Final exams in college is way too far back for me to remember.

And more often on different days

oh i remember how Marianna's final was literally unlimited time

that was good

I would prefer more time and more interesting questions

We had it 6:30PM so it couldn't quite have been unlimited time

But we had about 4 and a half hours instead of the usual 2

it seems like she was generally nicer to your year too

Though I was fasting on that day and dehydration was not fun

One of my former colleagues gave a final exam for his undergraduate real analysis class in the evening and let the students stay as late as they wanted. I think a few people needed rides home at midnight.

I don't believe in this, actually.

Oh yeah for sure, her tests weren't that bad, like our average was like, 40-50%

i acutally dont understand why she gave us unlimited time because her exam wasnt that bad

I mean at least one person stayed for 8 hours, no?

And actually figured something out in the last half hour

my intro algebra exam was so easy that the teaching assistant gave an apology because it would've been boring to the better students

yeah but that's because he used contour integration to figure out something that was really hard with complex analysis but trivial with fourier analysis

In any event she told us that she doesn't believe math is a race at all so she doesn't like the idea of having too heavy time constraints

I concur with that, Demonark, but allowing infinite suffering just isn't fair.

i mean i agree with that but my solution would be to give interesting take homes

That, I'd be down for

that's what Andre did and i really liked that exam

Especially with things like MSE, take-home exams can't be trusted at all any more.

that's true i guess

I think if the course size is small enough, oral exams are the best way to gauge understanding. (People still need to learn to write maths of course)

i would hope that students taking a class like honors analysis at uchi would be kind of intellectually honest

@XanderHenderson i.gyazo.com/30262acc106f3f276d752961d3fcfbb0.png someone did it to me :o

I had oral exams for qualifying exams in grad school. It's very time limited and you often only cover 3 or 4 questions ...

Eric, the temptation is just too strong, even for fundamentally honest people. Especially with grades riding on it.

hmm, we have 15 min preparation time and then 30 min time to talk about the questions and other stuff

Yeah same, and if you really couldn't trust students somehow, you can just try to give questions which are not standard, so that if such a post comes up it's a red flag

That's almost no time at all, Mathein.

I guess Alessandro has talked about having both written and oral. That sort of is ideal.

yes, I think having both written and oral is ideal

Like if someone posts about Besicovitch failing with constant 4 in the plane that's almost definitely Marianna

Demonark, red flag to whom? And even, if so, there are plenty of people here answering.

@TedShifrin yeah i guess youre right, it's sad though because ive really really liked all my take home exams

I caught my own diff geo students posting my homework questions here.

i had them in riemannian geo and algebraic top and you can just ask such interesting questions when you have that much time

I used to give take-home exams in honors multivariable calculus (the quarter version, after 2 quarters of Spivak). Also occasionally in diff top. Years ago.

In any event though, I do know some profs who did take home exams, like Calegari in complex, and the vibe I got is that students were quite diligent about not getting help

Again, I think homework is the place for really interesting questions, and exams should just be checking the basics and a tiny bit of medium stuff.

What I like about oral exams is that it gives flexibility. You can talk about basic stuff with the weaker students and if you see that the student knows his stuff you can talk about more interesting things

One time I was talking about a very similar problem to a friend and was asked to keep it down because it was a problem on his final and he didn't want spoilers

@Daminark i guess Danny does it for everything. idk if any of the other undergrads taking his class talked abt the exams with each other but i certainly didnt

@Mathein: The problem they used to have with the PhD students at Berkeley is that some of them literally turned into quivering jelly with nervousness with the oral exams. I personally loved it, but I already had a lot of teaching experience.

springer link doesn't work for me for some reason

what's the article?

@TedShifrin i guess i just personally like the idea of having hard hw and no exams because im not gonna cheat anyway, but i see how it can be a problem

I still think studying for exams/finals makes things sort out and ultimately even a good student like you learns something (even if the exams aren't super hard).

Oh, hold on, @MikeM.

Imagine putting a two-part question on a calculus exam, "(a) Differentiate $\ln(\sec x+\tan x)$. (b) What is $\int\sec x~dx$?"

A book called Topics on Real Analytic Spaces, by three Italians.

Here's the abstract: In this chapter we shall study, first locally and then globally, the complexification of a real analytic space (variety). The main results for the local models, which will be exposed in the first two paragraphs, are due to H. Cartan [2]. The existence of a complexification for a real analytic manifold was proved by F. Bruhat and H. Whitney [1], H.B. Shutrick [9], A. Haefliger [5] and, in the compact case, by C.B. Morrey [8].

The extension to real analytic spaces, which had been announced by H. Hironaka [6], was given by A. Tognoli [10].

I guess the consensus here would be, "no, do not do that"

"Don't try to teach them new things on an exam, do that on a homework or in class"

DogAteMy: It's more interesting to make you do part (b) by partial fraction techniques.

@TedShifrin yeah that's standard for most courses

@Ted what my diff top professor does is she gives an in class midterm but instead of a final you have to write a short exposition of a topic that you find interesting

Our second semester LA exam introduced new concepts, but not from linear algebra, but from commutative algebra for some reason

thanks

But how much of the grade depends of the written part and how much on the oral one depends a lot on the professor

Still, if you do do that, you'd want that to be either a homework problem or a bonus question on an exam

By such an advanced course, I'm fine with that, Eric. In my graduate diff geo courses I didn't give exams (except a few times a final at the end of the first quarter).

(Or both- first a bonus question, and then later done in class or as homework)

Sure thing @MikeM.

Luis just gave me a long list of crazy PDE problems and told me to do them and that was fun too but absolutely wouldnt work in a normal class

Freitag, who is retired but still teaches classes always does his courses now like this: Twice a week, he's giving a lecture and once a week there's a presentation of one of the students and the grade is determined by that presentation. He says he does that because he doesn't want to grade, but I think that's actually pretty cool

Some professor give you a mark and adjust it in a ±3 grades range (out of 30) with the oral exam, some only consider the written exam as "passed" or "not passed" and base your grade almost entirely on the oral part

Oh speaking of commutative @Mathein, in algebra today (just normal algebra) our prof told us one fun theorem that he said he wouldn't have time to prove

The name was like, Nullstensatz or Nullstellensatz, I forget which

Nullstellensatz ;)

nullenstellensatz

nuts, i was hoping I remembered that right

That's a huge theorem, Demonark, but not for undergraduate algebra. Artin does do a little of it in his book, however.

Null shtellin sats

null + stellen + satz

Press F to pay respects

I got interested in the Positivstellensatz a while back

zero+locus+theorem :P

satz = theorem, stellen = to place ... null you can figure out.

@TedShifrin Is it not pronounced with a sht sound?

grumph @Antonios

We proved it in our second algebra course, I agree that it's not a theorem for a first theorem

yes, it is, but so what?

i should study for E&M...

Antonios, how did your meeting go?

Bye, Eric

But yeah he is gonna prove the Hilbert basis theorem on Wednesday which is pretty nifty

@EricSilva I gotta write up the discussion problem for tomorrow and finish grading HW tonight

I didn't see that until I took commutative algebra, Demonark.

@TedShifrin Oh, I thought you were correcting me

We did Hilbert basis theorem in our second algebra course, but I did it before as an exercise in an alg geo book

sorry, internet died

No, I was breaking down the word into its pieces.

tethered to my phone now

LOL, Antonios.

How did your meeting go?

well, I'm in a café

but anyway, it was alright.

Ohhh ...

But yeah, the Hilbert basis theorem is pretty cool

I think she was a bit less sympathetic to the students' plight than I am.

Where's she visiting from?

Pittsburgh, but she's spent most of her time in Gdansk I believe

Pitt that is

It might even be null shtellin zats, I'm not sure

Ah, Europeans are often off in a different world vis-à-vis what students are.

DogAteMy: but null is pronounced "nool" with a liquid L. And not so much the "in."

To a certain extent, I agree. If the students haven't taken the time to spend a bit of time with the lecture notes/homework, then it's not necessarily on the professor.

"nool" sounds like a long vowel, it's like the oo in nool, but short

On the other hand, one should be sure the level is fair with respect to the prerequisites, I guess.

it's not like "oo" in "hook," Mathein.

more like "u" in "rule"

Oh right

Yes, @Antonios, I agree that plenty of students are too lazy and expect to be spoon-fed. But, still, a first theorem-proving class should be realistic about what they've learned even if they took the basic intro to proofs course.

Anyhow, @Antonios: if I can help, let me know. Otherwise, don't :)

Nutellensatz is the tastier version

I'll let you know, @TedShifrin. Thanks for the advice and all.

On the flip side I got praise from the students in the other course, apparently.

LOL, smacks lips at Alessandro

But they're on autopilot.

That's great, @Antonios. You're certainly more interested in the job than the average TA I've experienced.

I don't like half-doing things haha.

@AlessandroCodenotti I think I gave you the wrong reference, sorry

it's not in the field theory notes by Milne, but in the field theory notes by P.L. Clark

math.uga.edu/~pete/FieldTheory.pdf

For the archimedean fields thing?

yeah

I see, thanks a lot!

anyway, I'll be back in a few hours if anyone else is.

He even proves the embedding for archimedean ordered abelian groups

I have cappell's class @TedShifrin

At night? Wow. Have fun.

7:10-9...

good stuff.

@Mathein: My prolific former colleague :)

I was quite pleased to see that P.L. Clark formed the plural "Nullstellensätze" right

He's quite pedantic, yes.

But some of us Mericans actually know some furn languages.

@TedShifrin really? I love his expositions

He's good. I was just referring to his attention to detail.

His commutative algebra notes actually include some stuff that I find really interesting which is not in Eisenbud

and which is hard to find in the literature in general

mostly things about certain classes of non-Noetherian domains

I'm going to sleep now, I'll read the proof tomorrow, thanks @Mathei

@TedShifrin if you see him, say him thanks from me, his commutative algebra notes are top-notch

Oh, by the way, today marks the day that the Berlin Wall has been down as long as it's been up

@Mathein: Well, since I moved across the country I rarely see him. I actually didn't see him at all on my visit back a few months ago. But I'm sure he'd appreciate the kind words.

It stood 28 years and was torn down 28 years ago

Night, @Alessandro.

hi. Let $f_{n}(x)$ be a sequence of extended real valued function from a space $X$ and define $f(x) = inf f_{n}(x)$. Can someone help me understand why is $\{x \in X : f(x) \ge \alpha\} = \cap_{n \in N} \{x \in X : f_{n}(x) \ge \alpha \}$

Good night and see you @AlessandroCodenotti

You need dollar signs for us to read the LaTeX, @Shobhit.

@TedShifrin@MatheinBoulomenos may I know w ho you are talking about ?

I could use some notes of commutative algebra

i forgot

@Shobhit: That's just the definition of inf.

ok, thought so, ty :)

1 more question

$f(x)\le f_n(x)$ for all $n$ and there is no number $y$ satisfying $f(x)<y\le f_n(x)$.

@Jacksoja: Pete Clark (at UGA)

oh

@Jacksoja alpha.math.uga.edu/~pete/expositions2012.html there's the link to his commutative algebra notes, along with other good stuff

@MatheinBoulomenos@TedShifrin thanks

in my book we are required to prove that $f^{*}(x) = liminf f_{n}(x)$ is also a extended real valued function. In there proof, they just wrote liminf (thing) = sup (inf(thing)) and therefore the measurability is readily established. How to write this properly?

That's the intersection union stuff we were talking about a week ago, except you were doing limsup, not liminf. Just fix it.

@Shobhit you show that sup f_n and inf f_n are measurable if f_n is measurable and its then automatic

@MaryStar sounds good to me

oh ok, i'll try now

ty @TedShifrin @DavidReed

Anyone here well-read about measure theory? I'm wondering if there's an axiomatic approach to the Hausdorff measures (on R^n, if I have to) like the Lebesgue measure is (up to scaling) the unique Borel measure that measures the unit cube as positive size, the s-dimensional Hausdorff measure is the unique Borel measure that does something...

Here I actually have a screenshot of it....

@MarkS. Great! Thank you!!

Pete L. Clark is the guy who wrote up the thing on real induction, which I really enjoyed

Yeah, he and I argued about that in terms of pedagogy, DogAteMy. It's cute, but I didn't think it belonged in the Spivak course where only his top two students were following anything he did.

Hrm, perhaps not

See, the bad thing about pedagogy is that, I'd just have the instinct to show them all the cool things

You have to remember that your students aren't as gifted as you.

@MarkS. You need translation invariant for Lebesgue to be unique