Mathematics - 2018-03-05 (page 1 of 7)

Ted Shifrin

00:00

NOOO, check your exponent laws, @DarkRunner.

Antonios-Alexandros Robotis

first sem @TedShifrin

bout to read the others

Ted Shifrin

Oh, OK, @Antonios. You can see my style is far more concrete, but they still have to write real proofs.

Antonios-Alexandros Robotis

yeah that's probably a good thing

I guess I went about learning all these things very abstractly, just by the nature of my courses.

DarkRunner

@TedShifrin It's $(10^{ 10 })^{ 10 }$

Semiclassical

So if I can reduce the scenario to $a=c$, then I can characterize the image in a nice way.

DarkRunner

00:01

And then, you can repeat the same for the rest!

Ted Shifrin

Right, @DarkRunner, and you keep taking $10$th powers.

Antonios-Alexandros Robotis

I've found that this year I've been working on fully understanding examples, and it's really complemented my understanding quite well.

DarkRunner

Right, right

Ted Shifrin

Of course, @Antonios. But too many teachers fail to teach that way.

DarkRunner

thanks a lot, @TedShifrin

Ted Shifrin

00:01

Sure thing, @DarkRunner.

Especially in grad school, faculty tend to think that students can think up examples and work them out ... all on their own. Which very few can or do.

One needs to develop that skill.

Semiclassical

And the examples we come up with as grad students aren't always the best ones...

Ted Shifrin

Even in grad courses I assign (often quite hard) explicit computations and examples.

So, @Semiclassic, it's sorta interesting writing out that matrix.

Antonios-Alexandros Robotis

yeah, @TedShifrin. I think in my case, it was strange because I started math relatively late, and since so much of it built on the earlier material I should have been more comfortable with, I ended up crutching on "abstraction" in some way.

Semiclassical

Yeah.

Antonios-Alexandros Robotis

Of course, now that I'm beginning to see it both ways, I think it's really made life a lot easier.

Ted Shifrin

00:05

@Semiclassic: Ignoring the denominator, all the entries are sin of differences of the four angles.

Semiclassical

Right.

Ted Shifrin

And those differences can vary from $0$ to $\pi$ or something, non-inclusive.

I guess there's some relation, though, among $\beta-\gamma$, $\beta-\delta$, $\gamma-\alpha$, and $\delta-\alpha$.

Semiclassical

Yeah.

Ted Shifrin

Alternating sum is 0.

So three of the matrix entries determine the fourth.

Antonios-Alexandros Robotis

This philosophy of rings first is very interesting @TedShifrin.

Ted Shifrin

00:10

I can justify it in many ways, @Antonios, although I realize it's only about 25% popular among teachers.

For a student headed to math grad school, I'm fine with Artin and groups first :) And I love the linear algebra he intermingles throughout.

Antonios-Alexandros Robotis

Well, I do agree that rings are more intuitive coming from HS math.

[commutative rings*]

Ted Shifrin

But when a significant number of the students are going to teach high school and a significant number of the math majors are not super strong, rings are easier (despite the two operations) and more important [integers, mod arithmetic, polynomials, no worry about normal subgroups].

Right, although I did use matrices a bit, too. But, yes, commutative rings.

Antonios-Alexandros Robotis

It's interesting. I feel like a good highschool algebra course can actually prep you so well for basic ring theory

Ted Shifrin

My point was for the teachers to understand things like setting factors = 0 to find roots of a polynomial. And to understand where they're using integral domain, etc.

I also had fun asking them how we know 1/2 isn't an integer.

I could send you separate exams from my summer course, when 80% of my students were math ed. I biased that course a bit differently (especially cuz of the shorter time span, and generally weaker students).

A few of those students are still Facebook friends and occasionally thank me for training them :)

Antonios-Alexandros Robotis

Sure, the more exams the merrier. I've been doing undergrad exams a lot recently for personal edification/fun.

Ted Shifrin

00:15

Well, it might help you teach/TA better, although the course is out of your hands.

Antonios-Alexandros Robotis

Good to practice doing problems that you (should) know how to do.

I'm going to try to get her to take the max of the midterm scores.

Ted Shifrin

Might give you pedagogical ideas, though.

Antonios-Alexandros Robotis

Doesn't seem right to condemn students to a max grade of ~B after one test.

Ted Shifrin

None of her attitude sounded "fair."

Antonios-Alexandros Robotis

Yeah, the students don't have a very high opinion of her, it seems.

[the ones who are with it, mind you]

Ted Shifrin

00:16

Students pick up quickly on teachers who don't really give a damn about the students.

I pushed students pretty hard because they knew I cared and would help them try to succeed.

Antonios-Alexandros Robotis

Well it seems worse than that. She's always going on about how some of the students are just "stupid" or won't stop asking "stupid questions."

Ted Shifrin

As you can tell from talking with Alessandro, Kasmir, and Mathein (and others in Europe), that's not the prevalent attitude amongst profs in Europe, e.g.

Semiclassical

ugh

Ted Shifrin

She's an entitled asshole.

Antonios-Alexandros Robotis

Contrary to semi-conventional wisdom, I do agree that not all questions are created equal, but more in the sense that some are for office hours, rather than class.

Ted Shifrin

00:18

Of course.

Antonios-Alexandros Robotis

She just seems to judge the students immediately based on questions

Ted Shifrin

Maybe she should think about how that reflects on her presentation of material?

I mean, some of the students definitely have to be blamed for things, but ...

You can't just teach to the top 5% of the students.

Antonios-Alexandros Robotis

I want to sit in on the lecture

Ted Shifrin

The students will give up and quit asking, and she'll have succeeded.

Antonios-Alexandros Robotis

But I'm not sure what that's going to tell her...

Ted Shifrin

00:19

Well, you can be polite and say that you want to see what you should concentrate on in discussion section.

Antonios-Alexandros Robotis

Might be a good move.

PVAL-inactive

I'm not really brave enough to do this, but I think jokingly insulting the audience can be entertaining.

Ted Shifrin

I can do that because I've established a great rapport with my classes, almost always, PVAL, but without that, it will backfire badly.

Witness our dictator in chief.

PVAL-inactive

I mean he was joking. I think. I hope.

Ted Shifrin

He has negative sense of humor.

Antonios-Alexandros Robotis

00:21

what happened this time?

PVAL-inactive

if you are referring to his comments on china

Ted Shifrin

He thinks he should be anointed emperor. No joke.

Antonios-Alexandros Robotis

LOL

Semiclassical

Deprecating the audience works better when you (implicitly or explicitly) allow self-deprecation as well.

Ted Shifrin

As I said, empathy and rapport.

Semiclassical

00:22

Yep.

Joe Shmo

in good taste, i would appreciate it if the professor had pointed out that my question is idiotic

Faust

@TedShifrin

Joe Shmo

i would still expect an answer, of course.

Faust

Morning

Antonios-Alexandros Robotis

most people don't @JoeShmo

Semiclassical

00:23

I prefer "silly" to "idiotic"

Ted Shifrin

@JoeShmo: Did you want me to say that to you when you were stuck on that pullback?

Hi @Faust

Antonios-Alexandros Robotis

hi @Faust

Joe Shmo

in good taste

Ted Shifrin

I have excellent taste, thank you.

Joe Shmo

if he says that it's idiotic (which i probably already know) in a demeaning fashion, i wouldn't appreciate it

Ted Shifrin

00:24

The thing that Hippa memed years ago from my multivariable class was totally taken humorously by my students, but I'm not sure Hippa's meme quite carried that humor :P

Joe Shmo

yeah i'd seen you let them have it in class @TedShifrin

Ted Shifrin

@JoeShmo: I have been known to tell people in here — "You go figure that out." Or "you can do that yourself."

user2154420

Is it true that a simple extension of $\mathbb{F}_q$ of degree $m$, where $q=p^n$, is isomorphic to $\mathbb{F}_{p^{mn}}$?

Joe Shmo

@TedShifrin if you still answered it, sure :) i dont take it personally. and again, in good taste. and most questions out there are idiotic when you're learning new material. once you get to the good ones, you want to work on them yourself..

PVAL-inactive

There was a question on main, and I immediately commented "of course not." and my comment received like 3 or 4 upvotes.

Ted Shifrin

00:26

Yes @user2154420. Easier to think of it as $q^m$ :)

Antonios-Alexandros Robotis

I think a lot of people have a quite condescending attitude on MSE.

Ted Shifrin

@Antonios: I get angry when people give complete solutions to homework problems, and I get angry at those people if I'm engaged in hinting with the OP. But I assume you're not talking about me.

More Anonymous

@Antonios-AlexandrosRobotis agreed ...

Antonios-Alexandros Robotis

lol nope

Ted Shifrin

I think that because I've bitched in here, people like Balarka and Akiva (and even Leaky) have improved their pedagogy and patience.

Mathein isn't always condescending, just to me.

Antonios-Alexandros Robotis

00:28

loll

Seems most of the people in here are fairly well-behaved so to speak.

A lot of the grad students at Berkeley were very very negative towards undergrad students, as I recall.

Joe Shmo

ive developed thick skin over the years. some of my professors in college had a major pole up their behind

DarkRunner

@TedShifrin I honestly don't know what happened. After converting all the 10s to 3s, I found that the period for the powers of 3 was 4, and then found the powers of 10 up to 10 mod 4.

I got that 10 (mod 4) =2, and everything thereafter is 0, thus I did 2+1+1...+1, and got 11 mod 7, which is 4

Ted Shifrin

@Antonios: I spent a lot of time talking with and helping undergrads when I was at Berkeley.

Joe Shmo

one of my math professors congratulated me on having found "the worst possible question to ask in class". After protesting, he clarified -- "I'm not scolding you. I'm congratulating you. Congratulations, you found it" -- needless to say, that wasn't done in good taste

user2154420

Cool. So say I have extensions of $\mathbb{F}$ of degree $m_1$ and degree $m_2$. Does this mean that the composite of these extensions has degree $lcm(m_1,m_2)$?

More Anonymous

00:30

@JoeShmo I've had similar experience in my ex-university ... but my current one (Nottingham) they actually listen!!

user2154420

*$\mathbb{F}_q$

Antonios-Alexandros Robotis

There were DRP talks (directed reading program) at the end of each semester: these were programs where undergrads would pair up with grad students and read some stuff, then present on it at the end of the semester. A number of grad students would show up to the talks and absolutely roast the undergrads with questions they knew they wouldn't be able to answer.

DarkRunner

the correct answer is 5

Ted Shifrin

What a good way to feel superior.

Antonios-Alexandros Robotis

Qiaochu asked me a fairly tough question, but they were quite reasonable with me. The other students were not so lucky.

Ted Shifrin

00:31

Qiaochu knows more math than ten faculty put together.

PVAL-inactive

I usually ask what I'm curious about.

Antonios-Alexandros Robotis

It's quite disturbing.

Qiaochu is... impressive.

user2154420

Is what I said hogwash?

Antonios-Alexandros Robotis

Mariusz Wodzicki called him "a genius" which means a lot coming from his mouth.

Joe Shmo

... I should have congratulated him at the time as well, since he also found the worst possible answer to give to a student.

Ted Shifrin

00:32

@DarkRunner: Why are you doing mod 4?

DarkRunner

nvm, I got it

stupid mistake

Ted Shifrin

OK.

PVAL-inactive

The idea that you should only ask questions during a talk if you're convinced the speaker will be able to answer is silly.

Ted Shifrin

It's a matter of tone, PVAL, and this isn't a visiting research faculty member.

Antonios-Alexandros Robotis

Sure, but if you see a clearly nervous undergrad giving a talk, would you ask them about XYZ research topic if they're presenting a basic result like Stone-Weierstrass?

Ted Shifrin

00:34

I got asked plenty of questions in geometry seminar at Berkeley that were tough ... but that was a seminar presenting (someone else's) research.

@user2154420: Too many things to think about too quickly.

So we need the smallest exponent divisible by $m_j$ for both $j$. Isn't that what you have?

@MoreAnonymous: I find your post very hard to read. Is $s$ an indeterminate? If you're thinking of $\hat K$ as a linear mapping, you'd better specify the domain and range, for starters.

PVAL-inactive

I mean if you don't know the answer you say, "I don't know the answer". If this is really an issue the mentor should explain this to the student beforehand. I'd never ask a question in a situation like this where I already knew the answer though, so that means typically the questions are not going to be that easy.

Ted Shifrin

I think you should treat an undergrad giving his first talk a bit differently from the way you treat a 40-year old research seminar speaker.

It shouldn't just be about you.

Don't bother going if you don't want to be constructive.

user2154420

@TedShifrin I guess I ask this because of this special case:

Suppose $f$ and $g$ are two monic irreducible polynomials over $\mathbb{F}_q$ of degrees $16$ and $20$ respectively. Let $F$ be the splitting field of $fg$ over $\mathbb{F}_q$. Then I want to say that $F$ is really just the composite of $\mathbb{F}_{q^{16}}$ and $\mathbb{F}_{q^{20}}$, which I guess is $\mathbb{F}_{q^{80}}$? So this is a degree $80$ extension?

More Anonymous

@TedShifrin thank you for looking! s is a variable. I was just reading wikipedia I should have used the word transform perhaps?

Ted Shifrin

What are your proposed domain and range for $\hat K$?

Lozansky

00:40

If someone likes spotting mistakes, please see my question math.stackexchange.com/questions/2677073/…

Ted Shifrin

I think that's right, @user2154420. You could test it out with smaller numbers, too. :)

user2154420

Sweet, To sage I go

Ted Shifrin

@MoreAnonymous: So you're working with polynomials in $s$ where the coefficients are matrices?

More Anonymous

yes

user2154420

@TedShifrin Thanks!

Ted Shifrin

00:42

Sure ...

But you've only defined it in a very special case, @MoreAnonymous. How is it defined for a general expression $\sum f_j(x)s^j$?

Oh, I guess it's not even polynomials. It's certain formal power series in $s$.

Leyla Alkan

Hi all, could you show me how to get those upper limits of integrals that I marked with blue pen? I find all the other limits, but have problem with those marked ones.

More Anonymous

I would have to power expand $f_j(x)$ as well

Leyla Alkan

More Anonymous

for that

and then apply the algorithm

Ted Shifrin

@Leyla: What's the equation of the bounding tilted plane?

PVAL-inactive

00:46

@TedShifrin I think asking questions expressing my own personal ignorance and confusions are generally better than lobbing softball questions I know the answer to which test the ability of the student to answer, especially in a situation like this when I am not supposed to be evaluating the student.

Ted Shifrin

@MoreAnonymous: I don't see how you have defined $\hat K$ other than on one single expression. So I don't see how to abstract to a general situation.

Leyla Alkan

@TedShifrin I didnt understand, what are you asking exactly?

Ted Shifrin

@PVAL: I'm surmising your own personal ignorance won't be involved with the subject of the talk, but if it is, I don't preclude that as a reasonable question.

@Leyla: You have a 3D region there. It's bounded on the left, bottom, and back by coordinate planes. What is the top of the region?

PVAL-inactive

oh I'd never ask anything not involved with the subject of the talk.

Ted Shifrin

@PVAL: I'm sorry if I misunderstood. Antonios specifically mentioned something clearly out of line with the topic of the talk ...

Leyla Alkan

00:48

@TedShifrin Yeah, but I dont know that. Thats why I am asking :)

Ted Shifrin

It's the equation of a plane. $Ax+By+Cz=D$. How do you find the equation? Hint: You know three points on the plane.

PVAL-inactive

Now I'm trying to remember some of the questions I asked.

Ted Shifrin

LOL ... I won't try to do that, although I've sure asked lots more questions than you have :)

In Ph.D. orals and defenses I was notorious for asking for a specific example if the speaker hadn't given me one.

More Anonymous

@TedShifrin $hat K$ can only be defined by take the modulus difference of the powers of x (which is the coefficient of s^r, which in turn can be expanded) ... Followed by leaving only the positive coefficients after this process

Joe Shmo

@TedShifrin if c is a k-chain, ∂c is a (k-1)-chain

Leyla Alkan

00:51

@TedShifrin I dont know how to do really

Ted Shifrin

Yes, @JoeShmo. Sure.

ÍgjøgnumMeg

If $K_1 = \Bbb Q(\alpha_1)$ and $K_2 = \Bbb Q(\alpha_2)$, is the map $\sigma_2(\alpha_1) = \alpha_2$ necessarily $\Bbb Q$-linear, just because the powers of the $\alpha_i$ are a basis for $\Bbb Q(\alpha_i)$ as $\Bbb Q$-vector spaces, or does one have to give more information about the maps (i.e. that they are ring homomorphisms)?

Ted Shifrin

@Leyla: Tell me the three points on the plane. They have to satisfy that linear equation I wrote down. Plug them in and what do you find?

Joe Shmo

i am asked to verify the Stoke's theorem works, i.e. compute both sides of the equation by hand. i have the l.h.s and am praying to G-d it's actually equal 1/4.

PVAL-inactive

I remember a talk on Tychonoffs theorem where the student gave a proof of the theorem using the language of closed sets, and I asked if the proof was logically equivalent to one using open sets. The student couldn't answer that, but of course I couldn't either.

Joe Shmo

00:52

im not so sure how to proceed with the integral of alpha on the boundary

Ted Shifrin

THat's a perfectly reasonable question, though, PVAL.

So you did the integral of $d\omega$ on the $k$-chain, @JoeShmo?

(Hmm ... Have you watched my Stokes's theorem vids? :P)

Joe Shmo

yes

and that's presumably = 1/4

um, yeah, i think

Ted Shifrin

So what is the $k$-chain here?

Joe Shmo

im lying potentially

Ted Shifrin

I did LOTS of examples ... even some in 4D.

Joe Shmo

00:53

c: (t1,t2,t3) --> (t2t3, t1t3, t1t2)

3-chain

sorry 1-chain

Ted Shifrin

with domain a 3-cube or 3-simplex?

Huh? $1$-chain?

Joe Shmo

c: [0,1]^3 --> R^3, a 3-cube

More Anonymous

@TedShifrin sorry for the sloe reply I was thinking about what you had said previously

Joe Shmo

it's a 1-chain correct?

Ted Shifrin

@MoreAnonymous: I don't understand the construction well enough to even guess if it's

No, that's a $3$-chain, @JoeShmo.

Joe Shmo

00:55

well no its a 3-chain

Ted Shifrin

And they gave you a $2$-form $\omega$ somewhere.

Joe Shmo

right. he defines k-chains as linear combinations of k-cubes. and since we have a 3-cube, its a 3-chain

ok

Ted Shifrin

So you need to look at the 6 faces of the cube and pull back $\omega$ to each of those, integrate, and add up with correct signs.

The boundary of the cube has 6 faces.

Joe Shmo

they did... it's like you know

Ted Shifrin

I took a totally wild guess!

In 4 minutes I can get a martini and make wilder guesses.

Joe Shmo

00:56

alpha = x1 dx2 /\ dx3

DO IT

dalpha = dx1 /\ dx2 /\ dx3 (you taught me that, not the other guy)

Ted Shifrin

A bumblebee could have taught you that much.

Antonios-Alexandros Robotis

lol

Joe Shmo

hhaha...

Ted Shifrin

So you need to pull back $\alpha$ by $c$ and then restrict to the 6 faces.

Leyla Alkan

@TedShifrin I plugged $(2,0,0),(0,2,0),(0,0,1)$ and got $x+y+2z=2$

Ted Shifrin

00:58

BTW, it's \wedge

Joe Shmo

i don't actually see latex on here. do you?

Ted Shifrin

Perfect, @Leyla. Can you see how to finish now?

See the LaTeX in chat link over there >>>>>^^^^^^^ @JoeShmo

Antonios-Alexandros Robotis

@JoeShmo I couldn't get Mathjax in chat to work in chrome

works fine in other browsers

might have been my own incompetence, though.

Ted Shifrin

I'm always in Chrome in here.

Joe Shmo

lucky for me im on chrome

Leyla Alkan

00:59

yes, then I can get 1-(x+y)/2 . Okay thanks!

Ted Shifrin

I use Chrome on my desktop and on my iPad. Works fine.

You're welcome, @Leyla.

Antonios-Alexandros Robotis

Probably just me, then.

Transcript for

$Mathematics$ Mathematics