We will definitely meet :)

Formidable, Dodo :)

@Ted i'm going to watch some of your LA lectures

@Zach: Don't complain that I go too slowly. I'm trying to teach everyone to do mathematics carefully :)

Good choice @Zach. Given an opportunity, more linear algebra is always better

Yup, both linear algebra and multivariable calculus are not sufficiently well mastered.

@TedShifrin that's not too far from my hometown (Brescia) but it's not very close to Trento (where my uni is), when are you planning to travel there?

@Balarka told me this morning about how there's much more to multivariable calc than I (and many others too) think

@Alessandro: A few days in the middle of June is what their schedule says :P

It kinda bleeds in to more advanced subjects

@Daminark: You still haven't learn a lot of it! :D

That's pretty true

@ZachHauk Much more, as in, secretly there's a lot of stuff going on.

I didn't really get the Jacobian

Greeting @Balarka

well, why it works anyways

Hi @Ted!

Linear algebra, @Zach

at that point in my study i didn't know any linear algebra :P

That's the way colleges teach calculus in this country, but for a mathy person it's wrong.

LA before calculus?

"That" meaning multi before linear algebra?

My course it's all integrated. I think that's ideal.

Right, @Daminark.

Wait, I didn't realize you talked about ideals in your class

:P

was that an attempt at a joke

I have exams in June so I could have to be in Trento depending on the days. But there are also direct Trains from Trento to Munich and I've been told it's a beautiful city and I should visit it sometime so I might be able to arrange something

In principal, that's another class, @Daminark. And, yes, there are also principal directions in another class.

He must have if he talked about minimal polynomials ?

Well true

@Alessandro: But Munich will be way at the beginning of my wandering.

I am not embarrassed to admit that I learnt a lot of single variable calculus I skipped out before through Ted's multicalc course.

@Astyx: Actually, I don't in the first course. There's not enough time for all the analysis/calculus and doing more advanced linear algebra.

I do not recommend that though!

And Balarka no longer has contempt for analysis.

one of my friends is studying analysis

We're not talking about psychoanalysis, @Zach.

not IRL friends

Analysis is fun.

See ... changed his tune.

Debatable

@Ted Do the Soug approach and assign an ungodly number of linear algebra problems as homework each week so you don't have to worry about covering it in class.

:P

@Alessandro Certainly more fun than your god awful topological spaces.

:P

Hi, I have a tiny question: why do we have 0 < |x-b| < delta in the epsilon delta definition of limit, but we dont need the 0 < |x-b| part in the definition of continuity?

@Ted someone's phone went off during the lecture

LOL ... Thanks for the news update, @Zach.

:P

ugh, sorry but i could NEVER write on a chalkboard

Don't worry @Balarka you're managing to give me an headache with just $S^2$, no need for weird stuff :P

i would need a whiteboard if i was ever a professor

I absolutely detest whiteboards and the smelly markers that dry out when you look at them wrong.

Whoa whoa whoa @Zach have you never used Hagoromo chalk?

I can hold 4 colors of chalk in my hand. I can't do that with the damn markers.

Actually, @Daminark, I never got to. ....

:(

I was invited to give a lecture at an MAA meeting and they had no working markers in the room for me.

Hagoromo is the best

What about digital boards ?

by the time i'm a professor i can just use my fingers to write on some giant touchscreen thing :P

Yeah that

@Ted how does one become a math teacher at undergraduate/graduate level in the US? In France there's a very competitive written exam followed by thorough oral tests (l'agrégation)

In a school I went to in the UK, they had that in all the classrooms (almost)

@Alessandro Ah, those are the beautiful pictures. I'll tell you more about it after you figure it out (it might take some time; I didn't come up with this proof myself and I don't think I could have but let's see if you can)

There are zillions more math teachers/professors in the US than in France, Dodo. At least until our current administration dismantles all of education. Here anyone with a Masters degree is "allowed" to teach at lower level schools and anyone with a Ph.D. is "allowed" to teach at research universities and colleges.

@Ted for number 4 would you like me to consider the degenerate cases as well?

@Zach: That was the intent of the question, but there are more interesting questions if you keep going.

when two of the coefficients are 0, i get a plane

unless the constant is less than 0

nope.

What's the question ?

let me think this through before i blabber, one second

That's not very encouraging, but I'll think about it @Balarka

So much for watching LA lectures ...

i just want to finish this, first Ted!

:P

@Ted wait... what is it then? We'd have something of the form $ax_0^2 = d$

which has, wait

2 solutions.

it'd be a pair of planes?

@Alessandro Just ponder on it idly. I can always give a hint or two.

Yup, unless $d=0$ or unless $ad<0$.

@Balarka: Have you abandoned forms and differential geometry again? :P

ok so that's one of the cases

@Ted damn... by the way, is there a significant difference in the quality of courses and teachers at community colleges and private universities ? Is there a difference in how the teachers get paid as well ?

@TedShifrin I wasn't supposed to be working on forms and differential geometry? Although I do want to learn differential geometry again after this month - both the surface theory and Riemannian.

I should be working on foliations but I am not.

Got exams in 2 weeks.

Dodo: Some graduate schools try to actually train grad students to be competent teachers, but most do not. I'm proud to say that at U Georgia we worked hard at that (and I spent countless hours watching grad students teach, planning lectures, and helping write exams ... and giving not infinitesimal amounts of feedback).

Exams get priority, @Balarka.

when we have 1 zero, we either have no solutions ( both coefficients < 0 ), a hyperbola that goes infinitely up and down in one direction ( one coefficient < 0), or an ellipse (both coefficients > 0 )

now to consider the degenerate cases

Dodo, for the most part, people with just a masters make less money than people with a Ph.D. But some people with a Ph.D. end up doing adjunct work and it pays horridly.

@Zach: You're still messing up. It's always a surface.

I see Ted

ellipse going up and down infinitely*

infinite right cylinder

Lol I've heard that being an adjunct kinda sucks

Dodo: I think it is not unfair to say that I devoted more energy to students and teaching than the average professor at a research university. But I would never survive in today's world.

@Zach: Right, hyperbolic and/or elliptic cylinders.

ok, what about if it's something like $ax_0^2 + bx_1^2 = 0$?

That's what you just talked about?

Oh, 0.

So what is that?

wouldn't that just be the line $x_0 = 0, x_1 = 0$

NOOOO.

??

You aren't being careful.

but no matter what both those terms are positive or 0

so they have to both be 0

Huh? Why can't $a<0$ and $b>0$?

I'm considering the case where $ab > 0$

Since when? You were throwing hyperbolas around a moment ago.

since i decided :P

Was that made clear to the reader?

What are we doing

sorry, reader

@BalarkaSen classifying some surfaces

It was left as an exercise to the reader

Classifying euclidean quadric surfaces, @Balarka.

Guessing what hypothesis I made is left as an exercise to the reader.

Projectively thinking about it comes in a later exercise, which Zach should get to in a month or so :P

Oh I remember in 163 we did some of the quadric surface stuff

@Astyx: Perfect style for teaching, right?

Our prof didn't want to use Calc on Manifolds

@TedShifrin when $ab < 0$ we have the plane $ax_0 = bx_1$

Sounds like completing the square essentially

He felt that the topology within would be redundant with that which we would get analysis

No, @Zach. Geez.

No, @Balarka: We've already diagonalized, by the Spectral Thm.

Ah, great.

First time I've seen Theorem abbreviated Thm.

So he wanted to do stuff like quadric surfaces which we wouldn't likely see

I probably wrote Thm. on the board a thousand+ times in my teaching.

Should have done some projective geometry, too, @Daminark :P

I like it lol it's one of those "Where has this been all my life!"

fixed @Ted

@Ted In your Spectral Theorem lecture on youtube, do you not prove the theorem ?

Nope, @Zach, sloppily fixed.

Of course I do, @Astyx. Using Lagrange multipliers.

Haha, we weren't gonna go quite that far into the stuff, we just had to do an intro to multi in 5 weeks

fixed now?

I've also always written Thm (although I can't say I did it more than a thousand times)

Nope, @Zach.

@Astyx: Il faut attendre quelques années.

well, assume $a > 0$ WLOG

we get $ax_0^2 = bx_1^2$

@Ted Ou alors passer deux ou trois jours à recopier "Thm" sur des feuilles (ou un tableau pour être plus eco-friendly)

@Zach: You definitely need to take time to triple-check and question yourself. You're in too much of a hurry.

Peut-être pas, @Astyx.

oh but there's squares

And you lost a sign.

We only did the classification of quadratic curves in class, maybe I should do this exercise too at some point

Peut-être pas :p

what do you mean?

Look at your algebra.

i mean

for $ax_0^2 - bx_0^2 = 0$

@Ted the Master of Financial Engineering at Berkeley is 70k$ for one year, while the top notch master in financial math here costs 400$ a year. I've looked around and there's no grant available. Are they expecting people to take a loan or what ? The only reason I'd go for a masters in the US is the huge rep bonus on the CV. In my opinion, it's like buying the diploma

for that case

How do you prove the spectral theorem by Lagrange multipliers? For 2x2 matrices I can do this, by seeing the corresponding quad. form has a maximum and a minimum on the sphere.

And that's it.

Oh we did it that way actually

and $ab > 0$

@Alessandro: It's more interesting to understand (nonsingular) conics (curves) projectively and quadratic surfaces. In one case they're all projectively the same; in the other, that's false.

@BalarkaSen Induct?

@Ted "So on Friday we will prove the theorem" I guess I wasn't watching the right video

Yeah it's exactly that

Dodo: In math programs, most students get supported in the US. Financial engineering they're expecting people to get loans, I guess, and make a ton of money. Have you switched from math to finance?

Induction on what, exactly? Oh I see.

You have your quadratic form, maximize it on the sphere, Lagrange multipliers give you an eigenvalue/eigenvector pair

@Ted the coefficients must fall into 1 of 2 forms, $ax_0^2 + bx_1^2 = 0$ or $ax_0^2 - bx_1^2 = 0$, for $a > 0$, $b > 0$

OK, @Zach. Once again, you changed notation in mid-stream without alerting the reader.

Once you get that, you say alright, restrict to the orthogonal complement

Say your eigenvector is $x_0$

@Balarka: As Daminark just pointed out, the key thing is that symmetric linear maps that have an invariant subspace must also have the orthogonal complement as an invariant subspace :)

Nothing happened here no spoilers at all nope nope nope

Sshhhh @Daminark.

By symmetry of the matrix

@Ted so for the first case we get the line $x_0 = 0, x_1 = 0$

Sure, I agree.

@TedShifrin for the second, we get the intersection of two planes

@TedShifrin I got into an engineering Grande Ecole, not an Ecole Normale Supérieure, so most of what we do is applied math. My Ecole focuses on statistics, data science, finance and economics. Sometimes I miss pure math though :P

OK on the first, NOT on the second. GRRRR.

not intersection

i mean, 2 planes

What's the right word, @Zach?

union

:P

But yeah when you restrict to the orthogonal complement and prove it's an invariant subspace, you induct on the dimension

Oh, Dodo, I didn't know that. I have nothing against applied math, nothing whatsoever. It was just not what I had seen you talking about a few years ago.

You can produce your eigenvalue/eigenvector by using FTA

But the advantage with Lagrange multipliers is that you can generalize to infinite dimensions

@Daminark: So as to use my integration of calculus and linear algebra, I did that with Lagrange multipliers.

Hmm ... I haven't thought about how to generalize to infinite dimensions.

If you're going to use FTA, you need to use the Hermitian inner product and prove that the eigenvalues are real, @Daminark. More stuff.

We haven't done it yet, that'll be at the end

ok so far we have: Infinite right cylinder, hyperbolic cylinder thing, 2 parallel planes, 2 orthogonal planes, and a line. and no solutions

But basically you can use weak compactness of the unit ball in a Hilbert space

@Daminark does Schlag do that

I don't remember

I think so?

sound right, @Ted?

Ah, another UC person :) Hi, Eric.

I'll check Samir's notes lmao

@Zach: Where did two orthogonal planes come from?

hello

I need to clean my linear algebra rust one of these days.

the case we just did

we got the union of two planes, $x_0 = x_1$ and $x_0 = -x_1$

Um, where did $a$ and $b$ go?

they went out to dinner

at Zach's Eatery of Oblivion

Well, unless they bring me a good Kung Pao Chicken back with them, they're fired.

Yeah Samir's notes seem to suggest that it uses Alaoglu

ok so yeah they're not necessarily orthogonal

so those are all the cases for only one zero

now, for two zeros

we get two planes, like we did earlier, or one singular plane

or no solutions

Heyoo

for three zeroes we have all of $\Bbb R^3$

Right, @Zach.

now for the fun part

Heya @Krijn.

no zeroes! blows party horn

@TedShifrin How's life on a scale from 1 to 10?

@Krijn: I am currently sick, so I give a low mark.

@TedShifrin Just a flu or sick sick?

flu is worse than most things, actually ...

Ah yeah you can use weak compactness to construct a subsequence that maximizes your quadratic form in the limit and some algebra shows this is an eigenvector, then you mess around on the orthogonal complement and do some analysis to show the eigenvalues go to 0 using compactness or something @Daminark

Hi @Ted.

@Ted for all positive coefficients, we either get an ellipsoid, the origin, or no solutions

I have succesfully procrastinated all evening

Bye chat !

@Eric @Daminark: You doing compact operators or the general self-adjoint case?

Night, @Astyx.

compact operators

Compact

Hi @Fargle

for one negative, we get either a 2 sheeted hyperboloid or a 1 sheeted hyperboloid?

What about the $0$ constant case?

umm

1 sheeted hyperboloid

?

no

@BalarkaSen How's life on a scale from 1 to 10?

let me think about this for a bit

5. Always was and always ever will be.

the origin is included

@BalarkaSen Average would be a 5,5

That's subaverage

so it's this weird degenerate hyperboloid

@Balarka Locally or globally?

@Krijn Yeah, I know.

@TedShifrin it's like two paraboloids glued at their minima

@Zach: It is actually a cone over a hyperbola.

I don't think you have a surface of revolution any more.

@BalarkaSen :'(

by the way

@Daminark If I don't die a tragic, painful and slow death - globally.

@Krijn: There's more grade inflation, typically. Probably 7.5 or 8. is average.

are hyperboloids the intersection of some 4d cone thing with a hyperplane?

Yikes @Balarka

@TedShifrin Ehhh, visit my uni one day and tell 'em that

because hyperbolas are like the intersection of a conic with a plane

Well, what is it locally?

Yes, @Zach. You'll see more spatial projective stuff in the next section.

neat

anyways, i think that's all of the cases

It's enough for me, thanks.

because 2 negatives and 3 negatives is just negating the entire equation

Yup.

now that i finished that

i think ill go watch some LA

No Oscar awards for that.

"Best LA Lecture goes to, Ted Shifrin.... but it appears he could not make it."

I'm pretty sure not even close ...

well

i haven't watched any LA lectures before

On that note Tarkovsky's polaroids collection came out recently.

@BalarkaSen Have you received his fathers poetry already?

Yeah, I got it in December '16.

Have you read it as well?

Lots of it. I carry it around with me all the time.

I have quoted one in my MSE profile.

Yeah but then again, your profile says "Interested in number theory"

I am!

Balarka's interests have changed/expanded, but he has film and literary interests that aren't in his profile :P

Nice (easy) question on main: Two smooth, simple convex curves that intersect at infinitely many points.

I don't think that can happen.

This hierarchy of separation axioms is really intimidating.