Mathematics - 2015-01-31 (page 5 of 5)

Kaj Hansen

11:00 PM

@KevinDriscoll, that actually happened to me in high school

Danu

General foundations of algebra?

Kevin Driscoll

I always thought her work in math was realted ot the physics stuff

related*

Quaxton Hale

@TedShifrin Lol I've watched your videos and I have a copy of that text.

Danu

Why would anyone care about rationalizing denominators?

Quaxton Hale

Yes, I should probably remove that question now.

Danu

11:00 PM

@KevinDriscoll I don't know: I'm asking!

Mike Miller

Noether worked in a wide range of fields, but she's probably most remembered for her work in algebra, yes.

bd1251252

@TedShifrin I want to become a professor of mathematics someday. I realize that's a bit general, but what advice could you give me on how to achieve that goal?

Danu

General foundations, or some specific things? Any particularly notable results?

Kevin Driscoll

@Danu My understanding is that she worked on the calculus of variations and that's where the theorem comes from. But then also more things useful for physics about commutative rings, non-abelian group stuff

Danu

Hmkay

I'm really motivated to learn some algebra: If only I had a bit more free time :\

bd1251252

11:03 PM

@KevinDriscoll I got a book on Calculus of Variations and started reading it about a week ago. It's really neat

Ted Shifrin

@bd1251252, when you say "The Institute," to what are you referring?

Danu

In fact, I should get back to studying for my diff. geo. exams :(

Ted Shifrin

very cool, @Quaxton. I'm surprised you're talking to me :D

bd1251252

@TedShifrin The Rochester Institute of Technology

Danu

@TedShifrin IAS?

@bd1251252 lol, not what I was expecting

Kevin Driscoll

11:03 PM

@bd1251252 It is! And an area that I think draws comparatively little focus in math today. Maybe it's all settled, I don't know.

Ted Shifrin

ah, ok, @bd1251252. I went to another "Institute" many years ago :P

yes, @Danu, or else I'll ask you hard diff geo questions.

bd1251252

@Danu What were you expecting?

Ted Shifrin

He's a college kid, @Danu :P

bd1251252

Ohhhh

Wow I just caught on to that

Mike Miller

I don't know what general foundations means, @Danu. She's so well-known for her work on the theory of ideals there's a specific class of (very important, very common) rings named after her.

bd1251252

11:04 PM

My bad lol

Kevin Driscoll

@Danu I have found that the list of things one would like to learn as a graduate student is far too long to be completed even in 6 years. Neither you nor I will know it all when we graduate. It's almost as if postdocs and professors must be constantly learning things as well. Which is shocking, I know.

Ted Shifrin

@bd1251252: Work hard, don't lose your enthusiasm, and don't be scared to wrestle with very challenging problems.

@Kevin: I don't know anything any more.

Danu

@TedShifrin Throw something at me! :D I'm pretty sure I'm going to fail it :\

Ted Shifrin

What's the scope of the course/exam, @Danu?

Danu

...incomprehensible!

Ted Shifrin

11:06 PM

LOL, that doesn't help me.

Kevin Driscoll

@TedShifrin Wonderful! Think of all the exciting things you have to learn, then! You could watch your own videos series, for instance.

Danu

It was quite a disorganized course

Ted Shifrin

Physics course or math course?

bd1251252

@KevinDriscoll I study and read math all day long. I'm about 3-4 weeks ahead in my (lame) discrete math course, and I can say I've mastered the material we've covered so far in multivariable calc

Danu

@TedShifrin pure math

Ted Shifrin

11:06 PM

@Kevin: If I did that, I'd hurl things through the screen.

Danu

Among the topics discussed were...

Kevin Driscoll

@TedShifrin Danu's profile says hes a ........ mathematical physicist

Danu

@KevinDriscoll Are you mocking me? :P

Ted Shifrin

Don't take him seriously, @Danu.

He's a physicist who's jealous that he doesn't know more math.

Danu

the basic notions of tangent space etc etc, then some stuff on group actions, then some stuff on foliations & distributions, then some things on the cotangent bundle, exterior algebra & finally orientations and integration on manifolds (ending with Stokes)

Kevin Driscoll

11:07 PM

@Danu In this chat, I have to fight for my right to a lack of rigor and reliance on pretty pictures instead of proofs

Danu

...also a week of De Rham cohomology but I'm not sure that'll be on the exam.

Mike Miller

You have no right to that, @Kevin.

Ted Shifrin

Oh, @Danu, see, to me differential geometry is the next course (where you learn about metrics and connections). This is just basic manifolds. OK, let me see ...

bd1251252

@TedShifrin I like how active your classes are...in my experience in math classes (i.e. every single one I've ever had) the professor had to...I don't know...coerce kids into answering anything. there was always a pause after they asked things. It's like you're having a conversation with them but at the same time teaching them

Kevin Driscoll

@MikeMiller @Danu SEE WHAT I MEAN!?

Danu

11:08 PM

@TedShifrin Riemannian manifolds is a separate course here

Karlo

@PedroTamaroff the degree of vetrticle u for e.g is only possible in undirected graph and if it means all edges that have u as an beginning or end?

Ted Shifrin

@Danu: Agreed. I just don't call the first course differential geometry. I call it manifolds :P

I will miss my students, @bd1251252.

Pedro Tamaroff

@Karlo I don't see a problem with defining the multidegree as the number of edges that meet with $v$.

Quaxton Hale

@TedShifrin what are you implying here lol

bd1251252

@TedShifrin I've never heard a professor even use the word "manifold" in a class

Pedro Tamaroff

11:09 PM

But I've never studied multigraphs.

Karlo

@PedroTamaroff yes ofcourse

Danu

The material we covered corresponds roughly to Lee's book on smooth manifolds, ch. 1-11 (skipping 6 & 10), then 14-17 and 19,20

Kevin Driscoll

@Ted @bd1251252 Its almost as if there was this guy like 2000 years ago who developed this method of teaching through conversation and asking questions........

Mike Miller

The distinction between multigraphs and graphs is so minimal people usually just call them graphs.

Karlo

@PedroTamaroff otherwise the number will be max 2

Ted Shifrin

11:10 PM

@bd1251252: They don't normally show up in the undergraduate curriculum for most students.

Danu

all in very disorganized fashion, and not very in depth

bd1251252

@KevinDriscoll Ha yes, Socratic Dialectic

Danu

The professor is somewhat of a legend here... for his 'lecturing skills'

bd1251252

@TedShifrin I wish they did!

Ted Shifrin

@Danu, I am sorry. When I retire I may start LaTeXing my graduate course notes :P

Danu

11:11 PM

@TedShifrin I actually have a latexed version of this course's notes :)

Ted Shifrin

I sent them to @Mike but he claimed he couldn't read them.

bd1251252

@Danu Oh really? Gosh...I can't tell you how much of an honor it would be to take a course with Dr. Shifrin

Ted Shifrin

Well, my courses are never criticized for being disorganized :P Maybe vapid, but not disorganized.

oh hush, @bd1251252

Danu

@bd1251252 Not a course by Shifrin, I mean the one by my professor here in Munich

Mike Miller

Why, @bd? I wrote a book too, once.

Danu

11:12 PM

his name is Leeb

Ted Shifrin

Don't know him, @Danu.

bd1251252

@TedShifrin I don't see it as disorganized sitting here watching his lecture...

Danu

Oh, that reminds me: Is there any equivalent of INSPIRE-HEP for mathematicians?

bd1251252

@TedShifrin "Your" lecture, sry

Danu

I.e. some site where one can check out researcher profiles and find their papers, citations etc etc?

Ted Shifrin

11:12 PM

OK, can you prove that if $G$ is a compact Lie group acting on a manifold $M$, then any closed $k$-form on $M$ differs from some $G$-invariant form by an exact form?

Mike Miller

MathSciNet.

user159870

@TedShifrin Show that the ideal $\langle x^2+1 \rangle$ is a maximal ideal of $\mathbb{R}[x]$.

I took $\phi: \mathbb{R}[x] \to \mathbb{R}$ with $\phi(x^2)=-1$ and $\phi(z)=z, \forall z \in \mathbb{R}$.

Let a polynomial $p(x) \in \mathbb{R}[x]$. We apply the euclidean division of $p$ with $x^2+1$ and we have $p(x)=(x^2+1)g(x)+h(x)$ with $deg(h(x)) \leq 1$.

Let $p \in ker \phi$. Then $\phi(p(x))=0 \Rightarrow (\phi(x^2)+\phi(1)) \phi(g(x))+ \phi(h(x))=0 \Rightarrow \phi(h(x))=0$

Mike Miller

You love that theorem, @Ted, but for good reason.

Ted Shifrin

No, @bd1251252, I have failings, but being disorganized isn't one of them.

bd1251252

@MikeMiller I don't know...I'd say it's not the book, it's the process

Danu

11:13 PM

@TedShifrin Afraid we didn't talk about $G$-invariant forms

only about left-invariant vector fields

user159870

@TedShifrin How do we find $h(x)$ ?

Danu

(the section on forms was a little rushed :\ )

Ted Shifrin

That's my favorite part, @Danu ...

Danu

Mine too, actually

...because it makes a lot of sense to a physicist's mind

along with flows

Karlo

@PedroTamaroff why in directed and undirected graph the path and road resepctively between $(u_1,u_1)$ is 0 and 1 respectively

Ted Shifrin

11:14 PM

Do the division algorithm, @user159870

user159870

Of $h(x)$ with what? @TedShifrin

Ted Shifrin

Can you prove that every derivation on $C^\infty$ functions is of the form $\sum a_i\dfrac{\partial}{\partial x^i}$ for some $a_i$, @Danu?

@user159870: Hint: Your homomorphism should not map to $\Bbb R$. It should map to something else, or else $x$ won't have anywhere to go.

bd1251252

@TedShifrin the vector notation you use is exactly what my current professor told me not to use. Wow...

Danu

@TedShifrin The space of derivations (aka the tangent space) is just a (finite dimensional) vector space spanned by the $\frac{\partial}{\partial x_i}$

Ted Shifrin

@bd1251252: That's because I'm teaching a real math course, and I'm using both linear algebra and calculus ... But most multivariable courses use $(a,b,c)$ or $\langle a,b,c\rangle$. There are good reasons to do what I'm doing and what you did, but he won't like them.

Prove it, @Danu.

bd1251252

11:18 PM

Like to say, \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} -- yeah no, he told me he'd take off points if i used them any more

Ted Shifrin

@bd1251252, he's a twit.

But you want to get an A, however little you learn.

Danu

@TedShifrin I guess the way to go would be to just use the differential of the inverse of a chart and use the canonical basis vectors on $\mathbb{R}^n$

Ted Shifrin

No, you have to start with the definition of derivation.

bd1251252

@TedShifrin Haha. And yes, I totally recognize that so I will respect his way

Ted Shifrin

Prove it for a point in $\Bbb R^n$.

user159870

11:20 PM

Could we map the homomorphism to $\mathbb{C}$ ? @TedShifrin

Ted Shifrin

Good idea, @user159870

@bd1251252: I shouldn't ask this, but what level does your mom teach?

Danu

Right, I see what you're saying. Good exercise. I think, however, that in this exam we will be required to simply assume results such as this (which were, of course, proven in the lecture)... the professor seemed to imply as much when we discussed the upcoming exam last week. Honestly though, I'm not sure what to expect since this is my first pure math lecture.

user159870

Do we take then the homomorphism $\phi: \mathbb{R}[x] \to \mathbb{C}$ with $\phi(x)=i$ and $\phi(z)=z, \forall z \in \mathbb{R}$ @TedShifrin ?

Ted Shifrin

OH, ok, @Danu. It's hard for me to play this game in a fair manner.

Sounds good, @user159870. Now prove that the kernel is the ideal you want.

Danu

@TedShifrin It's really scary, not knowing what to expect...

I think there is a reasonably large chance that I'll fail the exam. There is a retake in about 2 months though, and I will study my ass off for that (here in Germany we get 2 months of holidays in spring, so plenty of time)

Ted Shifrin

11:24 PM

@Danu: One of my favorite exercises (even for undergraduates), crucial for differential geometry, is the Cartan lemma: Suppose $\omega_1,\dots,\omega_k$ are linearly independent $1$-forms on $\Bbb R^n$. Suppose $\theta_1,\dots,\theta_k$ are arbitrary $1$-forms satisfying $\sum \theta_i\wedge\omega_i = 0$. Prove that $\theta_i = \sum a_{ij}\omega_j$ for smooth functions $a_{ij}$ with $a_{ij}=a_{ji}$.

(Note that $k$ may be $<n$.)

Well, @danu, this is a course I've loved teaching, and it's essential stuff.

bd1251252

@TedShifrin Oh no it's fine. My mother is an adjunct professor of basic math, algebras and (soon to be) calculus. She hasn't been able to get a tenured position, however, so she's currently getting her PhD in math education from the University at Buffalo. I wish she could as our family badly needs the income and she's a very good professor, but so far it just hasn't come to fruition.

@TedShifrin In the mean time my dad couldn't get a position anywhere with his Bachelor's degree in music so he's working as a cleaner at Brockport State College, all the while going for his Master's in Library Science. So yeah...I have two older sisters in college and both my parents are too

Ted Shifrin

yikes, it's tough, @bd1251252, but don't be limited by your mom's view of mathematics.

I suspect she was defending your teacher just to make your life easier.

Danu

@TedShifrin I think I'd have a very hard time trying to prove this within the hour (or twice that, which is the time I'll get on my exam)

bd1251252

@TedShifrin I won't be. I like to think of myself as beyond all of that stuff...I've already told her I could never teach kids who couldn't graph lines on a plane

Ted Shifrin

@Danu: It's not hard. It's basic linear algebra. Hint: Extend $\omega_i$ to a basis.

Well, @bd1251252, make sure you learn everything well and don't just have an attitude.

Danu

11:28 PM

@TedShifrin Basic linear algebra is an elusive thing for us physicists ;)

Ted Shifrin

Linear algebra is universally important, @Danu. Shaddup:)

Danu

I am painfully aware of that fact

...and of the fact that my undergraduate degree only offered half a course on it

where we didn't get any further than basic properties of matrices.

bd1251252

@TedShifrin That too...I've been trying to not sound arrogant as that's certainly not my intention. Arrogance doesn't get you anywhere. But I suppose being a wimp doesn't either. (you can tell I'm that awkward unsocial person)

Danu

not even close to forms

Ted Shifrin

Here's one of my all-time favorites, @Danu, and it's physical :) The state space of a car may be taken (locally) to be $\R2
\times S^1\times S^1$, with coordinates $(x,y,\t,\vp)$: let $(x,y)$ be
the coordinates of the midpoint of the front axle, let $\t$ be the angle the
car makes with the positive $x$-axis, and let $\vp$ be the angle the
front wheels are turned.

%\vskip1in\hskip1.6in\special{illustration car.eps scaled 500}

\noindent
Consider the vector fields
\begin{align}
S &= \text{Steer } = \d{}{\vp} \qquad\text{and} \\

Danu

11:30 PM

@TedShifrin This is a little hard to read :P

user159870

$\phi: \mathbb{R}[x] \to \mathbb{C}$ with $\phi(x)=i$ and $\phi(z)=z, \forall z \in \mathbb{R}$.

Let a polynomial $p(x) \in \mathbb{R}[x]$. We apply the euclidean division of $p$ with $x^2+1$ and we have $p(x)=(x^2+1)g(x)+h(x)$ with $deg(h(x)) \leq 1$.

Then do we apply then the euclidean division for $h(x)$? with what? @TedShifrin

Ted Shifrin

Sorry, @Danu. Grr. Not enough time to edit the pasted text. I'll edit separately and paste.

No, @user159870, you use $h(x)$ to figure out what $\phi(p(x))$ should be.

user159870

$\phi(p(x))=0 \Rightarrow \phi(h(x))=0$

How we do go on from here? @TedShifrin

Danu

lol

I think I can read it

Ted Shifrin

The state space of a car may be taken (locally) to be $\Bbb R^2
\times S^1\times S^1$, with coordinates $(x,y,\theta,\phi)$: let $(x,y)$ be
the coordinates of the midpoint of the front axle, let $\theta$ be the angle the
car makes with the positive $x$-axis, and let $\phi$ be the angle the
front wheels are turned. Consider the vector fields
$S = \text{Steer } = \partial/\partial\phi$ and
$D = \text{Drive } = \cos(\theta+\phi)\partial/\partial x + \sin(\theta+\phi)\partial\partial y + \sin\phi\partial/\partial\theta$.

bd1251252

11:38 PM

Oh you use WebWork as well?

Ted Shifrin

Darn, still not right.

OK, @Danu. That was too stressful. :)

@bd1251252: Yes, I coded several hundred problems of my own for the class.

They also have written homework each week that's all proofs.

bd1251252

@TedShifrin Gosh...what are the prerequisites for your course?

Ted Shifrin

There are freshmen with 5 on the BC AP exam, and then some sophomores. They have to be super-motivated, @bd1251252. There are courses like this at places like Vanderbilt, Stanford, Yale, Harvard, etc.

Danu

Differential geometry (or "manifolds") in the first year?

Ted Shifrin

No, no, @Danu. He's talking about a different course!

Danu

11:41 PM

Heh, okay. I was starting to feel even more useless

bd1251252

@TedShifrin Okay...I guess that means I'm gonna have to study and study and study until I get the chance to do something like this. Thank you for the motivation... it may not have seemed like it, but this is exactly the kind of course I've always wanted to take

Ted Shifrin

No, the manifolds course is a first- or second-year graduate course, although I've had excellent undergrads take it.

@bd1251252: Most universities don't offer this course, but you should be able to take a serious real analysis and multivariable real analysis course senior year, so aim for that. But if you work through these lectures, you'll be way far ahead.

Danu

@TedShifrin I think the Lie bracket computation couldn't be too hard (the kind of stuff we have to do all the time in physics), but I'm not sure how one should use it to wiggle out of parking spots. I feel like I'm really missing that picture you had in there

bd1251252

@TedShifrin I'll save the links and watch them over the next few weeks

Danu

the $\theta$ angle would be the incline of a hill?

or should the wiggling become crystal clear once I actually did the computation?

Ted Shifrin

11:44 PM

It's a cool problem. I stole it. No, it's all in the plane. $\theta$ is the angle from the $x$-axis to the car (the axle from front to rear), and $\phi$ is the angle you're steering relative to that.

Danu

Ah, like that

Ted Shifrin

OK, @bd1251252. I hope you continue to be inspired to excel :P

Danu

I shouldve known from the cos and sin

classical mechanics 101 :P

Ted Shifrin

@bd1251252: I can email you exams if you want to see how you're doing.

Yup @Danu.

bd1251252

@TedShifrin Would you really?

Danu

11:46 PM

You seem to be a very nice and helpful person, Ted. Props to you :)

Ted Shifrin

Sure.

Thanks, @Danu. Well, after I retire this semester, I'll either disappear or have more time to be nice and helpful :P

Danu

On that note, what would your recommendations be for someone like me who is getting into mathematics and is in some trouble because of a lack of previous formal mathematics education?

In diffgeo, for instance, I really had a hard time following the more topologically oriented parts of the course

...I hadn't seen the formal definition of a continuous function at all before last July

Ted Shifrin

Yeah, you need basic point-set topology and multivariable real analysis (inverse function theorem) for sure.

Danu

I feel like it's mostly linear algebra and topology that I'm lacking

Topology I'll let slide for now since there is a supposedly excellent course on it that I will be able to take next year

Ted Shifrin

Yikes. Well, in all seriousness, my multivariable book/course that @bd1251252 is looking at would help, @Danu. I do the derivative as a linear map, inverse function theorem, basic open/closed/compact, etc., plus linear algebra in that course.

Danu

11:49 PM

Linear algebra, not so much. I don't think they can pass that off for graduate :P

@TedShifrin Hmkay

I'll have a look around, see if Ican get my hands on a copy

bd1251252

@TedShifrin Would you like me to email the email you have provided on your web page or post it here?

Danu

I've caught up to the basics by now

It's just that the foundations are all wobbly ;)

Ted Shifrin

No, @bd1251252, email me when you're ready :P

Danu

(I can use the lingo, just don't press me too hard on the defintions ;D)

Ted Shifrin

Yeah, @Danu, that's very hard. The graduate manifolds course puts a lot of sophisticated stuff together.

And you definitely need to be used to a certain amount of topology, for sure.

I wish you good luck. If I can help (within reason), let me know.

bd1251252

11:51 PM

@TedShifrin Okay, I'll email you in a few weeks...

Danu

@TedShifrin I'm able to cope, and a book like Lee's on smooth manifolds is not beyond me. Without his appendix on topology though, I wouldn't have made it very far in that book. But that's not really a good replacement for an actual course/book

Ted Shifrin

Lee's book is very well respected. I don't know it personally. However, he took a graduate course from me when he was a grad student, so I've known him for years :P

Danu

Any recommendations for books on topology? Do you recommend Munkres?

Lee's book is amazing. Please tell him he's awesome next time you see him.

Ted Shifrin

Munkres is the standard, yes, although you don't need all the esoteric counterexamples for your purposes.

Danu

...also whether he'd be kind enough to answer this:

13

Q: The history of different constructions of tangent spaces

In Lee's book 'Introduction to Smooth Manifolds', there is an interesting discussion (near the end of chapter three) of several different ways of viewing/constructing the notion of a tangent space to a point on a manifold. As it turns out, there are quite a few. Tangent vectors can be understood ...

...or maybe you could do that yourself! :)

Ted Shifrin

11:54 PM

Lee is very responsive to questions on MathStackExchange. (He and I often correct each other. :) ) I don't know all the history, but I would think that thinking of tangent spaces in terms of tangent vectors to curves might be a plausible starting point. Before the tensorial view.

Danu

@TedShifrin Before? Interesting. I thought Lee's remark in his book seemed to imply the opposite

Ted Shifrin

Well, I haven't studied the history. I presume he has.

OK, dinner time. It's been a pleasure, @Danu, @bd1251252. Have fun learning:)

Danu

It really has. Hope to see you around

I guess it's bedtime for me (1 AM)

Ted Shifrin

Grad students never sleep :D

Danu

:(