Mathematics - 2018-05-12 (page 1 of 3)

0celo7

12:00 AM

hi @TedShifrin

Ted Shifrin

Doing ok, skull, thanks — been trying to plan some travel later this summer.

rehi 0celo.

0celo7

so suppose we have some inner product $\langle -,-\rangle$ on $\mathfrak g$ (the Lie algebra)

this extends to a left invariant metric on $G$ in a natural way. Do you see it?

write down $h_b(x,y)$ coming from the inner product

Daminark

$h_b(x,y) = \langle d(L_b)^{-1}x, d(L_b)^{-1}y\rangle$?

0celo7

exactly

you know that the inverse exists there because $L_b$ is a diffeomorphism

So this gives you a rich class of metrics on any Lie group

Ted Shifrin

Better yet, use $L_{b^{-1}}$.

0celo7

12:03 AM

Or that

Daminark

Ah true

0celo7

Ok so the story proceeds by noting that given any vector field $X$ and a Riemannian metric $g$, $g(X,-)$ defines a 1-form

Daminark

Hmm, let me be sure of this

MatheinBoulomenos

@TedShifrin how do you prove with GAGA that Zariski-connected varieties over $\Bbb C$ have a connected analytification? I guess the idea is that if your space is disconnected in either category, every sheaf decomposes globally as a direct sum

Daminark

So a 1-form right now is taking in a vector field and spitting out a smooth function?

0celo7

12:06 AM

yes

Ted Shifrin

Agh, I dunno, @Mathein.

Daminark

Okay let me see this in $\mathbb{R}^n$

0celo7

urge to use index notation intensifies

Ted Shifrin

Nothing wrong with using coordinates to understand things. :)

But sometimes they obscure things.

Daminark

So, if you have a 1-form $\sum_{i=1}^n a_idx_i$, then it should act on $f:\mathbb{R}^n\to\mathbb{R}^n$ and spit out a function $g:\mathbb{R}^n\to\mathbb{R}$

MatheinBoulomenos

12:07 AM

okay, dumb question, are locally constant functions a coherent sheaf?

0celo7

g is probs a bad notation here

Daminark

Oh yeah earlier you used $g$ for the metric, let's say $r$

Ted Shifrin

of $\mathscr O_X$-modules?

0celo7

i used $h$ for a metric as well :D

not enough Latin letters for Lie groups

MatheinBoulomenos

yeah

probably not

Ted Shifrin

12:08 AM

Nope.

0celo7

@Daminark So more abstractly, a 1-form is a smoothly varying object in Hom(tangent space, R)

more abstractly but also handwavey

So you have a vector field $X$ and consider the 1-form defined by $Y\mapsto g(X,Y)$

This is pointwise a linear functional on tangent spaces, so is a 1-form

Daminark

Wait so maybe I'm being a bit of an idiot here, but my image of a differential 1-form is something which takes in a point and spits out a linear functional. I'm not completely clear on why this is the same sort of object as something which takes in a vector field and spits out a smooth function

Oh oh

Okay that makes sense, thanks!

Ted Shifrin

Of course, you have a backwards resolution $0\to\Bbb C\to \Omega^0\to\Omega^1\to \dots$, @Mathein.

0celo7

Ok, so the usual notation is $X^\flat=g(X,-)$

Daminark

You weren't lying when you said there aren't enough letters to go around

0celo7

12:12 AM

Now you have a frame $e_1,\dotsc, e_n$ of $G$ coming from the parallelization

Ted Shifrin

That's why Cyrillic beats English.

Daminark

Actually GMT uses subscript # to denote push forward and I'm like 0_0

Ted Shifrin

That's used in algebraic topology in general.

0celo7

Probably because * is already overused

Ted Shifrin

That's different from the geometers' sharps and flats.

I think $*$ is reserved for homology, not for chains.

Not that I berember.

MatheinBoulomenos

12:13 AM

@TedShifrin maybe we can just use $\mathscr O_X$ itself? At least in the affine case, $X$ is connected iff the ring of global sections (aka coordinate ring) is indecomposable

0celo7

I have $\#$ denoting pullback and dual in the thesis

should cause no confusion...

@Daminark Ok, some linear algebra for you. Consider a frame $e_1,\dotsc, e_n$ of $TM$. Show that these can be made orthonormal in the sense that $g(e_i,e_j)=\delta_{ij}$ everywhere

Ted Shifrin

But global sections in the projective case is constants, but that's still gonna be OK. Of course, projectively, it's hard to have a disconnected variety.

# for pullback? Say what?

MatheinBoulomenos

I'd be fine with the affine case, too

0celo7

*pushforward

Daminark

So we're basically trying to do a "smooth Gram-Schmidt"?

0celo7

12:15 AM

yeah

you just have to check the usual GS formula is smooth

Ted Shifrin

But it is smooth.

Yup.

That's smoothly deformation retracting $GL(n)$ to $O(n)$.

0celo7

So now assume we have an everywhere defined orthonormal frame of $TG$

Now define the 1-forms $\omega^i=e_i^\flat$

And finally consider $\Omega=\omega^1\wedge\cdots\wedge\omega^n$

That's your left-invariant volume form.

It's also the Riemannian volume form of $g$, but you probably don't know what that is yet.

MatheinBoulomenos

@Ted are connected components some $0$th sheaf cohomology (of a coherent sheaf)? GAGA also gives isomorphism of sheaf cohomology groups

Daminark

Okay, I see what you're doing here. I'll need some time to absorb this completely properly but I think I get the general idea!

Yeah I dunno that

0celo7

maybe modulo an n! or something stupid, but you get the point

MatheinBoulomenos

12:17 AM

ah lol, $0$-th sheaf cohomology are just global sections

but of course, the set of connected components won't be a module over the global sections of $\mathcal{O}_X$

Ted Shifrin

Hmm, I don't see that.

Well, it is in the projective case :)

MatheinBoulomenos

so projective is actually easier than affine

Daminark

Was that a snipe? I tabbed out for a second

0celo7

@Daminark This orientation business is in Bott and Tu, which you should probably read

Ted Shifrin

@0celo, no factorials unless you're stooopid and use the G&P conventions.

Bott & Tu is way beyond where Demonark is for now in topology.

But it's a great book.

0celo7

12:20 AM

I think he can read the first half

Ted Shifrin

But he hasn't learned homology or cohomology yet, so really no.

0celo7

oh

Ted Shifrin

He just has fancy abstract shit.

0celo7

I thought he did Dold-Thom with May

MatheinBoulomenos

$H^n(X,G)=[X,K(n,G)]$

Ted Shifrin

12:21 AM

LOL

MatheinBoulomenos

that's the definition, right?

Ted Shifrin

Yup.

MatheinBoulomenos

I think May did that

0celo7

Clearly $H^k$ is the number of harmonic $k$-forms

Ted Shifrin

Number?

Yeah, clearly.

0celo7

12:22 AM

linearly independent ones

MatheinBoulomenos

but of course Dold-Thom also gives a handy definition of cohomology

because everyone knows homotopy groups of some colimit of quotients are easy to compute and intuitive

Daminark

Yeah Dold-Thom is how I established Eilenberg-Maclane spaces, modulo I don't really remember too well how the construction of Moore spaces went (didn't really get it completely even then)

Ted Shifrin

as I said, Demonark needs to take an actual alg top class to learn what things are.

skull

When does your AoPS course end, professor?

0celo7

I think he should learn homology from Federer

MatheinBoulomenos

12:24 AM

my favourite way to show the existence of Eilenberg-Maclane space is via the Dold-Kan correspondence

Ted Shifrin

Yeah, that would be perfect, 0celo.

mid-June, skull.

skull

cool

Ted Shifrin

I hope they rethink stuff seriously.

I've bitched a lot. Imagine that.

Daminark

Lol, next fall I'm gonna take algebraic topology and finally put this saga to rest

MatheinBoulomenos

The Dold-Kan correspondence says that chain complexes of abelian groups are equivalent to simplicial abelian groups and the correspondence works in such a way that the homology groups of your chain complex are isomorphic to the homotopy groups of the geometric realization of the simplicial abelian group (just considered as a simplicial set)

Ted Shifrin

12:27 AM

Gesundheit! And then I sign off.

skull

Cya, professor

Daminark

Though there's a chance the class is gonna assume fundamental group and covering space theory so I'm reading (well... I'm saying I should read, though with classes time is not easy to find) that stuff on the side

Ted Shifrin

You have summer, Demonark.

MatheinBoulomenos

@Daminark so you did (quasi)fibrations without doing coverings first?

Daminark

Yeah

Ted Shifrin

12:28 AM

*rolls $16 + \pi/4$ eyes.

MatheinBoulomenos

and homotopy groups without talking a bit about $\pi_1$ in particular

Ted Shifrin

(Won't roll any further.)

Daminark

I mean I know what $\pi_1$ is, I had went through that a long time ago

0celo7

@Daminark Ah, do the Almgren and study homology via the homotopy theory of current groups

skull

@TedShifrin How irrational

Daminark

12:29 AM

I knew a covering space as a fiber bundle with discrete fiber, and even then my knowledge of fiber bundles is... iffy, at best

0celo7

lol

Ted Shifrin

But there's nothing wrong with the way UC teaches mathematics.

Daminark

Basically my AT at this point is just a bunch of bits and pieces kinda floating around.

Ted Shifrin

pages @EricSilva

0celo7

@TedShifrin Do the most general thing possible, then you only have to do proofs once!

Ted Shifrin

12:30 AM

yeah, glad that was never my teaching philosophy.

Quite the opposite.

Daminark

@Ted keep in mind that I've yet to take any topology-related classes here, this isn't the fault of classes so much as my getting excited about the topology from the REU

MatheinBoulomenos

fiber bundle with discrete fiber is not as terrible as it looks. it's just a rather direct reformulation of the usual definition

0celo7

@MatheinBoulomenos We know

Ted Shifrin

Interestingly, I just got a message second-hand from a guy who used my algebra book and said he'd gotten further and his students had understood more than when he'd taught out of one of the standard texts.

Daminark

I mean I guess I've taken difftop, much as that was technically not supposed to be a diff top class

ÍgjøgnumMeg

12:31 AM

Anyone got a favourite "first course" in topology that they can share with me? In PDF format or smth, looking to learn some topology over the summer before my masters lol

MatheinBoulomenos

When I asked for an algebra book as a freshman, I was recommended Lang

Ted Shifrin

Getting intuition for intersection numbers and degree will help you in a regular alg. top. class, Demonark.

Of course, Mathein.

0celo7

@ÍgjøgnumMeg why not Munkres

Ted Shifrin

what do you mean by topology, @ÍgjøgnumMeg

point-set or algebraic or geometric or ...

MatheinBoulomenos

@ÍgjøgnumMeg you know German, right? Laures-Szymik "Grundkurs Topologie"

Ted Shifrin

12:33 AM

I don't think Munkres is legally available in PDF, 0celo.

0celo7

inb4 it starts with sheaf cohomology

MatheinBoulomenos

also Jänich for intuition

though these are both regular books, not pdfs

ÍgjøgnumMeg

@MatheinBoulomenos Ich schau's nach, danke!

MatheinBoulomenos

@0celo7 there's a chapter on sheaves, actually

ÍgjøgnumMeg

@TedShifrin I think point-set

0celo7

12:33 AM

@TedShifrin I won't recommend textbook piracy in the same room as a textbook writer :P

Ted Shifrin

Ich hab' nichts mehr zu sagen.

Good. 0celo.

ÍgjøgnumMeg

@TedShifrin I.e. the stuff that comes before algebraic topology I suppose

MatheinBoulomenos

I just borrow stuff from the library

0celo7

@ÍgjøgnumMeg I really liked Lee's intro to top manifolds

MatheinBoulomenos

I also like to buy books

Ted Shifrin

12:34 AM

Well, it's needed for analysis and other things, too, @ÍgjøgnumMeg.

That's too advanced if you don't know point-set, 0celo.

I used to, too, @Mathein. I had zillions.

0celo7

@ÍgjøgnumMeg are you very comfortable with metric spaces from analysis?

Daminark

Lol I mostly go for library/free stuff online since I do not have much money to be throwing at textbooks

MatheinBoulomenos

But Lee's top manifolds book is written for people who don't know point set @Ted

ÍgjøgnumMeg

Nooope, I've basically spent the last year getting comfortable with algebraic number theory and algebra, so my analysis is essentially a first course in real analysis and a first course in complex analysis

Ted Shifrin

But why learn manifolds before you know what compact and connected mean?

That book is huge.

0celo7

12:35 AM

it's like 450 pages

Ted Shifrin

shrugs

MatheinBoulomenos

Lee introduces compact and connected in the first chapters

Daminark

We have this springer deal that gives some free pdfs so that's always nice

MatheinBoulomenos

(but he doesn't prove Tychonoff)

Ted Shifrin

I never proved Tychonoff in all the times I taught point-set.

0celo7

12:36 AM

It's not a prep for functional analysis, that's for sure

Ted Shifrin

Munkres did prove it when I took the course.

0celo7

He doesn't do Tietze either

@TedShifrin Morwen taught us Tychonoff

MatheinBoulomenos

there are some really unintuitive proofs, e.g. the one in Bourbaki which actually just uses the AoC directly not even Zorn and then if you do ultrafilters it's trivial once you set up all the machinery

@ÍgjøgnumMeg Laures-Szymik has the Zariski topology on $\operatorname{Spec}(R)$ as an example in an exercise, you'd like that

0celo7

The nice thing about Lee is that he has a whole chapter on constructions like products, gluing, quotients

Also surfaces are beat to death

Ted Shifrin

I've known Lee since his grad student days. He is a good expositor. I just don't agree with his choices in all his books, and I don't know them all that well.

ÍgjøgnumMeg

12:39 AM

@MatheinBoulomenos I shall have a look, there probably isn't a German copy in my library but if there's an English translation that might do!

MatheinBoulomenos

dunno if it's translated

0celo7

@ÍgjøgnumMeg Imo a good way to test what you know is to look at Appendix B of Lee's smooth manifolds and try to do all of the exercises.

MatheinBoulomenos

most of the top manifolds book by Lee is just a point-set topology book where most examples are manifolds. I like how he stresses the universal properties for all constructions, e.g. quotients, subspaces, products etc.

0celo7

It covers most of what you'd get in a first course in America

MatheinBoulomenos

later he does some fundamental groups and coverings and classification of surfaces iirc

ÍgjøgnumMeg

12:41 AM

Fair enough! Thanks for your suggestions

MatheinBoulomenos

and some homology, was it simplicial? not sure

0celo7

Yeah and homology, incl. Mayer-Vietoris

ÍgjøgnumMeg

@MatheinBoulomenos Jänich is in the library, that was one of your suggestions right?

MatheinBoulomenos

yes

0celo7

One of the most German names

MatheinBoulomenos

12:42 AM

don't use Jänich alone, you need a normal textbook for details

but it's super easy to read and gives a lot of intuition

Ted Shifrin

"normal textbook" ha!

skull

he likes classics

Ted Shifrin

I rather do, too ... if my own books count :P

MatheinBoulomenos

Jänich has no exercises and few proofs

but it's great

skull

sure they count

0celo7

12:44 AM

does Lang have a topology book

Ted Shifrin

To me (I've argued about this a lot), exercises are the most important part of a math text (except for graduate topics level).

No, 0celo.

But I like his real analysis.

0celo7

We used Fleming for analysis, it was pretty bad

MatheinBoulomenos

there's a standard subject on which Lang didn't write a book? seems unlikely

0celo7

I just read Jost instead

Ted Shifrin

That was the textbook for my second semester analysis back in 1972, also.

skull

12:45 AM

did you use your books to lecture from?

Ted Shifrin

It has its good points, but I would never use it as a text.

Yes, skull, but I didn't follow slavishly.

0celo7

The stuff on convexity is good, but that's about all I can say that was good

Ted Shifrin

It was innovative, trying to include multivariable and Lebesgue.

MatheinBoulomenos

we used the incredibly dense and idiosyncratic notes of my professor

0celo7

Ok, that's historical context lost on me in 2016 :P

Ted Shifrin

12:46 AM

Ironically, the one B I got in undergraduate was in that Fleming course. Most of that course turned out to be the beginnings of my field.

skull

a lot do that @MatheinBoulomenos

0celo7

My advisor used Dioudonne, I think

He took the exercises for our course from there

Ted Shifrin

It was also taught by one of the worst teachers I had ... a postdoc whose specialty was algebraic geometry.

Daminark

Hmm, I wonder if there are any people out there who wrote a book and maybe some time after didn't like it anymore

Ted Shifrin

Dieudonné is way too terse, but an amazing compendium.

I would rewrite my algebra book substantially, Demonark, but not the others.

0celo7

12:47 AM

@Daminark Hawking didn't like his one technical book

He thought it was too technical

It's the foundation of all of mathematical GR now

skull

Wow!

Daminark

Hmm, would you still lecture out of it now if you had to teach an algebra class? Also what would you change, if I may ask?

@0celo7 interesting

Ted Shifrin

(Demonark: The linear algebra book already was substantially revised, and I've continually revised the diff geo notes ...)

Yeah, I still taught out of it, but I reorganized and changed a lot of specifics and exercises, Demonark.

0celo7

It's sad because it's certainly one of the most profound books of the century

Ted Shifrin

Part of that was due to UGA's switching from quarters to semesters, tbh.

0celo7

12:49 AM

Black holes, big bang, expanding universe...that book laid the foundations

skull

RIP

MatheinBoulomenos

I should give you some quotes from Jänich, he's hilarious sometimes

0celo7

@Daminark Steven Weinberg wrote a classic GR book in the 1970s where he argued constantly that there was no geometry, and the metric was just another physical field consisting of particles

Sylent Nyte

Hey guys, whats the best way (computationally) to find out if two lines in a 3D space intersect?

0celo7

He has since called that an error of youth or something to that effect

Ted Shifrin

12:52 AM

@Sylent: Good question. How are you given the lines?

Sylent Nyte

@TedShifrin randomly generated

Ted Shifrin

parametrically?

If they're random, they're not gonna intersect.

Sylent Nyte

as in, the line is in vector form, with a position vector and direction vectors, but the actual position and direction are going to be random (fixed* throughout, but initialized randomly), if that makes sense

skull

@0celo7 have you looked at his Ph.D. Thesis?

Ted Shifrin

Right, so it's a set of measure 0 for the lines to intersect, so if they're random, it's probability 0 that they intersect.

0celo7

12:54 AM

I have but I don't remember what I saw

skull

ok

0celo7

Presumably it's where he proves his singularity theorem

Ted Shifrin

However, given actual lines, you should compute the distance. Pick $a_i$ in $\ell_i$, and compute $(a_1-a_2)\cdot (v_1\times v_2)$, where $v_i$ are direction vectors.

Sylent Nyte

sorry, my explanations on things are bad

Ted Shifrin

I answered it.

0celo7

12:56 AM

the equations are handwritten

yikes

ahhh, Kerr-Newman

skull

yup

0celo7

skull

I love how the demand for it crashed the servers.

0celo7

er, Newman-Penrose

Sylent Nyte

@TedShifrin how would computing the distance and the dot product help me?

0celo7

12:59 AM

it's also in the book

Ted Shifrin

If that formula I gave you comes out 0, they intersect. (And conversely.)

0celo7

Ted Shifrin

Some of the pages in my adviser's papers were amazingly formidable, 0celo.

MatheinBoulomenos

Zögernd nur nenne ich einen weiteren topologischen Begriff: Parakompaktheit.
Es gibt allzu viele solcher Begriffe! Ein A heißt B,
wenn es zu jedem C ein D gibt, so daß E gilt — das ist zunächst
einmal langweilig und bleibt es auch solange, bis wir einen Sinn
dahinter sehen können, " bis uns ein Geist aus diesen Chiffren
spricht". Wenn einer eine erste uninteressante Eigenschaft und
eine zweite uninteressante Eigenschaft definiert, nur um zu sagen,
daß aus der ersten uninteressanten Eigenschaft die zweite uninteressante

Wem allzu oft zugemutet wurde, Vorbereitungen zu unbekannten
Zwecken interessant zu finden, dem erkaltet schließlich
der Wunsch, diese Zwecke überhaupt noch kennenlernen zu wollen,
und ich fürchte, es verläßt manch einer die Universität, der
das eigentliche Zentralfeuer der Mathematik nirgends hat glühen
sehen und der nun sein Leben hindurch alle Berichte davon für
Märchen und das "Interesse" an der Mathematik für eine augenzwinkernd
getroffene Konvention hält. — Aber ich komme wohl
zu weit von meinem Thema ab.

Ted Shifrin

@Mathein: No paracompactness today.

Sylent Nyte

1:00 AM

@TedShifrin this will work for 3 dimensions also?

MatheinBoulomenos

@TedShifrin read the text

this is hilarious

Ted Shifrin

It was meant for 3D, @Sylent.

MatheinBoulomenos

that's the introduction to the section on paracompactness

Sylent Nyte

@TedShifrin ah! thank you! you saved my night sleep

Ted Shifrin

One paragraph suffices, @Mathein.

@Sylent: You owe me.

MatheinBoulomenos

1:02 AM

there's an English translation of Jänich's topology book I think

If you someone here has access via springer, look up the section on paracompatness

it's really great

just some random general rant

Ted Shifrin

And the reviewer of my algebra book complained when I had a word or two of humo(u)r.

Unbelievable.

skull

Must have been a prude.

Ted Shifrin

LOL ... I think I knew who it was, but I've forgotten and long since thrown things out

But yeah ... humor helps students engage.

skull

and remember

0celo7

Wonder what Federer's sense of humor was

Ted Shifrin

1:05 AM

skull, you need to learn stuff out of my books :P

0celo, (almost?) non-existent. The lecture series I saw him give at an AMS meeting was excruciating.

Sylent Nyte

@TedShifrin does the equation work if the position vector and direction vector of a line is 0..? confusing but its used to represent the path of particles

Ted Shifrin

You can't have a 0 direction vector, @Sylent. Then that's not a line.

skull

0 is special

Ted Shifrin

Direction vectors are nonzero by definition.

Sylent Nyte

yeah i know.. i even wrote an exception for it but i deleted it like 5 minutes ago, shouldnt do this tired

0celo7

1:07 AM

@TedShifrin there don't seem to be videos of him

Sylent Nyte

facepalm

Ted Shifrin

@0celo, thankfully :P

That AMS meeting was late 1970's ... were people filming then?

0celo7

by necessity I read the book alot... I want to know more about the man begind the madness

Ted Shifrin

I sat through the lecture series wishing Lawson were giving the lectures.

skull

@SylentNyte get some sleep and try again

Sylent Nyte

1:09 AM

@skull easier said then done, id just be thinking about the problem throughout the night not getting any sleep XD

Ted Shifrin

You can send me a reward later, @Sylent.

0celo7

@TedShifrin I've been trying to read his book and it seems harder!

MatheinBoulomenos

I'll translate the beginning maybe. "Only hesitantly I mention another topological notion: paracompactness. There too much such notions! An A is called B if for every C, there is a D such that E holds - that's boring at first and it stays that way until we can see a meaning behind it [...].

0celo7

That might be because I don't know K theory already

Ted Shifrin

depends which book ... you mean Spin Geometry?

0celo7

1:12 AM

yes

MatheinBoulomenos

If someone defines a first uninteresting property and a second uninteresting property only to say that the second uninteresting property follows from the first uninteresting property, but that there is an uninteresting example that has the second uninteresting property and doesn't have the first uninteresting property: One just wants to go crazy over something like that." [not sure how to translate that sentence, literally doesn't seem to work]

Ted Shifrin

It's challenging. Don't know what parts he wrote.

0celo7

The stuff on elliptic operators, indices, and scalar curvature I understand (of course)

Ted Shifrin

@Mathein: I worked as a translator one summer. Maybe you shouldn't. :)

0celo7

But the K-theory and characteristic classes...ugh

MatheinBoulomenos

1:13 AM

@TedShifrin people generally translate into their native language

I can translate English -> German fine

Ted Shifrin

LOL ... I did some technical translation into French, but you're right.

Ordinarily, your English is superb.

0celo7

Like trying to understand Milnor's $\hat{\mathcal A}$ invariant...

Ted Shifrin

I no longer have the book, 0celo, so I can't discuss rationally.

MatheinBoulomenos

Really? I was thinking that my English is quite bad sometimes

Ted Shifrin

No, it's really quite good.

MatheinBoulomenos

1:14 AM

thanks

0celo7

@TedShifrin Oh, I'm just rambling. If I had precise questions I would ask Mike.

Ted Shifrin

There are papers and other books, 0celo.

The courses I took from him (5 before he left Berkeley for StonyBrook) were superb.

0celo7

Probably. My advisor would tell me to read Milnor-Stasheff and Karoubi first

Ted Shifrin

I do not like Milnor-Stasheff at all. I don't know Karoubi.

0celo7

Everyone seems to love MS

Ted Shifrin

1:16 AM

Um, no, not everyone.

0celo7

Maybe it's by necessity

MatheinBoulomenos

I think the leap even from grad textbooks to papers is quite big. This semester I'm taking courses for the first time where there are no textbooks on the subject at all

Ted Shifrin

I've certainly taught courses where there weren't textbooks.

0celo7

I don't know if Lawson-Michelson is considered a textbook

Ted Shifrin

Sure.

For a second-year graduate course.

No exercises, of course.

MatheinBoulomenos

1:18 AM

For one course, a primary source are the original papers

Ted Shifrin

Lawson's CBMS lectures on gauge theory were superb. Check those out.

MatheinBoulomenos

and some unpolished French lecture notes

0celo7

Thanks, I will

For reference, I'm trying to understand Gromov-Lawson

Ted Shifrin

Most of the Harvey-Lawson stuff was well-written. It depends on co-authorship.

OK, going to cook dinner.

0celo7

bye

MatheinBoulomenos

1:20 AM

Guten Appetit!

Daminark

See you Ted!

0celo7

feddy...5.4.15, 5.2.19, 5.3.18, 5.4.6, 5.4.7 in two sentences

skull

cya

Sylent Nyte

@TedShifrin I have a contradiction

@TedShifrin [2,2,2] + t[2,2,2] and [2,2,1] + t[2,2,0]. So a_1 = [2,2,2] and a_2 = [2,2,1] => a_1 - a_2 = [0,0,1]. v_1 = [2,2,2] and v_2 = [2,2,0] => v_1 * v_2 = [-4,4,0]. [0,0,1] dot [-4,4,0] = 0 however they do not intersect

also i cant get mathjax to work on my pc so i cant use the scri[t

skull

math.ucla.edu/~robjohn/math/mathjax.html

8

A: How can I enable MathJax in chat?

Update 2017-05-01 The MathJax CDN retired and the javascript-URL idea is not so easy any more, because of browser security. (Chrome stips away any leading javascript: when pasting into the URL line. SE modified the javascript: link so that it does not work.) So here is my take. I modified the ...

Sylent Nyte

1:34 AM

but bookmark bars are ugly

skull

suit yourself :-)

Ted Shifrin

@Sylent: Learn cross product ...

Wow.

Harry Evans

2:18 AM

Good evening! I have a quick question and I need a hint

Let $X$ denote the vector space of continuous functions, $\varphi : [0,1] \rightarrow \mathbb{R}$ such that $\varphi (0) = 0.$ On $X$, consider the norm defined by
$$\left\Vert \varphi \right\Vert = \sup _{t \in [0,1]} \left| \varphi (t) \right|.$$
Define $T: X \rightarrow \mathbb{R}$ by
$$T \varphi = \int_0 ^ 1 \varphi(t) dt.$$

Show that there exists no $\varphi \in X$ such that $\left\Vert \varphi \right\Vert = 1$ and $T \varphi = 1$.

This is how I started it: Suppose not. Then, there exists $\varphi \in X$ such that $\left\Vert \varphi \right\Vert = 1$ and $T \varphi = 1.$ Let $f \in X$ such that $f'(t) = \varphi (t).$ Therefore
$$T\varphi = \int_0^1 \varphi (t) dt = f(1) - f(0).$$

My trouble is finding a contradiction. I think this would mean that $\varphi$ would not be continuous on $[0,1]$ but I'm not too sure

Daminark

I wouldn't quite approach it that way. So, since $\varphi$ is continuous and $\varphi(0) = 0$, you can find some $\delta$ such that $x < \delta \implies \varphi(x) < \frac{1}{2}$, right?

Harry Evans

Yes

Daminark

Okay, so you can write $T\varphi = \int_0^{\delta} \varphi(t)dt + \int_{\delta}^1 \varphi(t)dt$

Harry Evans

Yup

Daminark

Sorry I meant to say that $|\varphi(x)| < \frac{1}{2}$. But yeah so, you know that $\int_0^{\delta} \varphi(t)dt \le \int_0^{\delta} |\varphi(t)|dt < \frac{1}{2}\delta$

And then $\int_{\delta}^1 \varphi(t) dt \le \int_{\delta}^1 |\varphi(t)| dt \le 1-\delta$, so $T\varphi < 1 - \frac{\delta}{2}$

Harry Evans

2:28 AM

So $T\varphi \le \frac{1}{2} \delta + \int_{\delta}^1 \varphi (t) dt$

Yeah, I was suppsoed to say that $T\varphi$ will be less than 1

Great!!! Thanks

Daminark

So, I did that proof very formally, what's the idea there? Let's work with positive functions, since if a function is negative that just makes it even harder for the integral to be 1

gian

is the cone on the $n$-sphere always homeomorphic to the $n+1$-disk?

Harry Evans

Amazing, thanks!!!

@Daminark

Daminark

You're trying to say that the integral is less than 1 and the function is bounded in absolute value by 1 (sup norm). Well, you'd be good if you had the constant 1 function, but you don't, since $\varphi(0) = 0$, and you can't exactly "jump" immediately to 1 by continuity

Harry Evans

I was thinking of a function that would be discontinuous and that would be the contradiction

Daminark

2:31 AM

@gian hmm, so $CX = (X\times I)/(X\times \{1\})$, right?

gian

Yeah

It's definitely true for the 0-sphere and 1-shpere

Daminark

Hmm, so I feel like this is true and you'd prove that by basically taking the points of the sphere, scaling them down to 0 (this is the map out of $S^{n-1}\times I$), and then quotienting stuff out appropriately

Like, $f:S^{n-1}\times I \to D^n$ by $f(x,t) = (1-t)x$

gian

Thanks :)'

Daminark

No problem!

Joshua

I just noticed something very interesting. The Axiom of Finite Choice is provable in ZF. The Axiom of Choice is not. This gives the possibility of some kinds of supertasks behaving differently than their limits in finite math. I can prove that any such case is not in first order.

Ted Shifrin

3:13 AM

@gian: Draw the cone with base the sphere, and then notice you can collapse it down to the ball.

Harry Evans

3:53 AM

I have one last question

Let $C([0,1])$ denote the vector space of continuous functions $u : [0,1] \rightarrow \mathbb{R}$. On $C([0,1])$, consider the norm defined by
$$\left\Vert \varphi \right\Vert = \sup _{t \in [0,1]} \left| \varphi(t) \right| \quad (\forall \varphi \in C([0,1]).$$
Set
$$C := \bigg \{ u \in C([0,1]) : \int_0^1 \left| u(t) \right| ^2 dt < 1 \bigg \}$$

Show that $C$ is an open and convex subset of $C([0,1]).$

Is it correct if I use the definition of convexity here?

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