Mathematics - 2016-03-02 (page 1 of 3)

Integral

00:10

@GeorgeSmyridis You mean $\lim_{x\to 0} (d/2)^2 + x^2 = (d/2)^2$?

user19405892

00:21

hi

@AkivaWeinberger Oh you are taking the limit?

@AkivaWeinberger Precisely

Here is perhaps a better wording

A triangle $T$ is given in the plane with sides $300,400,500$. A second triangle $r_l(T)$ is created by reflecting $T$ about a line $l$. Find $\min\{[r_l(T) \cup T]\}$ where $[\cdot]$ denotes area.

user19405892

00:47

@AkivaWeinberger What do you mean by opposite orientation?

Ted Shifrin

@user19405892: Draw the triangle on a piece of paper and turn the piece of paper over.

user19405892

yes there are many possibilities right?

Ted Shifrin

I'm not sure what's going on — just answered your question :)

Akiva Weinberger

@user19405892 You can't get from $T$ to $r_l(T)$ just by translations and rotations

but you can using reflections.

Ted Shifrin

heya DogAteMy :0

Akiva Weinberger

00:52

That's what I mean by "opposite orientation" — you can only get from one to the other by (an odd number of) reflections.

user19405892

Right, so in a sense my question is asking what is the line closest to a line of symmetry for a scalene triangle

Akiva Weinberger

Hello!

Ted Shifrin

Closest in what metric?

user19405892

in just 2 d space

Akiva Weinberger

@TedShifrin I don't think s/he was trying to be precise there

Ted Shifrin

00:53

No, you're talking about some notion of "best line of symmetry" — how do you decide what "best" means?

user19405892

@TedShifrin Hmm, good point I was thinking the least possible union of areas

Akiva Weinberger

"Best" means "the one that minimizes the area of $r_l(T)\cup T$ :P

Ted Shifrin

ah, so greatest possible overlap.

user19405892

yes

Ted Shifrin

And the minimum does exist by a compactness argument.

Interesting question. Don't know how computable this is in general.

user19405892

00:55

@TedShifrin Me neither i didn't know how hard or easy it was going to be

Akiva Weinberger

So you don't know the answer

Ted Shifrin

Ah, actually, not so bad. Do you know about the formula for area of a polygon in terms of just its vertices?

user19405892

no what it is

Akiva Weinberger

I think it's called the Shoelace Formula or some such?

user19405892

ah right

Ted Shifrin

00:57

Cool exercise. It's like a piecewise-linear version of Green's Theorem in calculus. It's just a sum of determinants.

You basically put the coordinates of the consecutive vertices into 2x2 determinants and add 'em up.

user19405892

@TedShifrin You mean shoelace theorem?

Ted Shifrin

I don't know the name.

user19405892

do you think it is possible to answer the question using this?

Ted Shifrin

I actually discovered this from Green's Theorem and turned it into an exercise in one of my books ... then discovered it was classical.

Well, starting with the vertices, you can write down a formula for your reflection and compute what it does to the vertices, and then chain them all together (assuming the reflected vertices do not in fact coincide with the original ones).

Akiva Weinberger

You would probably have to do some sort of casework for that, though

Ted Shifrin

00:59

Correct, DogAteMy.

Akiva Weinberger

for how exactly the vertices of $r_\ell(T)$ line up between those of $T$

Ted Shifrin

But common sense will probably narrow it down ...

Still, it's something to play with, and something easy to explore with Mathematica or somethin'.

Akiva Weinberger

My naive guess is that $\ell$ is the altitude on the hypotenuse.

Ted Shifrin

Wait, I thought we had a general scalene triangle.

Are we doing right triangles?

Akiva Weinberger

It was 3-4-5, I thought

though the problem generalizes, of course

Ted Shifrin

01:01

Oh, specifically that one?

user19405892

here is the question again

A triangle $T$ is given in the plane with sides $300,400,500$. A second triangle $r_l(T)$ is created by reflecting $T$ about a line $l$. Find $\displaystyle \min_{l}\{[r_l(T) \cup T]\}$ where $[\cdot]$ denotes area.

Ted Shifrin

Oh, specifically that one.

user19405892

*you can change the 300,400,500 to 3,4,5 if you like but i have a feeling the answer is going to be not very nice

Akiva Weinberger

Well, something to think about

Ted Shifrin

I don' t like zeroes. :)

user19405892

01:03

A triangle $T$ is given in the plane with sides $3,4,5$. A second triangle $r_l(T)$ is created by reflecting $T$ about a line $l$. Find $\displaystyle \min_{l}\{[r_l(T) \cup T]\}$ where $[\cdot]$ denotes area.

Ted Shifrin

I don't know the answer, but I actually like my suggested approach. I'll let you kiddies play with it :)

Akiva Weinberger

I don't actually have paper near me at the moment but i'll think about it, it sounds interesting

(Why does autocorrect change "ill" to "i'll" but not "I'll"?)

(That seems wrong.)

Ted Shifrin

On iOS, autocorrect seems often to automatically change merged words to separate capitalized words; drives me mad because on my iPad it's not easy to just change one letter and I have to retype the (long) second word without the capital.

crocket

0

Q: How do I translate 'no philosopher student admires any rotten lecturer' into quantificational logic formula?

Let's assume that $Fx=x$ is a philosophy student, $Rx=x$ is a rotten lecturer, and $Mxy=x$ admires $y$. My translation of the sentence was $\forall x(Fx\supset\neg\forall y(Ry\supset Mxy))$, but my logic textbook translated it as $\neg\exists x(Fx\wedge\exists y(Ry\wedge Mxy))$. As far as I kno...

Mike Miller

Hi

Ted Shifrin

01:09

heya @MikeM. How did your G-B lesson go?

Mike Miller

I forgot to prep.

So I taught holonomy.

Ted Shifrin

Oy.

OK. Did you mention Ambrose-Singer?

user19405892

@AkivaWeinberger Playing with geogebra it looks like it might be the angle bisector

Ted Shifrin

Through the vertex opposite the hypotenuse, @user19405892?

user19405892

@TedShifrin no opposite the leg ill check the hyp now

Ted Shifrin

01:13

Opposite which leg?

The shorter?

user19405892

yeah the shorter

Ted Shifrin

You get some sort of winged origami shape ...

Interesting. Keep me posted, please. Cool question.

user19405892

yeah right like two small triangles attached

Mike Miller

I ran out of time :( I defined parallel transport, proved it's orthogonal for LC, defined holonomy, said de Rham's theorem, talked about how restricted holonomy lets you get properties like orientability, Kahler

And then I was out of time. I wanted to draw the big table of possible holonomy groups.

And then Ambrose-Singer.

Ted Shifrin

Yeah, that's a lot of stuff ... particularly if not carefully prepared.

Not sure they know enough actual geometry to appreciate it.

Semiclassical

01:23

evening

user19405892

hi

Ted Shifrin

hi @Semiclassic

Semiclassical

just got back from caucusing

@TedShifrin do you remember what it looks like, or know where I can see it? i'm curious now

Ted Shifrin

It's precisely what I said up there, @Semiclassic, with a factor of 1/2.

If the vertices are $x_1,\dots,x_n$, make $2\times 2$ determinants with $x_i,x_{i+1}$ as rows/columns, using $x_n,x_1$ for the last one.

Semiclassical

hmm

user19405892

01:28

@TedShifrin I think the shoelace theorem finds the area of a polygon given vertices

Ted Shifrin

@Semiclassic: Think about the Green's Theorem formula for area inside a curve given by $\dfrac12\oint_C (-y dx+x dy)$ :)

@user19405892: You keep saying this. I don't care about names.

Semiclassical

Shoelace formula

The shoelace formula or shoelace algorithm (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by ordered pairs in the plane. The user cross-multiplies corresponding coordinates to find the area encompassing the polygon, and subtracts it from the surrounding polygon to find the area of the polygon within. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like tying shoelaces. It is also sometimes called the shoelace method...

user19405892

hmm i think greens theorem may be computationally better

Ted Shifrin

No, not really.

Semiclassical

it includes the determinant form on there, btw

Ted Shifrin

01:29

This is Green's Theorem for a piecewise linear curve.

Semiclassical

i'll have to think about how that works

i mean, i can see the determinant-ish nature of the integrand in green's theorem (which isn't shocking, since i can think about the area as the z-component of $\mathbf{x}\times d\mathbf{x}$)

Ted Shifrin

Precisely correct, @Semiclassic :)

Semiclassical

but i'm not seeing the implication immediatley. i'm sure it'll emerge, though

L33ter

@TedShifrin I coudln't imagine how much applications were there for finite fields.

I am glad I took this applied algebra class

Ted Shifrin

Karim, yes, I encouraged you to take it ... just not the other :)

L33ter

01:33

yeah

Semiclassical

it also kind've reminds me of the sorts of formulas one sees in projective geometry, though i'm really know nothing about those

L33ter

Yeah

Ted Shifrin

One of my colleagues at UGA who retired just before I got there designed a round-robin tennis tournament scheduler (copyrighted) using finite fields (block designs, I guess).

L33ter

cool

very cool

Ted Shifrin

Not quite sure what you're thinking of there, @Semiclassic.

Semiclassical

01:34

nor i, if i'm honest. it's a vague recollection.

Akiva Weinberger

Projective geometry doesn't really do areas, I don't think

Ted Shifrin

I mean, I've done projective geometry stuff most of my career, but I don't think I can guess what you're thinking of.

DogAteMy, nope.

Semiclassical

i may be remembering something else

shrugs

Ted Shifrin

To quote one of my dear friends, @Semiclassic, "Wait 'til you're my age."

Semiclassical

hahaha

Ted Shifrin

01:36

(He's 2 years older than I am, @Semiclassic :) )

Semiclassical

ahh

L33ter

I like algebra and topology a lot

@TedShifrin I haven't heard from other universities yet should I just accept university of alberta offer

?

or should I wait 1 more month ?

Ted Shifrin

It's still early, Karim. If it's really your top choice, then, yes, accept.

L33ter

no my top choice is UBC I will just wait

I hate waiting it sucks

Ted Shifrin

You don't hear from others because they're waiting to hear from their first-round offers, who are likewise sitting on their rear ends.

Semiclassical

01:39

oh, i see it now. that's quite cute

L33ter

I see

Ted Shifrin

Isn't it? @Semiclassic

Semiclassical

what i did to see it was consider a particular line segment parametrized as $x=ut+v(1-t)$ where $u,v$ are the position vectors

Ted Shifrin

You can prove it purely geometrically by dividing the polygon into triangles and using what we know about determinant and area of triangle/parallelogram.

Semiclassical

and then doing $(x\times dx)_z$ gives $(u\times v)_z\,dt$

yeah, i can believe it. but the algebra was cute enough that i like this one.

Ted Shifrin

01:41

Right, @Semiclassic, if you assume Green's Theorem. :)

Semiclassical

heh, true

i suspect there's another approach, though far more sophisticated than necessary

L33ter

@TedShifrin I am doing elliptic curve cryptography btw

Ted Shifrin

I don't know any of that stuff, Karim, but I know it's cool.

Semiclassical

namely, by thinking about Christoffel-Darboux (i think that's the name?) conformal mappings

L33ter

yeah it is pretty cool

Ted Shifrin

01:44

Say what, @Semiclassic?

Semiclassical

wait, that's not the right name. christoffel-darboux is an orthogonal polynomials result

christoffel-something, though

Ted Shifrin

Schwarz-Christoffel

Semiclassical

right

mostly i just like the idea of finding an explicit representation of the region inside

Ted Shifrin

Yeah, those mappings are, of course, conformal away from the vertices. But too much hassle for me :)

L33ter

I like the elliptic curve group

I have never imagined such things existed

Ted Shifrin

01:48

Have you seen the picture for what the group law is? I gave a lecture to undergraduates on that when I was in grad school.

Semiclassical

sure. hence why i said it was far more sophisticated than necessary :)

L33ter

yeah

I am currently reading from this intro @TedShifrin

Semiclassical

if i remember right, isn't the hard part of that showing that the group law is associative?

L33ter

law.upenn.edu/cf/faculty/jvagle/workingpapers/…

yeah

Ted Shifrin

Yes, @Semiclassic. That's basically the theorem (a version of the Residue Theorem) that if a cubic passes through 8 of the 9 points of intersection of two other cubics, then it passes through the 9th as well.

Semiclassical

01:49

hmm!

L33ter

@TedShifrin would you mind sharing the lecture note ?

oh you probably don't have it

Ted Shifrin

Long gone, Karim. I threw all that stuff out when I emptied out my office. But I had it up to then.

Semiclassical

I'd be curious in what sense it's a version of the Residue Theorem

L33ter

I am gonna Introduce the elliptic curve set prove that it is a group, and then introduce implemention of it and introduce attacks

as well as quantum attacks :D

my talk will be cool

Ted Shifrin

There's a beautiful paper by Griffiths called "Variations on a Theorem of Abel," too, which talks about Abel's Theorem in basic algebraic curve theory and how it generalizes the addition law for trig functions and $\wp$ functions. Both Karim and @Semiclassic should look at it sometime.

Semiclassical

01:51

i mean, what my immediate hope would be is that it's related to elliptic curves

L33ter

cool

Semiclassical

right.

i remember reading a bit about abel's theorem

Ted Shifrin

Griffiths's paper starts out very concretely... beautiful stuff.

Semiclassical

if memory serves, don't weierstrass functions have a relation to conformal mappings?

"starts out"?

Ted Shifrin

They're how you parametrize elliptic curves in the projective plane.

well, it gets quite a bit fancier.

Semiclassical

01:52

right. the word uniformize enters there too, i think?

Ted Shifrin

shrug

PVAL

Hi @Semi @Ted

Ted Shifrin

heya @PVAL :)

Semiclassical

hi @pval

PVAL

I am slowly but steadily becoming more and more concerned with my candidacy...

I blame @Semiclassical

Semiclassical

01:54

say whaaaat

(alternatively: oh snap)

Ted Shifrin

candidacy for what or whom? you could beat out Trump, for sure.

PVAL

I am also pretty convinced I proved something that my advisor knew how to prove and deleted from a paper.

For a doctorate at some point in the future (hopefully <17 years).

Ted Shifrin

@PVAL: In total seriousness, I am to this day convinced that my Berkeley thesis (which got published in Transactions) was something Chern could have done in an afternoon. Don't put yourself down unnecessarily.

Semiclassical

how did i become a scapegoat in this? :/

PVAL

I think even if I talk to him and he says he didn't know about this. I'll probably just assume he forgot.

Ted Shifrin

01:57

Better a scapegoat than a billy goat?

@PVAL: Don't be that way.

You're a smart dude.

PVAL

@Semi You stressed needlessly over a similar endeavor forcing me to do the same.

Ted Shifrin

clobbers both

Semiclassical

...dude, that wasn't an invitation to do so

PVAL

@TedShifrin Like I can honestly point to the paragraph in his paper where it would be.

Semiclassical

(also, stressed is past tense.)

Ted Shifrin

01:59

@Semiclassic is asking for future perfect tense.

@PVAL: Don't second-guess yourself or him.

Semiclassical

in the sense that i perfectly know my immediate future to be tense if i can't make progress on this paper :)

on this pointless, pointless oral exam paper

Ted Shifrin

I am not saying a word.

Semiclassical

this week it's just been tough for me to focus on it

not for any other reason than i just find it unpleasant, though

Ted Shifrin

@Semiclassic: Do you have to write a paper, or do you just have to deliver a lecture?

PVAL

I have recently read a few papers that got published in decent places (say Topology and Geometry or even Duke or JDG) which have about 4 pages of new content (which usually proves pretty nice statements) and 16 pages everyone in the field already knows.

Semiclassical

02:02

write a paper, do a presentation

not exactly a lecture since it's only in front of my committee

and the paper is basically just "take stuff i did a two years ago and package it into a research report"

Ted Shifrin

@Semiclassic: They want the lecture (which, presumably, you're not freaked about). The point of the paper is, presumably, to give you some practice at writing so that you can do the next step.

PVAL

There's this wonderful invention called a stapler :D

Ted Shifrin

@PVAL: I personally applaud papers that have some exposition and aren't just readable by the top two or three experts.

@PVAL: How do you staple in LaTeX? :D

PVAL

Presumably you print out the papers first.

Ted Shifrin

Ohhhh, I missed that.

PVAL

02:06

I don't know if I think they are poor papers or anything. It's just interesting to me how little of the work is original (now that I have a decent understanding of known things and how to read these papers).

Past theses are even worse on this count.

Ted Shifrin

@PVAL: At one point when I was ready to quit on my Ph.D., I went to the library and looked through some of the theses that Chern had approved. Some were pretty shabby. That emboldened me to say, "I can do better than this." And I did.

Semiclassical

the thing that's frustrating for me is that i don't really stress out over delivering the lecture per se. or at least i don't organize my anxiety around that

Ted Shifrin

Right, that's why I asked specifically.

Oh oh, DogAteMy returneth.

Semiclassical

i organize it around the fact that 1) i find preparing papers and presentations to be really frustrating and unpleasant, 2) i know I do, so i know i'm likely to avoid doing it, 3) knowing i'm likely to avoid doing it, i don't want to make commitments i can't fulfilll

and if that seems circular and/or self-fulfilling

well, yes

Ted Shifrin

@Semiclassic: I think you have a lot of talent. I truly think you need some help from a counselor/psychologist/psychiatrist to get over this hump ... or to get pointers.

Semiclassical

02:11

i've been doing that for a while, though

Ted Shifrin

Maybe a different person, then?

Semiclassical

eh, i've had different people. a number, in fact, though not so many lately

PVAL

I feel like if I tried to prepare presentations front to back they'd be really be difficult. Usually there is some small part of it that I know I need to describe and can do so easily. Then I fill in the background and the other interesting parts fall out of it.

Semiclassical

i suspect one basic issue is that i feel resigned to it

not necessarily in a "oh foolish fool that i am" kind've way

PVAL

Like there are likely very easy parts of your preparing that you could do now, and the other parts would become easier after doing so.

Semiclassical

02:14

but more of a "shrug it's what i do"

Ted Shifrin

@PVAL: We all have our personal styles for preparing lectures and writing papers. Each of us needs to figure out what works best for him/her.

PVAL

I just think breaking up a seemingly insurmountable task into smaller tasks helps me a lot.

Ted Shifrin

Absitively.

Semiclassical

there's a line out of Emerson which has stuck with me for a long, long time

"What use heroic vows of amendment if the same old law-breaker is to keep them?"

Ted Shifrin

@Semiclassic: Did you know about this foible before you started grad school?

That line seems too Trump-like :P

Semiclassical

02:16

to some extent. but i'd managed to push through it before.

grad school is just such a different environment. for the first two years i lived by myself, and for the rest with my parents. compare that to undergrad where lived with friends.

it all seems to blend together into one frustrating mess

Ted Shifrin

Grad school is not the social thing that undergrad is. Maybe isolating yourself and turning into a hermit is something you should change, if possible.

Semiclassical

can't say i disagree

Akiva Weinberger

So I haven't had time to read more into homology, but I know the Hopf fibration is a thing and that it's the reason that $\pi_3(S^2)\ne0$

Ted Shifrin

Some grad departments are a bit more nurturing than others. But, ultimately, you have to make your life work and try to be successful. We're rooting for you, @Semiclassic.

Akiva Weinberger

And I'm not sure why that doesn't imply that $H_3(S^2)\ne0$

Ted Shifrin

02:20

Yeah, DogAteMe, the Hopf fibration is cool. Not a homology thing.

Because $H_n(X) =0$ when $n>\dim X$.

Akiva Weinberger

What if you triangulate the 3-sphere (I think you need five simplices?), and apply the Hopf fibration to them?

Ted Shifrin

Homotopy classes of maps from $S^3$ to $X$ is different from studying $3$-cycles in $X$.

Akiva Weinberger

That would give you a 3-cycle in $S^2$. Is it just also a boundary?

Ted Shifrin

It's a degenerate $3$-cycle, yes.

Semiclassical

for purely literary interest, here's where that line of Emerson is from. the whole paragraph has stuck with me, really

PVAL

02:21

A precise relation between homology and homotopy groups is given by the Hurewicz (who accidentally fell off a Mayan pyramid at a conference and died) theorem.

Ted Shifrin

LOL @PVAL for injection of history.

Akiva Weinberger

@PVAL Wow

(W)ow

Ted Shifrin

smacks DogAteMy

Akiva Weinberger

ow

Ted Shifrin

:)

Semiclassical

02:24

$H_1(X)\cong \pi_1(X)/[\pi_1(X),\pi_1(X)]$ is fun, yeah (though i know that's just the $n=1$ case)

Ted Shifrin

Yeah, @Semiclassic. Higher $\pi_n$ are all abelian, so there's no such thing.

Semiclassical

right.

Ted Shifrin

DogAteMy, have you studied point-set topology already?

Semiclassical

i imagine that there's still a homology $\subset$ homotopy statement, though?

Ted Shifrin

No, not really, but I'll defer to @PVAL.

Akiva Weinberger

02:26

Not too closely, but I read a book on it

Semiclassical

hmm

Akiva Weinberger

(It was a small book)

Ted Shifrin

DogAteMy, if you're interested, I can send the midterm I just gave my two long-distance students back at UGA. But it's based on Munkres's book, so maybe you won't want it.

Akiva Weinberger

Sure

Semiclassical

i'd have expected from the $n=1$ case that, at the very least, the $n$th homology group would be isomorphic to a subgroup of the $n$th homotopy group.

Ted Shifrin

02:27

Email me your email address (my addy is in my profile).

Akiva Weinberger

I'm not seeing your email

Ted Shifrin

Really?

PVAL

@Semiclassical No a $K(G,1)$ for instance has 2nd cohomology of $G$ (which is rarely even finitely generated) but vanishing $\pi_2$.

L33ter

@TedShifrin do you know any good book on elliptic curve group the resource I am reading from only does stuff shallowly

??

Semiclassical

huh.

L33ter

02:29

why is my words repeated twice

weird

Akiva Weinberger

Yeah

L33ter

maybe it is on my end

Semiclassical

that makes it sound like "homology is simpler than homotopy" is a slogan that's really only true for $n=1$

Akiva Weinberger

Your posts aren't repeated on my end, @L33ter

L33ter

homology is actually pretty cool

I am liking it much more than homotopy

Akiva Weinberger

02:30

@L33ter So people keep telling me

PVAL

The 1st homology of the Klein bottle isnt even isomorphic to a subgroup of $\pi_1$ (which is actually torsion free). There's an answer I wrote on this site which sketches this and leads down a rabbit hole I recently sent and undergraduate down.

Ted Shifrin

Look at Miles Reid's book, Karim.

Akiva Weinberger

@TedShifrin Yeah, I can't find your email

PVAL

Heres the answer math.stackexchange.com/questions/1546624/…

Semiclassical

wait, what? doesn't that contradict the Hurewicz theorem?

PVAL

02:31

The rabbit hole is the phrase "left orderable"

It's naturally a quotient not naturally a subgroup.

Ted Shifrin

wow, DogAteMy, MSE has changed things. It used to be there. [email protected] will work.

Semiclassical

hrm. i guess i'm conflating those. the commutator group is certainly a subgroup, but quotienting out by that needn't be

Akiva Weinberger

@TedShifrin Email sent

Ted Shifrin

OK.

PVAL

I think if all quotients were automatically isomorphic to subgroups you might not have too many non-abelian groups.

Ted Shifrin

02:34

Or everything would be a semi-direct product ...

L33ter

thansk @TedShifrin

Akiva Weinberger

It's a homomorphic image, I think

L33ter

I am very behind in everything

I will probably sleep like 6 hr next couple of days

Semiclassical

that's a good point. after all, loops are contractible on a sphere, but not upon identifying antipodes to get $RP^2$

Ted Shifrin

DogAteMy: I sent. See you guys later. :)

Akiva Weinberger

02:36

Received. Thank you!

Ted Shifrin

Keep me posted :)

Akiva Weinberger

Remind me what $\Bbb R_\ell$ is?

Some weird topology on the reals, I think.

Never mind, got it

Lower limit topology

Ted Shifrin

Yup. That's it.

Semiclassical

i'm forgetting, actually. what's the short exact sequence for quotients---$A\to B\to B/A$, where $A$ is a subgroup of $B$?

Akiva Weinberger

Inclusion and quotient map, no? EDIT: And zeroes at the end

$0\to A\to B\to B/A\to 0$

PVAL

02:41

@Semiclassical Sure if $A$ is normal in $B$

Semiclassical

i forget what normal means, but i imagine it's sensible

i forget why one includes the zeros in there, though.

Akiva Weinberger

To make the first map injective (and thus an inclusion)

and to make the last thing surjective

PVAL

Nothing interesting is happening. Saying $0\to A \to B$ is exact is the same as saying $A\to B$ is injective.

Akiva Weinberger

(Distinguishing between isomorphic objects is "evil" in category theory, right?)

In any case, a surjective homomorphism is a quotient map, I think, so it makes the last thing a quotient map

user19405892

hi

PVAL

02:45

@AkivaWeinberger I am certain that I don't care.

Akiva Weinberger

Hi

user19405892

@AkivaWeinberger I was thinking of that question and after playing with geogebra it seems like an angle bisector might give the min

hard to prove that, though

Akiva Weinberger

Have you figure out the area you get from that?

user19405892

i am using the case with sides of 3,4,5

ill tell in a sec

I'm getting around 7.52

7.69 to be exact

This sort of seems like a statistics question: if we can get the residual to be as small as possible between the area of the triangle = 6, that minimizes the union

Akiva Weinberger

You should be able to figure that out exactly

user19405892

02:59

ok

6.67

exactly

seems very close to 6

Akiva Weinberger

6 is the area of $T$ by itself

user19405892

yep

if this does turn out to be the minimum, i wonder if this is the case for other types of triangles as well

Mike Miller

03:29

@Ted: It was carefully prepared. I just chose it because I could write a good lesson and cover it then. I didn't read the GB proof and would be uncomfortable presenting it even with good notes.

PVAL

Hi @Mike

Mike Miller

@PVAL: I'm pretty sure what I'm doing now would be trivial for the people who care. It's reasonably distressing.

PVAL

I'm convinced I've proven something my adviser figured out, maybe even wrote in the first draft of a paper, and then decided it was something that wasn't worth publishing.

I can even point out where and in what paper it would be.

Down to the paragraph.

Mike Miller

Yea I was responding to the previous discussion. You mean your surgery result I assume.

I talked to some students here and they liked it.

PVAL

Ya it only works for +1 though.

It's false for -1

Mike Miller

03:39

I'm going to ask Ko probably if you don't mind.

If he knew this.

PVAL

in a very small amount of cases that I know of.

Mike Miller

(I wasn't going to ask unless you said it was OK.)

PVAL

If anyone did, I'm sure its Bob. There's like a few pages where he discusses this.

It's a rather a simple proof.

I'd rather it not become folklore if it isn't already.

Mike Miller

Ok, I wont. The students I talked about are unlikely to repeat it.

PVAL

I've been trying to think about obstructions to a contact manifold being a double branched contact cover.

I think Giroux showed everyone is a triple branched contact cover, but I have no idea what are the known obstructions (if any) to being a double branched cover.

"everyone"

It seems open whether if there is a natural analogue to Montesino's trick.

Unrelatedly, I gave an example here math.stackexchange.com/questions/1676715/… Where I didn't have to do any computations with the fundamental group of a knot.

Mike Miller

03:50

I would read the triple beanched cover paper.

I don't think I have anything sexy to tell you.

PVAL

Giroux papers are hard.

and French..

Mike Miller

I like him.

PVAL

I'll probably try and see if Etynre has any notes which cover this.

I think I need to read every expository article (and do all the exercises) he wrote.

What have you learned about L-spaces

?

Stan Shunpike

How do I plot 1/z? $z \in \Bbb C$

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