I mean if you have any reference book I will be delighted to stop bothering you
but I only know hatcher from Algebraic Topology and his 3-manifold notes are quite an advanced starting point (at least for my pov) and don't even cover this topic
He's well into graduate work now, plus making presentations as usual. Not clear to me what field he's going to actually settle into completely. But he's certainly more of a geometer than algebraist :)
is there any generalization of in-betweeness that doesn't require case work? e.g., in $\mathbb{R}$, $s$ is in-between $x,y$ if (and then we have two cases) if $x<y$, then $x<s<y$, and if $y<x$, then $y<s<x$
I've basically got the outline of that proof up to 4 black boxes. Lindenstrauss showed that if $\Gamma$ is a certain kind of lattice in $SL(2,\mathbb{R})$, and $\mu$ measure on $\Gamma\setminus SL(2,\mathbb{R})$ which is invariant under geodesic flow, satisfies some "Hecke recurrence property" (not entirely sure what this means), and whose ergodic components wrt geodesic flow have positive entropy
Then $\mu$ is just the "Haar measure"
And my advisor said to focus on understanding how he proves the positive entropy hypothesis, and that hey may have other problems along those lines
@TedShifrin i'm trying to find a way to do that, and to prove this implies there exists an analogous 2-case proof with inequalities. i'm guessing there must be such a general proof out there in terms of order properties of the real line
oh, right, signs aren't relevant here, thanks hehe. I'm trying to get at the idea of direction, somehow. for instance, the proof that $\frac{1}{2}(a+b)$ is in-between $a,b$ requires that we first do a proof supposing $a<b$, and then another supposing $b<a$. but these two proofs are so similar, there has to be a way to do them both in a single shot with a more general notion of direction
Also been toying on and off with whether or not to do an internship this summer. On the one hand I don't want to lose too much research time, but on the other hand I should have some kind of prep for industry there so I don't risk getting caught with my pants down come postdoc apps
Note that $\frac12(a+b) = a+\frac12(b-a) = b + \frac12(a-b)$. It's the midpoint of the line segment, whichever way you go.
What you want to think about is $\lambda x + (1-\lambda)y$. As $\lambda$ varies you get the entire line. When $0<\lambda<1$, you are in the line segment joining $x$ and $y$.
One of my former students who finished his PhD at UCSD in applied stuff got a job making $200K+$ from a company he had interned with one summer!!!!
Yeah, in particular an undergrad I know here interned at Jane Street and is coming back next year for a job, I applied for there and am currently waiting to be scheduled for a phone interview
Affine geometry is very cool, @shin. With more points, you get all sorts of interesting things. You can see a little bit of it if you go to my profile and from there to my webpage and click on my MAA lecture. Part of it was on these things.
But part of me hopes that, esp if my undergrad friend is right that the place he interned at is just 9-6 and that's it, you don't bring your work home with you
I don't know if that's the most interesting direction I could go, compared to e.g. data science. But if he's not wrong about the work not creeping home too much, then hopefully it's not disinteresting work, and it gives me free time to have a life out of my job lol.
there definitely are jobs where you get fired if you take it home with you. some trading stuff might be like that.
if they take your cell phone, wrap it in tin foil, and lock it in a lead safe every time you show up in the office, you probably are OK if you're out of the office. we should all get jobs like that.
Although sometimes it got tedious, I felt like grading homeworks and exams was an integral part of my effectiveness as a teacher. Of course, I traded off some research productivity for that.
Recall that a strategy is totally mixed iff it assigns a strictly positive probability to every pure strategy. A profile is totally mixed iff every strategy in it is totally mixed. Hereafter, let game refer to a two-player zero-sum normal-form game.
The problem is this: Given $n$ and $m$, constru...
some of the arguments in euclid's elements implicitly rely upon notions of betweenness that are not made explicit in the text. you can axiomatize those, hilbert offered one set of examples
for R and stuff based on R, you could see it as arising from the trichotomy property of the order relation
or ordered fields in general i guess
all of this is a special case of the infinity-pushout of the cohomology of cohomologies of a derived co-group
although i do not know what "infinity-pushout of the cohomology of cohomologies of a derived co-group" means, it looks like the sort of thing of which most things are a special case
another nice definition: $s$ is in-between if $|x-s| + |y-s| = |x-y|$
order theory looks fun, i need to find an application to agricultural economics to justify studying it
Uhm maybe. I guess that technically that's a percolation. The one about amenability iff the probability of having many infinite connected components or one are distinct
If you have a lot of percolation clusters with positive probability then it has to be amenable, otherwise you'd be able to tube these clusters to a single point with low cost (aka probability)
I need to revisit Lyons-Peres. I need to do infinitely many things.
@TedShifrin Once upon a long time ago, you gave me a lot of trouble for conflating the $k$-fold iterated tangent bundle $T^{(k)} X$ with the $k$-jet bundle $J^k X$. Do you remember what your main distinction between these two was? You and a fellow user pointed out that they fundamentally have different dimensions.
I think this isn't quite it because these are obviously different object. $J^1 X$ is $T^*X \times \Bbb R$.
Well, here's the question. What's wrong with defining a $k$-jet as a section of $\pi^{(i)}: T^{(i)}(X \times \Bbb R) \to T^{(i)} X$ for each $0 \leq i \leq k$, all compatible under the various projection maps $T^{(i)} X \to T^{(i-1)} X$, $T^{(i)} (X \times \Bbb R) \to T^{(i-1)}(X \times \Bbb R)$.
A function is just a section of $X \times \Bbb R \to X$.
just googling to learn these words i stumbled across math.stackexchange.com/questions/1383782/… where in the comments there's a link to a paper comparing 2-jets and the 2-iterated tangent bundle in some situation.
there's even a dimension count in the abstract. it's all nonsense to me.
Iterated tangent bundle is a very bizarre and nonstandard thing to do, I would like as much as possible to avoid thinking in terms of these crazy objects.
@JoeShmo You mean why $\int_0^\infty e^{-x^2}\,dx = \sqrt\pi/2$?
It's derived from polar coordinates. Yes, in my book (with rigor, even).
@Balarka: Ah, that's not how I defined higher-order tangent bundles. The original paper, as I recall, was another Chern student, William Pohl. Here you go.
