Consider the north pole and the south pole on the sphere, and the height function on it. One can consider the set of all gradient flowlines from the north pole to the south pole
there's an action of $\Bbb R$ on this space, sending $\gamma(t)$ to $\gamma(t+s)$
(reparameterizing the flowline); mod out by this and you have a nice space of unparameterized trajectories
for this example, the space is $S^1$ - gradient flowlines are in bijection with points on the equator (sending a gradient flowline to the one it goes through)
a more complicated setting is the height function on the following "tilted torus"
between, say, the bottom-most and second-to-bottom (or third-to-bottom) point, there are precisely two gradient flowlines
now consider the gradient flowlines from the very top point to the very bottom point. (i actually don't know what this space is, though i have good guesses.)
unlike the previous example, this space will not be compact - you could have a sequence $\gamma_n$ of gradient flow lines that go closer and closer to one of the critical points in between the top and bottom
so imagine a sequence of trajectories that, in the limit, starts at the top, goes down the "top slightly to the right of the hole" trajectory, then down the "bottom front" trajectory
so I said earlier that the space of trajectories, due to this, is not compact. but we can add these in (so that we allow 'broken trajectories' to be points in the space). then the result is compact.
right right
given as fact: the original space of trajectories (the noncompact one) is a manifold.
i kind've feel like one should be able to get that in a simple way by using the broken trajectories to define a cell structure, and then take the dual structure of that (so that all flowlines which are locally 'the same' are identified)
let's label our points a,b,c,d. a is the top, d the bottom, b,c in between
so the answer is that it's a manifold with boundary, and the boundary is in bijection with $\mathcal M(a,b) \times \mathcal M(b,d) \sqcup \mathcal M(a,c) \times \mathcal M(c,d)$
How long can I sit on someone's error without writing them an email?
user147690
You can edit your messages by hovering over them and on the left clicking the down arrow and edit. (Don't know anything about your question though sorry)
They posted a preprint with what at this point I'm relatively sure is a fatal error.
I looked at it, and found it, but I got other things to do..
user147690
No idea what to do there sorry. If you don't care about attaching your name to the email, you can just email from a dummy account, and then not worry about it again
@PVAL Just do it, as an author I would prefer it the sooner someone did, if they intend to do it.
@SemiC Anyway "gluing" is the process by which you take a broken trajectory and "glue them together" to show that there's an honest trajectory nearby that goes from a to c, and that the space of trajectories is a manifold with boundarh with appropriate boundary set. In higher dimensions you get the structure of a manifold with corners instead.
In the gauge theory context you would more or less like to get similar results but the answers tend to be much more gross. Eg there's no reason you couldn't get a Hawaiian earring.
There's enough structure that you can work with it but it's still bad.
Suppose you have a draft paper that you think is pretty good, and people tell you that you should submit it to a top journal. How do you work out where to send it to?
Coming up with a shortlist isn't very hard. If you look for generalist journals, it probably begins:
Journal of the American Ma...
I don't know exactly where Topology and its Applications ranks. It's certainly a reputable journal but not on the level of Geometry and Topology or JDG. There's maybe 2 or 3 people on the board I recognize compared to literally every single one on G&T.
@ForeverMozart The best place to put it is on the arxiv
Essentially that the Pin-(2) invariants don't obstruct embeddings into S^4 in any way that wasn't already known for Seifert fibred spaces over S^2. I.e. if an older gauge-theoretic invariant vanishes these do as well.
Well there is in the sense that "Count the number of degree $n$ covering maps onto $S^1$" ends up giving the number of partitions of $n$, but this is not a super interesting result. It's only interesting that the number it gives you is maybe momentarily surprising.
Let me add, as a corollary to the point I make in the post, is that we shouldn't be impressed with a difficult mathematical argument simply because it is difficult. Rather, let us criticise difficult arguments for being obscure and try to improve them.
wow so smart
I remember being envious of carpenters and artists, who after an achievement can point at what they've done, its value obvious; while in mathematics our very achievements can undermine our perception of their value.
this is very true in my situation
Metaphysics has become mathematics, ready to form the material for a treatise whose icy beauty no longer has the power to move us."- Andre Weil
If you're talking about classification of simple group, I think it takes 200 pages each major group theorist working on that at the time I can think of.
The original CSG proof is ~10k pages. I think that considerable progress has been made in reducing that number, but I don't think it's in triple-digits yet.
Mozart, do you think the same for the four-color theorem, then?
@Semiclassical as long as you've chosen the function you're thinking about carefully enough (the "Morse-Smale conditions"; pretend this just means that the space of flowlines is a manifold), then you always get a "manifold with corners" as the compactification
and the strata of this thing (for instance, the "boundary" - the codimension 1 part of the corners - corresponds to trajectories that break once; the codim 2 part corresponds to twice-broken trajectories; etc) are what you would expect from the above parenthetical
@Brody I'd like to be poked constantly, but it's good to think through thinks yourself and/or come up with good and thoughtful questions instead of bad ones.
You'll find a lot of food for thought in Ted's book (and his lectures!)
yeah, i believe that. this gluing theory shows up in a lot of modern contexts. whenever you have a moduli space of some flavor (for symplectic manifolds, moduli spaces of holomorphic curves) you'll have some "breaking" into simpler kinds of curves; that's more or less the same phenomenon
and one needs a gluing theory there too. there are other, more complicated phenomena in this setting, though
He's been away from the chat for a while. I e-mailed him a couple days ago - he said he'll come back soon. You should grab a conversation with him, perhaps.
Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose gradient descent from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(Q1.)
What is the class of functions/surfaces
whose gradient-descent paths are geodesics?
Certainly if ...
my undergrad advisor's office was legendary. Stacks of books and papers up to 6 feet tall, and only a narrow path from the door to his desk. My current advisor's office is not much different
@Semiclassical the comments are pretty insightful. they point out that, in particular, $f$ must be rotationally symmetric. i think this should mean that the only shapes you have are spheres and $\Bbb R^2$-type things.
Suppose you have a draft paper that you think is pretty good, and people tell you that you should submit it to a top journal. How do you work out where to send it to?
Coming up with a shortlist isn't very hard. If you look for generalist journals, it probably begins:
Journal of the American Ma...
