but you cannot triangulate arbitrary compact subsets of it (the boundary sphere in the above example will not be smooth)
rekt
if you think of the end here as being homeomorphic to S^3 x [0,infty) the smooth structure we gave E8 \pt is exotic w/r/t the smooth structure endowed by any such homeomorphism
Note something special about the smooth structure on S^3 x [0,infty): it's "cylindrical", there's a smooth action of the monoid [0,infty) shifting it downwards
any smooth structure on E8 \ pt with a cylindrical end gives a smooth structure on E8 (you identify the end with B^4 \ pt smoothly, then add the obvious chart)
reasonably behaved probably means something like saying that you actually have a pair of covers $\mathcal U, \mathcal V$, with $B^n \cong \overline V_i \subset U_i$, and then the induction is on $\cup V_i$
Say I have triangulated in a big ball $B \subset U$. $V$ intersects $U$ in $U \cap V$ and say $B \cap V$ is nonempty so we're actually on the danger zone to extend the triangulation on $B \cap V$ to $V$
I'm trying to suggest that the technicalities don't really matter about the covers, I'm just trying to make sure we don't need to deal with infinite objects
So here's my picture: if the transition maps are wild, the boundary simplices of the old triangulation may be too wild to know that I can draw a simplex on the "other side"
Theorem: (Change of Variables Theorem). Let $\Omega\subset\Bbb R^n$ be a region and let $U$ be an open set containing $\Omega$ so that ${\bf g}:U\to\Bbb R^n$ is one-to-one and ${\cal C}^1$ with invertible derivative at each point. Suppose $f:{\bf g}(\Omega)\to\Bbb R$ and $(f\circ{\bf g})\lvert...
Could someone talk with me about summer REU programs for a moment? I'm still in lower division coursework. Next semester I will be in vector calculus and differential equations. I have yet to take a linear algebra course, real analysis, or anything of a higher caliber. I'm going to apply anyways, because it doesn't hurt, but am I correct in feeling like I stand almost no chance of being accepted?
so I guess it means that if $X$ is the corresponding simplicial complex, the homeomorphism from the realization $|X| \to M$ is a diffeomorphism on each simplex, considered as a compact manifold with corners
@CookieToast: Pretty much. I hate to be the bearer of bad tidings, but it's quite competitive even with several upper-division courses finished. But you lose nothing by applying.
also, so many people I know have developed serious depression and anxiety in grad school, partly as a consequence of the competitive way academia is built
it's such a hard game to win even when you have an amazing hand
I think I would have trouble finding the time if it wasn't my job :) But I think I could probably keep up with reading research outside of academic math
To be honest, I really only cared about using it to make sure integration of differential forms was well-defined, and for that we need diffeomorphisms a.e.
A tiny percentage of people who graduate with a math degree go to grad school, and only a small percentage of those admitted to grad school end up in the job queue later in academia.
that being said, I shared an office this semester with an oldtimer who got his first academic job in mathematics at the same time as the collapse of the Soviet Union
Here are explicit equations for nonsmoothable manifolds (all of which admit triangulations). I do not know if these are the "easiest" but they are surely much more explicit than a description of the E8-manifolds, which is constructed as a result of some infinite, and very implicit, process (Free...
The requirements algebraic topology imposes on the intersection form of an oriented closed 4-manifold (the cup product pairing $H^2(X;\Bbb Z) \otimes H^2(X;\Bbb Z) \to \Bbb Z$) are that it's unimodular (the det of the matrix defining it is pm 1) and symmetric
I think the Cayley-Hamilton theorem has a big gap between "what you want to do" and "what you need to do". You want to just toss your matrix where lambda goes, but you can't, because that would be a type error.
@user21820 @amWhy @ZacharySelk @Did Now, I know this question is literally the second math.SE question, so yeah yeah historic value, but I really don't think we need this. Would appreciate it if it could be downvoted and deleted.
The question currently has 13 upvotes, so deleting it will be quite difficult. For anyone who agrees with my opinion, please assist.
Theorem: (Change of Variables Theorem). Let $\Omega\subset\Bbb R^n$ be a region and let $U$ be an open set containing $\Omega$ so that ${\bf g}:U\to\Bbb R^n$ is one-to-one and ${\cal C}^1$ with invertible derivative at each point. Suppose $f:{\bf g}(\Omega)\to\Bbb R$ and $(f\circ{\bf g})\lvert...
