nhaliday + topology   39

gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
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october 2017 by nhaliday
Best Topology Olympiad ***EVER*** - Affine Mess - Quora
Most people take courses in topology, algebraic topology, knot theory, differential topology and what have you without once doing anything with a finite topological space. There may have been some quirky questions about such spaces early on in a point-set topology course, but most of us come out of these courses thinking that finite topological spaces are either discrete or only useful as an exotic counterexample to some standard separation property. The mere idea of calculating the fundamental group for a 4-point space seems ludicrous.

Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!
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october 2017 by nhaliday
co.combinatorics - Classification of Platonic solids - MathOverflow
My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula v−e+f=2v−e+f=2 to show that there are exactly five possible triples (v,e,f)(v,e,f). But of course this is not a complete proof because it does not rule out the possibility of different configurations or deformations. Has anyone ever written up a complete proof of this statement?!

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This is a classical question. Here is my reading of it: Why is there a unique polytope with given combinatorics of faces, which are all regular polygons? Of course, for simple polytopes (tetrahedron, cube, dodecahedron) this is clear, but for the octahedron and icosahedron this is less clear.

The answer lies in the Cauchy's theorem. It was Legendre, while writing his Elements of Geometry and Trigonometry, noticed that Euclid's proof is incomplete in the Elements. Curiously, Euclid finds both radii of inscribed and circumscribed spheres (correctly) without ever explaining why they exist. Cauchy worked out a proof while still a student in 1813, more or less specifically for this purpose. The proof also had a technical gap which was found and patched up by Steinitz in 1920s.

The complete (corrected) proof can be found in the celebrated Proofs from the Book, or in Marcel Berger's Geometry. My book gives a bit more of historical context and some soft arguments (ch. 19). It's worth comparing this proof with (an erroneous) pre-Steinitz exposition, say in Hadamard's Leçons de Géométrie Elémentaire II, or with an early post-Steinitz correct but tedious proof given in (otherwise, excellent) Alexandrov's monograph (see also ch.26 in my book which compares all the approaches).

P.S. Note that Coxeter in Regular Polytopes can completely avoid this issue but taking a different (modern) definition of the regular polytopes (which are symmetric under group actions). For a modern exposition and the state of art of this approach, see McMullen and Schulte's Abstract Regular Polytopes.

https://en.wikipedia.org/wiki/Platonic_solid#Classification
https://mathoverflow.net/questions/46502/on-the-number-of-archimedean-solids
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july 2017 by nhaliday
Lecture 6: Nash Equilibrum Existence
pf:
- For mixed strategy profile p ∈ Δ(A), let g_ij(p) = gain for player i to switch to pure strategy j.
- Define y: Δ(A) -> Δ(A) by y_ij(p) ∝ p_ij + g_ij(p) (normalizing constant = 1 + ∑_k g_ik(p)).
- Look at fixed point of y.
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june 2017 by nhaliday
general topology - What should be the intuition when working with compactness? - Mathematics Stack Exchange
http://math.stackexchange.com/questions/485822/why-is-compactness-so-important

The situation with compactness is sort of like the above. It turns out that finiteness, which you think of as one concept (in the same way that you think of "Foo" as one concept above), is really two concepts: discreteness and compactness. You've never seen these concepts separated before, though. When people say that compactness is like finiteness, they mean that compactness captures part of what it means to be finite in the same way that shortness captures part of what it means to be Foo.

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As many have said, compactness is sort of a topological generalization of finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often "care about how something behaves on an open set", and for compact spaces this means that there are only finitely many possible behaviors.

--

Compactness does for continuous functions what finiteness does for functions in general.

If a set A is finite then every function f:A→R has a max and a min, and every function f:A→R^n is bounded. If A is compact, the every continuous function from A to R has a max and a min and every continuous function from A to R^n is bounded.

If A is finite then every sequence of members of A has a subsequence that is eventually constant, and "eventually constant" is the only kind of convergence you can talk about without talking about a topology on the set. If A is compact, then every sequence of members of A has a convergent subsequence.
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january 2017 by nhaliday
Mikhail Leonidovich Gromov - Wikipedia
Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.

Gromov is also interested in mathematical biology, the structure of the brain and the thinking process, and the way scientific ideas evolve.
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january 2017 by nhaliday
Covering space - Wikipedia
A covering space of X is a topological space C together with a continuous surjective map p: C -> X such that for every x ∈ X, there exists an open neighborhood U of x, such that p^−1(U) (the inverse image of U under p) is a union of disjoint open sets in C, each of which is mapped homeomorphically onto U by p.
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january 2017 by nhaliday
set theory - What are interesting families of subsets of a given set? - MathOverflow
This fascinating essay by Gromov discusses the issue of "interesting" substructures in a very general way.
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january 2017 by nhaliday
"Surely You're Joking, Mr. Feynman!": Adventures of a Curious Character ... - Richard P. Feynman - Google Books
Actually, there was a certain amount of genuine quality to my guesses. I had a scheme, which I still use today when somebody is explaining something that l’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball)—disjoint (two balls). Then the balls tum colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!"
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january 2017 by nhaliday
soft question - A Book You Would Like to Write - MathOverflow
- The Differential Topology of Loop Spaces
- Knot Theory: Kawaii examples for topological machines
- An Introduction to Forcing (for people who don't care about foundations.)
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october 2016 by nhaliday
Math attic
includes a nice visualization of implications between properties of topological spaces
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march 2016 by nhaliday

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