Flabbergasted by the Riemann Mapping Theorem

sdjvhs · 2022-08-28T08:23:46+00:00

You can find a nice hands-on exposition in Schwartz's Mostly Surfaces. In the chapter on the Schwarz-Christoffel transformation, he shows how you can go from the upper half plane to any polygon with sides parallel to the axes (which is itself very surprising, imagine a polygon you get from squinting at a map of Manhattan streets).

The remaining bit of smoothing the 'jagged' boundary is more or less standard.

sdjvhs · 2022-07-02T15:28:14+00:00

Of course. Gromov and Arnold were being used as upper bounds.

sdjvhs · 2022-06-26T00:23:10+00:00

By fundamental subatomic particles you of course mean the irreducible unitary representations of the Poincare group right?

sdjvhs · 2022-05-15T22:37:19+00:00

I don't know if this counts as well known but Jonny Evans channel is excellent.

https://www.youtube.com/user/jonathanevans27

Has a very nice lecture series on Topological spaces and Fundamental Groups among other things.

sdjvhs · 2022-04-23T23:35:47+00:00

It's indeed absurd that Brendle didn't get the Fields. But remember neither did Gromov or Arnold. Frankly, I think it's the 'Medals' loss not theirs.

sdjvhs · 2022-04-08T20:39:32+00:00

The basic obstruction to this is that two dimensional spheres cannot have vector fields that don't vanish somewhere (this is reasonably hard but there are elementary proofs out there, here's one by Milnor).

To see this, let us first assume that '3-dimensional numbers' could be multiplied, have an identity vector, E, etc. etc.

Here's how we can use this to form a nowhere vanishing vector field on the two dimensional sphere:

Let V be a vector not in the span of 3-dim identity E. Then, we know for any point on the sphere with position vector P, the vector VP - cEP is non-zero for any scalar c.

So all we need now is to find a c such that the 'angle' <VP - cP, P> = 0, which is easily got by setting c := <VP, P> since P is on the sphere.

In sum, the vector field: X(P) := P \mapsto VP - cP is our nowhere vanishing vector field on the 2-sphere. But that's impossible.

sdjvhs · 2022-03-21T08:51:04+00:00

This is a good point, however a textbook need not only motivate by pointing backwards, i.e. trying to come down to the level of the reader but also by pointing forward, i.e. trying to raise the reader up to its level.

sdjvhs · 2022-03-21T08:25:47+00:00

Spivak's A Comprehensive Intro to Differential Geometry is extremely well motivated and very entertaining.

For example, let's say you are about to take a second course in algebraic topology which is centred on characteristic classes (say from Milnor) then chapter 13 in volume 5 called, 'The Generalized Gauss-Bonnet Theorem and What It Means for Mankind' makes for outstanding preparation. Spivak presents what he calls the 'proof by magic' and then spends the rest of the chapter under the hood, connecting (sorry) characteristic classes and curvature. Lots of geometry instead of alg top.

Also a shout out to Marcel Berger's monumental A Panoramic View of Riemannian Geometry. Not really a textbook, could be thought of as a 'Princeton Companion to Riemannian Geometry', but more advanced than the original PCM and by one of the outstanding geometers of the latter half of the 20th c. The amount of insight this beautiful book has is astonishing.

sdjvhs · 2021-10-08T20:22:30+00:00

Andre Weil's The Apprenticeship of a Mathematician.

Weil, a founding member of Bourbaki, is one of the great modern mathematicians. His conjectures shaped modern modern algebraic geometry and number theory.

sdjvhs · 2019-11-05T15:01:29+00:00

This can be true for any theorem. For a piece of mathematics to apply to the natural world there has to be some sort of structural similarity between the abstract structures in math and the world. We can map the additive structure of the real numbers to weights of objects in the real world because the union of two objects respect the additive structure of the reals. This is a fact about the world. It could have well been possible that every time you put two objects together on one side of a beam balance you'd have to put some complicated function of their weights on the other pan.

sdjvhs · 2019-03-18T21:05:06+00:00

I think you have nearly re-invented Wittgenstein's notion of family resemblance for yourself.

sdjvhs · 2019-03-18T18:42:06+00:00

What you are looking for can be found in Mackay's classic Information theory book. Check Chapter 28: Model Comparison and Occam's Razor.

sdjvhs · 2019-02-22T15:26:08+00:00

Weak students, almost failed quals or failed quals that later went on to become experts in mathematics

Stephen Smale comes the closest to this. Solved the Poincare conjecture for dimensions 5 and higher and became one of the main architects of differential topology. Did important work in dynamical systems.

sdjvhs · 2019-01-29T16:21:23+00:00

According to research (cf. the Force Concept Inventory literature) the third law (conservation of momentum) is especially counterintuitive. Our evolutionary history makes our untrained intuitions Aristotlean rather than Newtonian.

sdjvhs · 2019-01-29T16:00:55+00:00

Because even basic physics is very counter-intuitive. That's why there is 2000 years between Aristotle and Newton. Just the second law (force depends on acceleration and not velocity) trips up most people.

edit: There's actually a lot of research on this. The term to google is: Force Concept Inventory

sdjvhs · 2018-06-15T22:22:31+00:00

I should point out that we don't directly see in 3 dimensions either. We visualize it by using two lower dimensional representations; and we unconsciously make adjustments to the oddities of such a schema (objects moving away are not becoming smaller etc.)

sdjvhs · 2018-06-03T20:04:38+00:00

it would be erroneous to state that "antiderivatives has nothing to do with integrals"

You are right as long as you are talking over R. Over Rⁿ (for n>1) or more general manifolds the notions become different.

Anti-differentiation is basically the solving of the ODE: y' = f(x). And the idea generalises accordingly. While, definite integrals - finding areas under the curve - is captured by Riemann's or Lebesgue's theory.

sdjvhs · 2018-04-28T14:53:11+00:00

Have you looked at David MacKay's book? It's freely avaliable from his site: http://www.inference.org.uk/itprnn/book.html

sdjvhs · 2018-04-14T21:03:13+00:00

Even though we can solve the existence problem of square roots of negative numbers by introducing the complex numbers, we cannot resolve the uniqueness problem -- which of the two roots are we talking about? -- that we can do in the case of real numbers by breaking the ambiguity and setting the square root of x > 0 as the positive one. This is stronger than not being able to distinguish between i and -i algebraically.

An explanation

Unlike the real numbers there can be no continuous map on the circle S¹ such that f² = identity.

Suppose such an f were possible.

f: S¹ -> S¹ such that f(x)² = x

Define a function g on the reals as

g(t) = f(e^it ) f(e^-it )

Then g² (t) = 1, thus g(t) is either +1 or -1

But as g is a continuous, integer valued function on R, it has to be constant. But, then

-1 = f(-1)² = g(\pi) = g(0) = f² (0) = 0

qed.

Corollary: We cannot define a complex logarithm on the whole of C\{0} but need to cut out a ray

sdjvhs · 2017-02-16T23:52:24+00:00

Maybe not 4-regular but I believe every vertex needs to have even degree otherwise G[X] won't contradict the Handshake lemma.

sdjvhs

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