Ciro Santilli @cirosantilli 34

 Incoming links: Manifold

Atlas (topology)  Updated 2024-12-15  +Created 1970-01-01

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Collection of coordinate charts.

The key element in the definition of a manifold.

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Covariant derivative  Updated 2024-12-15  +Created 1970-01-01

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A generalized definition of derivative that works on manifolds.

TODO: how does it maintain a single value even across different coordinate charts?

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Product definition of the exponential function  Updated 2024-12-15  +Created 1970-01-01

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e^{x} = lim_{n \to \infty} (1 + \frac{x}{n})^{n}

(1)

The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.

More precisely, we know that for any base b, exponentiation satisfies:

$b^{x + y} = b^{x} b^{y}$ .
$b^{0} = 1$ .

And we also know that for

b = e

in particular that we satisfy the exponential function differential equation and so:

\frac{d e ^{x}}{d x} (0) = 1

(2)

One interesting fact is that the only thing we use from the exponential function differential equation is the value around

x = 0

, which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.

Now suppose that we want to calculate

e^{1}

. The idea is to start from

e^{0}

and then then to use the first order of the Taylor series to extend the known value of

e^{0}

e^{1}

E.g., if we split into 2 parts, we know that:

e^{1} = e^{1 / 2} e^{1 / 2}

(3)

or in three parts:

e^{1} = e^{1 / 3} e^{1 / 3} e^{1 / 3}

(4)

so we can just use arbitrarily many parts

e^{1 / n}

that are arbitrarily close to

x = 0

e^{1} = (e^{1 / n})^{n}

(5)

and more generally for any

x

we have:

e^{x} = (e^{x / n})^{n}

(6)

Let's see what happens with the Taylor series. We have near

y = 0

in little-o notation:

e^{y} = 1 + y + o (y)

(7)

Therefore, for

y = x / n

, which is near

y = 0

for any fixed

x

e^{x / n} = 1 + x / n + o (1 / n)

(8)

and therefore:

e^{x} = (e^{x / n})^{n} = (1 + x / n + o (1 / n))^{n}

(9)

which is basically the formula tha we wanted. We just have to convince ourselves that at

lim_{n \to \infty}

, the

o (1 / n)

disappears, i.e.:

(1 + x / n + o (1 / n))^{n} = (1 + x / n)^{n}

(10)

To do that, let's multiply

e^{y}

by itself once:

e^{y} e^{y} = (1 + y + o (y)) (1 + y + o (y)) = 1 + 2 y + o (y)

(11)

and multiplying a third time:

e^{y} e^{y} e^{y} = (1 + 2 y + o (y)) (1 + y + o (y)) = 1 + 3 y + o (y)

(12)

TODO conclude.

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The beauty of mathematics  Updated 2024-12-15  +Created 1970-01-01

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Ciro Santilli intends to move his beauty list here little by little: github.com/cirosantilli/mathematics/blob/master/beauty.md

The most beautiful things in mathematics are results that are:

simple to state but hard to prove:
- Fermat's Last Theorem
- transcendental number conjectures, e.g. is $e + π$ transcendental?
- basically any conjecture involving prime numbers:
  - twin prime conjecture
- many combinatorial game questions, e.g.:
  - first-move advantage in chess
surprising results: we had intuitive reasons to believe something as possible or not, but a theorem shatters that conviction and brings us on our knees, sometimes via pathological counter-examples. General surprise themes include:
- classification of potentially infinite sets like: compact manifolds, etc.
  - classification of finite simple groups
  - classification of regular polytopes
  - classification of closed surfaces, and more generalized Poincaré conjectures
  - classification of wallpaper groups and space groups
- problems that are more complicated in low dimensions than high like:
  - generalized Poincaré conjectures. It is also fun to see how in many cases complexity peaks out at 4 dimensions.
  - classification of regular polytopes
- unpredictable magic constants:
  - why is the lowest dimension for an exotic sphere 7?
  - why is 4 the largest degree of an equation with explicit solution? Abel-Ruffini theorem
- undecidable problems, especially simple to state ones:
  - mortal matrix problem
  - sharp frontiers between solvable and unsolvable are also cool:
    attempts at determining specific values of the Busy beaver function for Turing machines with a given number of states and symbols
    related to Diophantine equations:
    decidability of Hilbert's tenth problem of a given degree and number of variables
    Hilbert's tenth problem over other rings
Lists:
- math.stackexchange.com/questions/139699/what-are-some-examples-of-a-mathematical-result-being-counterintuitive
- math.stackexchange.com/questions/2040811/what-are-some-counter-intuitive-results-in-mathematics-that-involve-only-finite/2055458#2055458
applications: make life easier and/or modeling some phenomena well, e.g. in physics. See also: explain how to make money with the lesson

Good lists of such problems Lists of mathematical problems.

Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:

Am I achieving insight, or am I just memorizing definitions?

Unfortunately, due to how man books are written, it is not really possible to reach insight without first doing a bit of memorization. The better the book, the more insight is spread out, and less you have to learn before reaching each insight.

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