/sci/ - Science & Math

[Anonymous No.5434667]

What do you think are some of the greatest theorems in mathematics?

You can take "greatest" to mean whatever you'd like, i.e. by usefulness, scientific applications, elegance/beauty, etc.

I suppose I'll start by getting the standard "fundamentals" out of the way:
>Fundamental Theorem of Arithmatic
>Fundamental Theorem of Algebra
>Fundamental Theorem of Calculus
>Fundamental Theorem of Galois Theory

[Anonymous No.5434668]

1 = 0.999...

[Anonymous No.5434672]

<span class="math">\int_{\partial M} \omega = \int_M d \omega[/spoiler] of course.

[Anonymous No.5434678]

Pythagoras.

[Anonymous No.5434683]

>>5434672
This easily

[Anonymous No.5434710]

en.wikipedia.org/wiki/Recursion#The_recursion_theorem

[Anonymous No.5434711]

>>5434667
pythagorean theorem and fundamental theorem of arithmetic are without a doubt the two best theorems in all of math

[Anonymous No.5434720]

As I've learned more and more math, I've come to care less about theorems. Definitions are what it's all about.

[Anonymous No.5434724]

>>5434720
that's dangerous

[Anonymous No.5434768]

Cauchy's Integral Formula

[Anonymous No.5434773]

I don't necessarily think these are the best, just... really really good.
Theorema Egregium
Hilbert's Nullstellensatz
Fundamental Theorem of Modules over a PID
And personally I think the Isomorphism theorems for rings and groups and the Chinese remainder theorem for rings are all pretty baller as well

[Anonymous No.5434801]

Goodstein's Theorem

(2deep4/sci/, though)

[Anonymous No.5434837]

I like beautiful and simple proofs.

Fermat little theorem
p is prime, a is integer not divisible by p
prove a^{p-1}=1(mod p)

proof
{a,2a,...,(p-1)a}={1,2,...,p-1}(mod p)

[Anonymous No.5434854]

I like beautiful and simple proofs.

Fermat little theorem
p is prime, a is integer not divisible by p
prove a^{p-1}=1(mod p)

proof
{a,2a,...,(p-1)a}={1,2,...,p-1}(mod p)
(p-1)!a^{p-1}=(p-1)!(mod p)
a^{p-1}=1(mod p)

[Anonymous No.5434868]

>>5434801
how is it deep

[Anonymous No.5434873]

Once they're proven, the Birch–Swinnerton-Dyer conjecture, the Riemann hypothesis, and the abc conjecture will be.

Since those are still open... how about the prime number theorem and the Mordell–Weil theorem?

[Anonymous No.5434877]

Just ate out a girl and went for the Schrödinger equation.
Regarding the time derivative, I naturally decided to use Newton's dot notation <span class="math">\dot\psi[/spoiler]. Removing the tongue from the clit for a moment turned out to be appreciated and led to new ideas.
Thanks Newton!

[Anonymous No.5434879]

>>5434873
$10 says if they weren't so famous, the 3 "once they're proven" conjectures you listed would look totally unimpressive to the average /sci/tard. Go to an alternate universe where Riemann never stated his hypothesis; post a thread where you state the hypothesis; nobody will be remotely interested

[Anonymous No.5434883]

>>5434873
>implying /sci/ understands the implication of any of those

[Anonymous No.5434884]

>>5434879
Confirmed for high schooler who has no clue about the background regarding the Riemann hypothesis.

[Anonymous No.5434893]

>>5434879
All you have to understand to see why RH and abc are important is that anything that gives serious insight into how the prime numbers work is bloody awesome.

[Anonymous No.5434902]

>>5434854
I can't tell what your proof is saying so here's a better proof: Consider the length p tuples of integers between 1 and <span class="math">a[/spoiler] inclusive, for some a.
Put an equivalence class on them by cyclic rotation of the entries. So for example <span class="math">(2,1,0) ~ (0,2,1) ~ (1,0,2)[/spoiler] is an equivalence class for p=3. There are precisely a one element equivalence classes, and every other equivalence class has p elements. It follows that p divides <span class="math">a^k - a \cong 0[/spoiler]

[Anonymous No.5434904]

>>5434902
Oops, ignore the <span class="math">\cong 0[/spoiler]

[Anonymous No.5434915]

>>5434893
RH goes further than that. If Montgomery's/Dyson's observation is true, it might answer deep questions about quantum physics via random matrix theory.

In other words, the fabric of our universe might be encoded within that function.

But you know, it's not too important I guess.

[Anonymous No.5434916]

>>5434893
Indeed, but without context it is not easy to go directly from RH/abc to prime number applications. We can contrive theorems which look very similar to RH/abc but which really are totally uninteresting. A slightly transparent example:

"Every zero of the function z-1/2 is either a negative even integer or has real part 1/2."

Now, the above example is blatantly transparent, but you could further obfuscate it (e.g., replace x-1/2 with x+e^{i*pi}/2, etc. etc.)

[Anonymous No.5434920]

>>5434915
> modern day astrology and numerology masquerading as mathematics and science
math fan detected

[Anonymous No.5434943]

<span class="math">\displaystyle\mathrm{Ind}\displaystyle(D)=\displaystyle\int_{T^\ast M}\mathrm{ch}\displaystyle([\sigma_m(D)])\displaystyle\smile \displaystyle\mathrm{Td}\displaystyle(T^\ast M \otimes \mathbb{C})[/spoiler]

[Anonymous No.5434946]

>>5434943
How come it's always the differential geometry fags that try to needlessly show off.

[Anonymous No.5434949]

>>5434667
I always like le hospital's rule, since he just bought the rights to it from someone else.

[Anonymous No.5434957]

>>5434946
I think the algebraic topology fags are worse.

[Anonymous No.5434965]

>>5434957
500nodecommutativediagram.jpg

[Anonymous No.5434967]

>>5434943
What does this even mean? ch()? That smiley?

[Anonymous No.5434968]

>>5434957
>>5434946
> implying category theory isn't the all-time low

[Anonymous No.5434989]

>>5434968

[Anonymous No.5435013]

I like those complex numbers because you can use those complex numbers to do that probabilistic quantum sciences.

[Anonymous No.5435037]

>>5434943
Where'd you people learn about the index theorems?

[Anonymous No.5435049]

Don't you hate it when a nobody mathematician gets credit for an awesome theorem?

i.e. Hadamard/de la Vallee Poussin

[Anonymous No.5435058]

>>5434943
Name of the theorem?

[Anonymous No.5435095]

A couple of my favourites are Lawvere's fixed point theroem and Grothendieck's reformulation of Galois theory.

[Anonymous No.5435174]

>>5434877
> thanks newton!
Did you hang her for counterfeiting afterwards?

[Anonymous No.5435182]

>>5435013
OMG me too did you know that waves are complex? so it's like the universe is half imaginary....

[Anonymous No.5435266]

>>5434667
Minkowski's theorem probably isn't one of the common greatest, but I think it's my favorite because it's surprisingly powerful. In a simple two dimensional form,
"Given a lattice with fundamental volume D, any region that is convex and symmetric with respect to the origin that has area greater than D*2^k, where k is the dimension of the lattice, has a nontrivial lattice point."
With this you can prove that any natural number is the sum of 4 integral squares, and I think you can even do quadratic reciprocity but I wouldn't remember how it's done.

[Anonymous No.5435278]

>>5434949
He's a smart fucking business man. You can't really put a price on a legacy. It doesn't mean much to him, but still.

[Anonymous No.5435281]

>>5434916
Holy shit, have you ever even taken a class in analytic number theory?

[Anonymous No.5435304]

>>5435266
Minkowski's theorem is pretty great; I would have listed it if it'd come to mind. Its proof, on the other hand, is pretty annoying. Remember that giant, messy string of integrals you have to compute in the process?

[Anonymous No.5435307]

>>5435304
Most of the proofs I've seen don't really deal with any integrals. Do you have a reference? I've seen it with the dimension of a lattice defined by the determinant, etc. and from there it's pretty straightforward to construct a point that works. (not the previous poster, I also forgot about Minkowski, sadly)

[Anonymous No.5435312]

>>5435304
I saw it in high school, and we were somewhat informal in taking a congruence modulo the lattice and preservation of area, but other than that the proof isn't technical at all.

[Anonymous No.5435318]

>>5435307
Wait, never mind. I was thinking of Minkowski's bound for the ideal class group, which makes use of Minkowski's theorem, but also involves a rather convoluted volume computation.

[Anonymous No.5435319]

>>5435318
Oh yes. That one is a doozy, but an example of the power of minkowski's theorem for convex regions!

[Anonymous No.5435328]

Fibonacci Number

The troll proof that 1=2 using algebra. It fails by dividing by zero but it's still one of the most elegant troll proofs ever constructed.

Also agree with the Fundamental Theorem of Algebra. I love Algebra.

[Anonymous No.5435337]

Squeeze theroem

[Anonymous No.5435362]

>>5435312
What kind of high school covers this?

[Anonymous No.5435377]

>>5434667
How about the Fundamental Theorem of Combinatorial Game Theory? Proven by Zermelo (it's not that hard either) and used to motivate categorizations of positions as "P" or "N", it ensures that every finite combinatorial game has a non-losing strategy (meaning a winner is ensured for zero-sum games).

Basically, you start by assuming that you can reach a position, say p, without a winning strategy in a game. p can't move to a winning position or else moving there would be the winning strategy, so p can only move to another non-winning position, say p'. p is non-terminal because moving into it would be considered winning and from there p would be, by default, a losing position, so you can always move to some non-winning position p' from a non-winning position p. This creates an infinite chain of positions in your game, which contradicts finiteness.

This means that at any position p, either a move can be made to a non-losing position, in which case p is called a "P-position" (meaning the Previous player wins), or p must move into a position that is losing, in which p is called an "N-position" (because the Next player wins).

Essentially the Fun. Thm. of CGT simultaneously verifies the existence of and classifies the winning strategies for combinatorial games, thus by itself motivating Sprague-Grundy Theory and applications to poset games like Nim. It's a shame that its not popular, but CGT is still a young and growing field after all.

[IQ fundie !!8XTTCUzK2no No.5435474]

>>5434943
That is quite hard to beat ;-), I would have to opt for the non-commutative analogue or

<div class="math">\Omega^k(M)=\rm{im}\Delta_k\oplus\ker\Delta_k=\Delta(\Omega^k(M))\oplus\mathcal H^k;~~\forall\alpha\in\Omega^k(M),\;\exists\omega\in\Omega^k(M):\Delta\omega=\alpha\Longleftrightarrow \alpha\perp\mathcal H^k</div>
For complex M, pic very related.
>>5435037
Not him, but I found Higson and Roe's "Analytic K-Homology" to have a decent and modern review in terms the aspects of algebraic K-theory. Nothing is better than the original papers, however.
>>5435058
It is the Atiyah-Singer index theorem, a beautiful result in the study of manifolds with endless applications. In general, the theorem allows us to calculate certain analytic data - the indices i.e. sums and differences of the numbers of independent solutions to differential equations of various kinds - in terms of topological data about the base manifold, e.g. the Betti numbers or the number of holes in the manifolds. The index of the Dirac operator for example is important to physicists as it determines the number of generations of leptons and quarks in the conventional compactifications of string theory with 6 internal dimensions, among numerous other things.

[IQ fundie !!8XTTCUzK2no No.5435476]

>>5434943
That is quite hard to beat ;-), I would have to opt for the non-commutative analogue or

<div class="math">\mathcal H^r(\mathcal M)=\bigoplus_{p+q=r}\mathcal H^{p,q}(\mathcal M),\;\mathcal H^{p,q}(\mathcal M)=\overline{\mathcal H^{q,p}(\mathcal M)}</div>
For complex M, pic very related.
>>5435037
Not him, but I found Higson and Roe's "Analytic K-Homology" to have a decent and modern review in terms of the algebraic K-theory. Nothing is better than the original papers, however.
>>5435058
It is the Atiyah-Singer index theorem, a beautiful result in the study of manifolds with endless applications. In general, the theorem allows us to calculate certain analytic data - the indices i.e. sums and differences of the numbers of independent solutions to differential equations of various kinds - in terms of topological data about the base manifold, e.g. the Betti numbers or the number of holes in the manifolds. The index of the Dirac operator for example is important to physicists as it determines the number of generations of leptons and quarks in the conventional compactifications of string theory with 6 internal dimensions, among numerous other things.

[Anonymous No.5435549]

eww, why all this number theory..

[Anonymous No.5435558]

>>5434949
>le hospital
>le
get back to reddit

[Anonymous No.5435581]

>>5435312
>inb4 asians and brazilians

[Anonymous No.5435594]

Nullstellensatz

[Anonymous No.5435614]

>>5434946
Because no other branch has any real applications?

[Anonymous No.5435616]

>>5434946
Because probability theory fags are too busy making 300k starting at wall street.

[Anonymous No.5435667]

I also really like

<span class="math">\int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_{n-1}} f(\sigma_{n}) \, \mathrm{d}\sigma_{n} \cdots \, \mathrm{d}\sigma_2 \, \mathrm{d}\sigma_1 = \frac{1}{(n-1)!} \int_a^x\left(x-t\right)^{n-1} f(t)[/spoiler]

(Cauchy formula for repeated integration)

[Anonymous No.5435672]

<span class="math">\mathrm{d}t[/spoiler]

[Anonymous No.5435681]

>>5435667
>babby's first calculus

[Anonymous No.5436064]

>>5435616
Or because their theorems suck.