I can just imagine that scenario happening to me. It'd be a group of awkward introverts with at least 2 females and they'd have pointless and moronic conversations like that. I'd just have walked away, who the fuck talks about their 'favourite' theorems besides undergrads and pop-sci faggots?
Christopher Campbell
>be me >bring Riemann Hypothesis to the table >anarchy breaks loose >we kill each other trying to solve it >tfw all I wanted was to be cool
Cameron Thomas
are you really equating knowing about math with being an intellectual?
Michael Barnes
1. Fundamental Theorem of Algebra 2. Pythagorean Theorem 3. Binomial Theorem 4. The Irrationality of the Square Root of 2 5. Pi is Trancendental
Take that back... Take that back right now... *no sleep*
Chase Clark
Easy, no particular order: Bolzano-Weierstrass theorem Helmholtz's theorem Stokes' theorem Cauchy integral theorem Chinese remainder theorem
Austin Martin
Fundamental Theorem of Forcing Gödel's Incompleteness Kunen Inconsistency Yoneda Lemma Compactness Theorem
Noah Kelly
>Stokes Theorem >Hodge Theorem >Nullstellensatz >Chow's Theorem >Cohomology Thm. (For any SES..., there exists a LES... etc.)
Jason Ramirez
Spanish hotel lemma Monotone convergence theorem L'hospital's theorem Existence of a primative Mean value theorem
Charles Torres
>Sylow >Nullstellensatz >Cauchy's Integral >Stoke >Riemann's Uniformization
Austin Sanders
Explain at least one of them, (very easily) if u can?
Luke Diaz
A group is a set with one inversible operation. Think the integers with +. The operation need not be associative though.
Finite groups are interesting for many reasons, that is, groups with a finite amount of elements. Think permutations: the permutations of a set of n elements for a group under composition. For example, with n=3, you have permutations like p1 = (1->2),(2->3),(3->1) [written as (123)] p2 = (1->2)(2->1) [written as (12)] p1 o p2 = (123)(12) = (13)
A subgroup of a group is a subset of a set of a group that is closed under the operation. For example, even numbers with + are a subgroup of integers with +.
The Sylow theorems are a deep result about subgroups of a group. Namely, if p divides the number of elements in the group, and p^n is the highest power of p that still divides it, then there's a subgroup of size p^n (and you can also know how many of them and how they are related by permutation actions, but that's more intricate).
Matthew Morales
>The operation need not be associative though. I think you meant commutative
Jonathan Parker
I am >Stokes Theorem Integral of diff. form over boundary = Integral of derivative of diff. form over full space
>Hodge Theorem Every diff. form decomposes into the sum of exact, coexact, and harmonic pieces.
>Nullstellensatz Zariski-Closed subsets of affine n-space (in the classical sense) over an algebraically closed field k radical ideals of k[x_1,..,x_n]. This also leads to the important relation, Points of Affine n-space = Maximal ideals of polynomial ring
>Chow's Theorem
Complex Projective Varieties are Algebraic (Leads to a duality between certain types of complex manifolds and certain types of algebraic varieties.)
>(Co)homology Thm.
Given a SES of objects in a abelian category, there exists a LES of (co)homology.
Wyatt Wilson
Cauchy's Integral Theorem is a fundamental result of Complex Analysis, that is, the analysis (think advanced calculus) of functions in the complex numbers.
We say a function is holomorphic if it's C-differentiable. That is, the limit lim(h->0) (f(z+h) - f(z) )/ h exists. This is harder to fulfill for complex numbers than for real numbers, because you can approach 0 from any direction (think derivatives in R^2) and you're dividing the actual numbers, not just their norms.
We say a function is analytic (around a point a) if it can be expressed (around point a) as a power series. For example exp is analytic in all C (we call these "entire"), the power series representation at 0 is 1 + x/1! + x^2/2! + x^3/3! + ..... In particular, analytic functions are infinitely differentiable, and coincide with their power series.
Cauchy's integral theorem might seem a little whachy at first depending on the formulation. If you integrate a holomorphic function along two paths, and you can "deform" one of them into the other one (transform it into the other one continuously) then they will give you the same value. Basically, the integral is independent of the path you take (given some nice conditions). This allows you to define a primitive of the function, an antiderivative. This quickly leads up to a fundamental theorem: all holomorphic functions are analytic.
Matthew Jenkins
Mathematics or Mathematics & Statistics
I'm asking for a friend
Julian Sanders
> Spanish Hotel Theorem Every sequence has a monotonic (either always growing or always shrinking) sequence. > Monotone Convergence Theorem Every monotonic sequence (in a complete space) is convergent if and only if it is bounded. > L'hospitals Limit magick > Mean Value theorem Derivative magick > Existence of a primitive Every continuous function has a primitive (a function that differentiates to be the continuous function) This is one of the Fundamental Theorems of Calculus
you replied to one of the two shittiest answers (the other being )
many of the others had Stoke's Theorem included which, as you should know, is a much stronger form of the FTC
Nathaniel Rivera
Fundamental theorem of arithmetic Fundamental theorem of algebra Fundamental theorem of derivatives Fundamental theorem of integrals Fundamental theorem of line integrals
I don't know if I've articulated my question clearly but anyway are there any interesting results you can prove assuming those theorem as a hypothesis? Or is there consequence/implications of those theorems?
Juan Price
Courcelles Theorem Horn Locality of FO Schützenbergers Theorem Halting Problem The one theorem that FO is the biggest Logic that is comapct and has the Löwenheim property
Henry Smith
Can you explain one of them?
Adrian Fisher
>stone's representation theorem (and Priestley duality) >every monad arises from the "extreme adjunctions" given by the Kleisi and Moore categories >yoneda functor is actually an embedding >fundamental theorem of finite abelian groups >girauds theorem
also cute: every construct is equivalent to a uniquely transportable construct
Wyatt Gonzalez
and nullstellensatz, fo sho
Camden Wilson
Give me some time, and I'll explain all and their uses ( here)
Liam Green
>fundamental theorem of finite abelian groups >not classification of finitely generated modules over a PID
Caleb Diaz
>Derivative magick Not magic at all. Draw a line between f(a) and f(b) and drag it vertically until it is tangent to your graph.
Adam Green
>Courcelles Theorem MSO is decidable on graphs of bound tree-width. The proof is even constructive and gives you an algorithm how to process the formula, but has a non-elementary bound on the size of the resulting formula. It inspired a lot of similar results, if I remember correctly the "biggest" class in that line is that every non-trivial set of graphs that is closed under minors is FO decidable
>Horn Locality of FO Any structure is a model for a FO formula iff a restriction of that structure is one, where the restricted structure has a way smaller size and may be disconnected (the actual statement is very technical).
>Schützenbergers Theorem A language is regular iff its syntactic monoid is finite. I like it because the connection between languages and monoids was very suprising for me when I first heard it
>The one theorem that FO is the biggest Logic that is compact and has the Löwenheim property I looked it up, it's called Lindströms Theorem. It shows that FO really is a sweet spot in expressive power
Blake Lewis
The Sylow theorems tell you everything you could ever want to know about groups in some sense. You can completely classify groups of a certain order very easily (assuming the order doesn't have too many prime divisors). You can also prove that the only simple groups of order less than 60 have prime order.
The Nullstellensatz is more or less responsible for algebraic geometry.
Cauchy's Integral Theorem lets you calculate integrals. I'm not sure how you're not getting this.
(Co)homology Thm helps us to calculate (co)homology groups without too much trouble.
Jose Reyes
also robinson's principle (?) I think it's called is one of those things that is easy to prove but unsettling
the one that says anything provable in the language of fields which holds in an ACF of char 0 holds in some ACF of prime characteristic as well (actually infinitely many), and vice-versa
relatedly, los's theorem is cool
Lucas Cruz
I'm surprised at how many people here are analysts. Everyone I know has been going into algebra and nt. Here's my list. I'll try to include ones that have not been mentioned. >poincaire lemma (not conjecture) >egorov (because i green-tao (i don't care if you don't like tt; this theorem is cool) >holder (how i knew that i was going into analysis) >rice-shapiro (cs because why tf not)
Jacob Young
Banach fixed point theorem Fundamental theorem of algebra Riesz representation theorem Spectral theorem (any) Picard-Lindelöf theorem
Jayden Allen
Let's be friends.
Joshua Kelly
I've taken far more analysis than algebra desu senpai. I'm actually fairly self-conscious about it, so I'm going through Maclane and Birkoff for review.
>Baire Category Theorem Given a complete metric space, if there is a collection of open dense sets, then their intersection is dense. This is also true for locally compact Hausdorff spaces.
>Riemann mapping theorem Given any region (open, simply connected set) in the complex plane that is not the plane itself, there exists a biholomorphic map from the region to the open unit disk. This mapping is conformal, which can come in handy.
>Dominated convergence theorem If you can "dominate" (essentially bound) a sequence of functions and their limit by a non-negative, L1 integrable function, then the limit of the integral of the sequence of functions is equal to the integral of the sequence of functions' limit. You really only need to dominate a subsequence though, as it's assumed that they all have the same limit (and that the limit exists).
>Central limit theorem The arithmetic mean of a sequence of n independent, identically distributed random variables has an approximately normal distribution, and this approximation becomes better as n goes to infinity.
>Pumping lemma Given a regular language, there exists a positive integer n such that for all strings of length > n, there is a substring that can be repeated an arbitrary amount of times inside the string, and this will produce a new string that is an element of our language.
I would've used tex to make some of this more clear, but I'm unfortunately on my phone and won't be near a computer for a long time. Don't want to bother with editing that stuff on my phone.
Dominic Gonzalez
Meant to reference instead of myself there.
Sebastian Myers
I don't allow geometry in my proofs. ur mean
Eli Perry
So I'll give some handwavy ways to think about these:
>Stone's theorem
It justifies why you need self adjoint hamiltonians in quantum mechanics by connecting that to the conservation of probablity.
>Hahn-Banach theorem
With this fundamental theorem you can do cool stuff like extend some functionals and prove that there exist functionals with certain properties. AAAALL kindz of functional analysis.
>Cauchy-Schwarz inequality
It is one of the most fundamental and oft-used inequalities in analysis, and analysis is basically inequalities.
>Kato-Rellich theorem
With this you have a nice way to prove self adjointness. Used a lot for perturbations of the Laplacian, i.e. for quantum mechanics.
>Lebesgue dominated convergence theorem and friends
These are rigorous results which allow you to exchange limit and integral, limit and sum! How handy!
Gavin Walker
I'll learn your list just in case someone asks.
Jacob Williams
>I don't allow geometry in my proofs. Are you mentally deficient? It's not a proof, but a reason why the statement should be expected to hold true. It is fucking far from being "magic"
