Theorems in convex geometry | Continuous mappings | Fixed-point theorems | Theorems in topology
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a convex compact subset of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem.This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu. The theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the n-dimensional closed ball was first proved in 1910 by Jacques Hadamard and the general case for continuous mappings by Brouwer in 1911. (Wikipedia).
A beautiful combinatorical proof of the Brouwer Fixed Point Theorem - Via Sperner's Lemma
Using a simple combinatorical argument, we can prove an important theorem in topology without any sophisticated machinery. Brouwer's Fixed Point Theorem: Every continuous mapping f(p) from between closed balls of the same dimension have a fixed point where f(p)=p. Sperner's Lemma: Ever
From playlist Cool Math Series
Proving Brouwer's Fixed Point Theorem | Infinite Series
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi There is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra. Tweet at us! @pbsinfinite Facebook: facebook.com/pbs
From playlist An Infinite Playlist
Three Hard Theorems in Topology #some2
An exposition of some elementary proofs of three intuitive yet notoriously difficult theorems in topology. John Milnor's note on the hairy ball and Brouwer's fixed point theorems: https://www.jstor.org/stable/2320860 Terry Tao's blog post: https://terrytao.wordpress.com/2011/06/13/brouw
From playlist Summer of Math Exposition 2 videos
Ulrich Berger: On the Computational content of Brouwer's Theorem
The lecture was held within the framework of the Hausdorff Trimester Program: Constructive Mathematics. Abstract: The usual formulation of Brouwer's Theorem ('every bar is inductive')involves quantification over infinite sequences of natural numbers. We propose an alternative formulation
From playlist Workshop: "Constructive Mathematics"
Lefschetz Fixed Point Theorem example
Here we give an example of how to use the Lefschetz fixed point theorem. These notes were really useful as a graduate student, some of them are down now, but I think these notes I had came from here: http://mathsci.kaist.ac.kr/~jinhyun/useful.html
From playlist Riemann Hypothesis
Paul Shafer:Reverse mathematics of Caristi's fixed point theorem and Ekeland's variational principle
The lecture was held within the framework of the Hausdorff Trimester Program: Types, Sets and Constructions. Abstract: Caristi's fixed point theorem is a fixed point theorem for functions that are controlled by continuous functions but are necessarily continuous themselves. Let a 'Caristi
From playlist Workshop: "Proofs and Computation"
In this video, I prove a very neat result about fixed points and give some cool applications. This is a must-see for calculus lovers, enjoy! Old Fixed Point Video: https://youtu.be/zEe5J3X6ISE Banach Fixed Point Theorem: https://youtu.be/9jL8iHw0ans Continuity Playlist: https://www.youtu
From playlist Calculus
In this video we prove the Lefschetz fixed point theorem assuming some properties of our cohomology theory. These notes were really useful as a graduate student, some of them are down now, but I think these notes I had came from here: http://mathsci.kaist.ac.kr/~jinhyun/useful.html
From playlist Riemann Hypothesis
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw The Curiosity Box by Vsauce: https://www.curiositybox.com/ LINKS TO SOURCES BELOW! My twitter: https://twitter.com/tweetsauce My instagram: https://www.instagram.com/electricpants DONG: https://www.youtube.com/dong M
From playlist Knowledge
The deterministic communication complexity of approximate fixed point - Weinstein
Computer Science/Discrete Mathematics Seminar Topic: The deterministic communication complexity of approximate fixed point Speaker: Omri Weinstein Date: Monday, February 22 We study the two-party communication complexity of the geometric problem of finding an approximate Brouwer fixed-po
From playlist Mathematics
Are a line and a square the same shape? | Introduction to Topology #SoME2
Is a line really different than a square? Watch the video to find out! In this video, you will learn exactly what topology is all about through dimension! This video was created for the 2022 Summer of Math Exposition. 00:00 Introduction: Is a Line Equal to a Square? 1:22 The Dimension
From playlist Summer of Math Exposition 2 videos
NYT: Sperner's lemma defeats the rental harmony problem
TRICKY PROBLEM: A couple of friends want to rent an apartment. The rooms are quite different and the friends have different preferences and different ideas about what's worth what. Is there a way to split the rent and assign rooms to the friends so that everybody ends up being happy? In t
From playlist Recent videos
AlgTop13: More applications of winding numbers
We define the degree of a function from the circle to the circle, and use that to show that there is no retraction from the disk to the circle, the Brouwer fixed point theorem, and a Lemma of Borsuk. This is the 13th lecture of this beginner's course in Algebraic Topology, given by Assoc
From playlist Algebraic Topology: a beginner's course - N J Wildberger
Brill-Noether part 4: Noether's Theorem
From playlist Brill-Noether
Intro to the Fundamental Group // Algebraic Topology with @TomRocksMaths
In this video I teach the amazing @TomRocksMaths a little bit of algebraic topology, specifically the fundamental group. Tom also taught me some really cool fluid dynamics and you can find our collab over at his channel here: ►►► https://www.youtube.com/watch?v=bpeCfwY4qa0&ab_channel=TomR
From playlist Collaborations