Geometric topology | Riemannian geometry | 3-manifolds | Conjectures

Geometrization conjecture

In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic).In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print. Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server. Perelman's papers were studied by several independent groups that produced books and online manuscripts filling in the complete details of his arguments. Verification was essentially complete in time for Perelman to be awarded the 2006 Fields Medal for his work, and in 2010 the Clay Mathematics Institute awarded him its 1 million USD prize for solving the Poincare conjecture, though Perelman declined to accept either award. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture. (Wikipedia).

Video thumbnail

Circle and Tangent Phenomenon

Here’s a neat phenomenon that takes place in the context of a circle & a line drawn tangent to it. How can we prove one segment to be the geometric mean of the other two? 🤔 Source: Antonio Gutierrez. geogebra.org/m/DERWQcdF #GeoGebra

From playlist Geometry: Challenge Problems

Video thumbnail

Intersection of Planes on Geogebra

In this video, we look at a strategy for finding the intersection of planes on Geogebra.

From playlist Geogebra

Video thumbnail

Math in Geogebra Circumference of Circle

the video going to explain about circumference of circle. created by onwardono

From playlist Go Geogebra

Video thumbnail

Constructing an ISOSCELES TRIANGLE: GeoGebra Beginner Exercise 5A

Screencast shows how to EASILY CONSTRUCT an ISOSCELES TRIANGLE in GeoGebra. Here, we build an isosceles triangle to informally discover the Isosceles Triangle Theorem.

From playlist GeoGebra Geometry & Graphing Calculator: BEGINNER Tutorial Series

Video thumbnail

Regular Polygon Phenomena!

GeoGebra Link: https://www.geogebra.org/m/ketkkfuj

From playlist Geometry: Challenge Problems

Video thumbnail

A Constant Surprise!

Here, a #GeoGebra animation of Vincent Pantaloni's problem posed earlier this week: www.geogebra.org/m/waqpbrac. How can we formally prove the sum of these 2 square areas = 4*R^2, no matter where the large points lie? 🤔 #MTBoS #ITeachMath #geometry #math #maths #FigureThat #EdTech

From playlist Geometry: Challenge Problems

Video thumbnail

Concavity and Parametric Equations Example

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Concavity and Parametric Equations Example. We find the open t-intervals on which the graph of the parametric equations is concave upward and concave downward.

From playlist Calculus

Video thumbnail

Ian Agol, Lecture 3: Applications of Kleinian Groups to 3-Manifold Topology

24th Workshop in Geometric Topology, Calvin College, June 30, 2007

From playlist Ian Agol: 24th Workshop in Geometric Topology

Video thumbnail

Live CEOing Ep 197: Geometry in Wolfram Language

Watch Stephen Wolfram and teams of developers in a live, working, language design meeting. This episode is about Geometry in the Wolfram Language.

From playlist Behind the Scenes in Real-Life Software Design

Video thumbnail

Automated Planar Geometry

We present updates to the automated geometric functionality of the Wolfram Language introduced in Version 12 and display new functionality for automated geometric reasoning.

From playlist Wolfram Technology Conference 2021

Video thumbnail

Serre's Conjecture for GL_2 over Totally Real Fields (Lecture 4) by Fred Diamond

Program Recent developments around p-adic modular forms (ONLINE) ORGANIZERS: Debargha Banerjee (IISER Pune, India) and Denis Benois (University of Bordeaux, France) DATE: 30 November 2020 to 04 December 2020 VENUE: Online This is a follow up of the conference organized last year arou

From playlist Recent Developments Around P-adic Modular Forms (Online)

Video thumbnail

Towards a Geometric Analogue of Sarnak's Conjecture - Will Sawin

Workshop on Additive Combinatorics and Algebraic Connections Topic: Towards a Geometric Analogue of Sarnak's Conjecture Speaker: Will Sawin Affiliation: Columbia University Date: October 28, 2022 Work of Mark Shusterman and myself has proven an analogue of Chowla's conjecture for polynom

From playlist Mathematics

Video thumbnail

Serre's Conjecture for GL_2 over Totally Real Fields (Lecture 2) by Fred Diamond

Program Recent developments around p-adic modular forms (ONLINE) ORGANIZERS: Debargha Banerjee (IISER Pune, India) and Denis Benois (University of Bordeaux, France) DATE: 30 November 2020 to 04 December 2020 VENUE: Online This is a follow up of the conference organized last year arou

From playlist Recent Developments Around P-adic Modular Forms (Online)

Video thumbnail

Ariyan Javanpeykar - Albanese maps and fundamental groups of varieties with many rational... - WAGON

Albanese maps and fundamental groups of varieties with many rational points over function fields In this talk we will discuss topological properties of varieties with many rational points over a function field, and present joint work-in-progress with Erwan Rousseau. More precisely, we def

From playlist WAGON

Video thumbnail

Fred Diamond, Geometric Serre weight conjectures and theta operators

VaNTAGe Seminar, April 26, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in the talk: Ash-Sinott: https://arxiv.org/abs/math/9906216 Ash-Doud-Pollack: https://arxiv.org/abs/math/0102233 Buzzard-Diamond-Jarvis: https://www.ma.imperial.ac.uk/~buzzard/maths/research/paper

From playlist Modularity and Serre's conjecture (in memory of Bas Edixhoven)

Video thumbnail

Live CEOing Ep 97: Geometry in Wolfram Language

Watch Stephen Wolfram and teams of developers in a live, working, language design meeting. This episode is about Geometry in the Wolfram Language.

From playlist Behind the Scenes in Real-Life Software Design

Video thumbnail

ASA Triangle Congruence Theorem: Proof Without Words

Link: https://www.geogebra.org/m/WKJJ2uPa

From playlist Geometry: Dynamic Interactives!

Video thumbnail

Conjecturing Theorems in Euclidean Geometry

To learn more about Wolfram Technology Conference, please visit: https://www.wolfram.com/events/technology-conference/ Speaker: James Mulnix, Ian Ford, Dan McDonald, Nicolas Robles, Lin Cong Wolfram developers and colleagues discussed the latest in innovative technologies for cloud compu

From playlist Wolfram Technology Conference 2017

Related pages

Heisenberg group | Bianchi classification | 3-sphere | Graph manifold | Lie group | Norm (mathematics) | Euclidean geometry | Hyperbolization theorem | Spherical 3-manifold | Topological space | Uniformization theorem | Weeks manifold | 3-manifold | Surgery theory | Torus bundle | Poincaré conjecture | Automorphism | Index of a subgroup | Klein bottle | Dehn twist | Discrete group | Hyperbolic geometry | Thurston elliptization conjecture | Link (knot theory) | Mapping torus | JSJ decomposition | Torus | Spherical geometry | Connected sum | SL2(R) | Geometric topology | Riemann surface | Spherical space form conjecture | Atoroidal | Ricci flow | Diffeomorphism | Prime manifold | William Thurston | Orientability | Interior (topology) | Compact space | Fundamental group | Homology sphere | Haar measure | Manifold | Seifert–Weber space | Ricci curvature | Seifert fiber space | Solvable group | Lens space | Haken manifold | Projective plane | Geometry | Surface (topology) | Dehn surgery | Simply connected space