An interesting homotopy (in fact, an ambient isotopy) of two surfaces.
From playlist Algebraic Topology
Homotopy elements in the homotopy group π₂(S²) ≅ ℤ. Roman Gassmann and Tabea Méndez suggested some improvements to my original ideas.
From playlist Algebraic Topology
Homotopy type theory: working invariantly in homotopy theory -Guillaume Brunerie
Short talks by postdoctoral members Topic: Homotopy type theory: working invariantly in homotopy theory Speaker: Guillaume Brunerie Affiliation: Member, School of Mathematics Date: September 26, 2017 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Homotopy Group - (1)Dan Licata, (2)Guillaume Brunerie, (3)Peter Lumsdaine
(1)Carnegie Mellon Univ.; Member, School of Math, (2)School of Math., IAS, (3)Dalhousie Univ.; Member, School of Math April 11, 2013 In this general survey talk, we will describe an approach to doing homotopy theory within Univalent Foundations. Whereas classical homotopy theory may be des
From playlist Mathematics
Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS
The goal of this series is to develop homotopy theory from a categorical perspective, alongside the theory of model categories. We do this with the hope of eventually developing stable homotopy theory, a personal goal a passion of mine. I'm going to follow nLab's notes, but I hope to add t
From playlist Introduction to Homotopy Theory
Lie Groups and Lie Algebras: Lesson 34 -Introduction to Homotopy
Lie Groups and Lie Algebras: Introduction to Homotopy In order to proceed with Gilmore's study of Lie groups and Lie algebras we now need a concept from algebraic topology. That concept is the notion of homotopy and the Fundamental Group of a topological space. In this lecture we provide
From playlist Lie Groups and Lie Algebras
Introduction to Homotopy Theory- Part 5- Transition to Abstract Homotopy Theory
Credits: nLab: https://ncatlab.org/nlab/show/Introdu... Animation library: https://github.com/3b1b/manim Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • YouTube Track Link: https://bit.ly/31Ma5s0 • Spotify Track Link: https://spoti.fi/
From playlist Introduction to Homotopy Theory
Algebraic Topology - 11.3 - Homotopy Equivalence
We sketch why that the homotopy category is a category.
From playlist Algebraic Topology
Computing homology groups | Algebraic Topology | NJ Wildberger
The definition of the homology groups H_n(X) of a space X, say a simplicial complex, is quite abstract: we consider the complex of abelian groups generated by vertices, edges, 2-dim faces etc, then define boundary maps between them, then take the quotient of kernels mod boundaries at each
From playlist Algebraic Topology
Strange Pattern in symmetries - Bott periodicity
A strange repeating pattern in the symmetries of circles, spheres and higher dimensional spheres called Bott periodicity. We will learn about symmetries of spheres, homotopy groups, the orthogonal groups and, finally, Bott periodicity. Produced by Connect films https://www.connectfi
From playlist Summer of Math Exposition Youtube Videos
Stable Homotopy Seminar, 6: Homotopy Groups of Spectra (D. Zack Garza)
In this episode, D. Zack Garza gives an overview of stable homotopy theory and the types of problems it was designed to solve. He defines the homotopy groups of a spectrum and computes them in the fundamental case of an Eilenberg-MacLane spectrum. ~~~~~~~~~~~~~~~~======================~~~
From playlist Stable Homotopy Seminar
Michael Hopkins: Bernoulli numbers, homotopy groups, and Milnor
Abstract: In his address at the 1958 International Congress of Mathematicians Milnor described his joint work with Kervaire, relating Bernoulli numbers, homotopy groups, and the theory of manifolds. These ideas soon led them to one of the most remarkable formulas in mathematics, relating f
From playlist Abel Lectures
Even spaces and motivic resolutions - Michael Hopkins
Vladimir Voevodsky Memorial Conference Topic: Even spaces and motivic resolutions Speaker: Michael Hopkins Affiliation: Harvard University Date: September 13, 2018 For more video please visit http://video.ias.edu
From playlist Mathematics
Higher Algebra 12: The Tate construction
In this video we introduce the Tate construction and especially Tate spectra. This is defined as the cofibre of a certain norm map, which we introduced for completely general group objects and stable infinity categories. We then also explain what it has to do with Poncaré duality and that
From playlist Higher Algebra
Henry Adams (8/30/21): Vietoris-Rips complexes of hypercube graphs
Questions about Vietoris-Rips complexes of hypercube graphs arise naturally from problems in genetic recombination, and also from Kunneth formulas for persistent homology with the sum metric. We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale param
From playlist Beyond TDA - Persistent functions and its applications in data sciences, 2021
Stable Homotopy Seminar, 1: Introduction and Motivation
We describe some features that the category of spectra is expected to have, and some ideas from topology it's expected to generalize. Along the way, we review the Freudenthal suspension theorem, and the definition of a generalized cohomology theory. ~~~~~~~~~~~~~~~~======================
From playlist Stable Homotopy Seminar
Fedor Manin (3/19/22): Linear nullhomotopies of maps to spheres
I will explain some aspects of how to build (null)homotopies of maps to simply connected spaces with controlled Lipschitz constant. Most of the difficulties appear already in the case of maps between spheres, where the result is as follows: every nullhomotopic, $L$-Lipschitz map $S^m \to
From playlist Vietoris-Rips Seminar
Homomorphisms in abstract algebra
In this video we add some more definition to our toolbox before we go any further in our study into group theory and abstract algebra. The definition at hand is the homomorphism. A homomorphism is a function that maps the elements for one group to another whilst maintaining their structu
From playlist Abstract algebra
Ling Zhou (1/21/22): Persistent homotopy groups of metric spaces
In this talk, I will quickly overview previous work on discrete homotopy groups by Plaut et al. and Barcelo et al., and work blending homotopy groups with persistence, including those by Frosini and Mulazzani, Letscher, Jardine, Blumberg and Lesnick, and by Bantan et al. By capturing both
From playlist Vietoris-Rips Seminar