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
An interesting homotopy (in fact, an ambient isotopy) of two surfaces.
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 elements in the homotopy group π₂(S²) ≅ ℤ. Roman Gassmann and Tabea Méndez suggested some improvements to my original ideas.
From playlist Algebraic Topology
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
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
Stable Homotopy Seminar, 2: Fiber and Cofiber Sequences
We review some unstable homotopy theory, especially the construction of fiber and cofiber sequences of spaces, and how they induce long exact sequences on homotopy and homology/cohomology. (There's a mistake pointed out by Jeff Carlson: when I take a CW-approximation at one point, I have
From playlist Stable Homotopy Seminar
Stable Homotopy Seminar, 11: Stable Model Categories and Triangulated Categories
(Note: I messed up the first recording and had to re-record the first 20 minutes of this.) I show that cofiber sequences agree with fiber sequences in Spectra, or indeed in any pointed model category where suspension is invertible. The homotopy category of such a model category is a highly
From playlist Stable Homotopy Seminar
Christoph Winges: Automorphisms of manifolds and the Farrell Jones conjectures
The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Building on previous work of Bartels, Lück, Reich and others studying the algebraic K-theory and L-theory of discrete group rings, the validity of the Farrell-Jones Conjecture has be
From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
Algebraic Topology - 11.3 - Homotopy Equivalence
We sketch why that the homotopy category is a category.
From playlist Algebraic Topology
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
Basic Homotopy Theory by Samik Basu
PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
From playlist Dualities in Topology and Algebra (Online)
Lecture 17: Frobenius lifts and group rings
In this video, we "compute" TC of spherical group rings and more generally cyclotomic spectra with Frobenius lifts. Feel free to post comments and questions at our public forum at https://www.uni-muenster.de/TopologyQA/index.php?qa=tc-lecture Homepage with further information: https://
From playlist Topological Cyclic Homology
Stable Homotopy Seminar, 9: Infinite Loop Spaces, and Homotopy Colimits
The fibrant spectra are the Ω-spectra, and we can give an elegant explicit description of the fibrant replacement. The "infinite loop space" functor, which is the derived right adjoint to the suspension spectrum, is then given by taking the 0th space of an equivalent Ω-spectrum. This allow
From playlist Stable Homotopy Seminar
Stable Homotopy Seminar, 10: Fiber and Cofiber Sequences of Spectra
(Note: some internet issues forced me to do the second half of this on the blackboard, so the quality's a bit worse.) We generalize the notion of fiber & cofiber sequence to arbitrary pointed model categories, using the notion of homotopy limit and colimit defined last time. Cofiber sequ
From playlist Stable Homotopy Seminar
Stable Homotopy Seminar, 17: Universal Coefficient Theorem, Moore Spectra, and Limits
We finish constructing the universal coefficient spectral sequence, and look at some classical examples involving Moore spectra. As it turns out, it's really easy in stable homotopy theory to invert or localize at a prime. In particular, *rational* stable homotopy theory is completely alge
From playlist Stable Homotopy Seminar
Francesca Tombari (5/9/22): What's behind the homotopical decomposition of a simplicial complex
Decomposing a simplicial complex by taking a covering of its vertices does not necessarily preserves the homotopy type of the original one. Thus, there is no hope in general to retrieve the homotopy type of the Vietoris-Rips complex of a metric space, just by studying Vietoris-Rips complex
From playlist Bridging Applied and Quantitative Topology 2022
Univalent Foundations of Mathematics - Vladimir Voevodsky
Univalent Foundations of Mathematics - Vladimir Voevodsky Institute for Advanced Study December 10, 2010 The correspondence between homotopy types and higher categorical analogs of groupoids which was first conjectured by Alexander Grothendieck naturally leads to a view of mathematics wh
From playlist Mathematics
Introduction to Homotopy Theory: Part 8- Homotopy in Model Categories
Credits: nLab: https://ncatlab.org/nlab/show/Introduction+to+Homotopy+Theory#homotopy_2 Animation library: https://github.com/3b1b/manim My own code/modified library: https://github.com/treemcgee42/youtube Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remembe
From playlist Introduction to Homotopy Theory