Homotopical connectivity

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

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

Homotopy animation

An interesting homotopy (in fact, an ambient isotopy) of two surfaces.

From playlist Algebraic Topology

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

Homotopy

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

Group Homomorphisms - Abstract Algebra

A group homomorphism is a function between two groups that identifies similarities between them. This essential tool in abstract algebra lets you find two groups which are identical (but may not appear to be), only similar, or completely different from one another. Homomorphisms will be

From playlist Abstract Algebra

Johannes Ebert - Rigidity theorems for the diffeomorphism action on spaces of metrics of (...)

The diffeomorphism group $\mathrm{Diff}(M)$ of a closed manifold acts on the space $\mathcal{R}^+ (M)$ of positive scalar curvature metrics. For a basepoint $g$, we obtain an orbit map $\sigma_g: \mathrm{Diff}(M) \to \mathcal{R}^ (M)$ which induces a map $(\sigma_g)_*:\pi_*( \mathrm{Diff}(

From playlist Not Only Scalar Curvature Seminar

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

Ling Zhou (1/15/21): Persistent Homotopy Groups of Metric Spaces

From playlist ATMCS/AATRN 2020

Localization of Spaces by Somnath 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)

Jennifer WILSON - High dimensional cohomology of SL_n(Z) and its principal congruence subgroups 3

Group cohomology of arithmetic groups is ubiquitous in the study of arithmetic K-theory and algebraic number theory. Rationally, SL_n(Z) and its finite index subgroups don't have cohomology above dimension n choose 2. Using Borel-Serre duality, one has access to the high dimensions. Church

From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory

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

Higher Algebra 8: Spectra

In this video, we introduce and discuss spectra (in the sense of homotopy theory). We explain how they generalise abelian groups. 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 informa

From playlist Higher Algebra

Ling Zhou (5/10/22): Persistent homotopy groups of metric spaces

By capturing both geometric and topological features of datasets, persistent homology has shown its promise in applications. Motivated by the fact that homotopy in general contains more information than homology, we study notions of persistent homotopy groups of compact metric spaces, toge

From playlist Bridging Applied and Quantitative Topology 2022

Bernhard Hanke - Surgery, bordism and scalar curvature

One of the most influential results in scalar curvature geometry, due to Gromov-Lawson and Schoen-Yau, is the construction of metrics with positive scalar curvature by surgery. Combined with powerful tools from geometric topology, this has strong implications for the classification of suc

From playlist Not Only Scalar Curvature Seminar

Introduction to Homotopy Theory- Part 4: Fibrations

Wow! This one was a lot more detailed than usual, so I'd really recommend going through the proofs with the nLab in hand. I tried to elucidate some of their explanations, but it's still good to have both, so hopefully in between both of our presentations you can find understanding. And as

From playlist Introduction to Homotopy Theory

Erin Wolf Chambers (5/12/22): Computing optimal homotopies

The question of how to measure similarity between curves in various settings has received much attention recently, motivated by applications in GIS data analysis, medical imaging, and computer graphics. Geometric measures such as Hausdorff and Fr\'echet distance have efficient algorithms,

From playlist Bridging Applied and Quantitative Topology 2022

Homomorphisms (Abstract Algebra)

A homomorphism is a function between two groups. It's a way to compare two groups for structural similarities. Homomorphisms are a powerful tool for studying and cataloging groups. Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ W

From playlist Abstract Algebra