Simplicial homology

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

Simplices and simplicial complexes | Algebraic Topology | NJ Wildberger

Simplices are higher dimensional analogs of line segments and triangle, such as a tetrahedron. We begin this lecture by discussing convex combinations and convex hulls, and showing a natural hierarchy from point to line segment to triangle to tetrahedron. Each of these also has a standard

From playlist Algebraic Topology

Emilie Purvine (5/2/21): Homology of Graphs and Hypergraphs

Graphs and hypergraphs are typically studied from a combinatorial perspective. A graph being a collection of vertices and pairwise relationships (edges) among the vertices, and a hypergraph capturing multi-way or groupwise relationships (hyperedges) among the vertices. But both of these ob

From playlist TDA: Tutte Institute & Western University - 2021

Simplicial complexes as expanders - Ori Parzanchevski

Ori Parzanchevski Institute for Advanced Study; Member, School of Mathematics February 4, 2014 Expanders are highly connected sparse graphs. Simplicial complexes are a natural generalization of graphs to higher dimension, and the notions of connectedness and expansion turn out to have inte

From playlist Mathematics

Omer Bobrowski: Random Simplicial Complexes II

A simplicial complex is a collection of vertices, edges, triangles, tetrahedra and higher dimensional simplexes glued together. In other words, it is a higher-dimensional generalization of a graph. In recent years there has been a growing effort in developing the theory of random simplicia

From playlist Workshop: High dimensional spatial random systems

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

Homotopy animation

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

From playlist Algebraic Topology

Lecture 6: HKR and the cotangent complex

In this video, we discuss the cotangent complex and give a proof of the HKR theorem (in its affine version) 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://www.uni-m

From playlist Topological Cyclic Homology

Thorben Kastenholz: Simplicial Volume of Total Spaces of Fiber Bundles

Thorben Kastenholz, University of Goettingen Title: Simplicial Volume of Total Spaces of Fiber Bundles It is a classical result that manifolds that are total spaces of fiber bundles, whose fiber has amenable fundamental group, have vanishing simplicial volume. In this talk I will explore t

From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022

Lecture 3: Classical Hochschild Homology

In this video, we introduce classical Hochschild homology and discuss the HKR theorem. 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://www.uni-muenster.de/IVV5WS/Web

From playlist Topological Cyclic Homology

Omer Bobrowski: Random Simplicial Complexes, Lecture I

A simplicial complex is a collection of vertices, edges, triangles, tetrahedra and higher dimensional simplexes glued together. In other words, it is a higher-dimensional generalization of a graph. In recent years there has been a growing effort in developing the theory of random simplicia

From playlist Workshop: High dimensional spatial random systems

Graham ELLIS - Computational group theory, cohomology of groups and topological methods 4

The lecture series will give an introduction to the computer algebra system GAP, focussing on calculations involving cohomology. We will describe the mathematics underlying the algorithms, and how to use them within GAP. Alexander Hulpke's lectures will being with some general computation

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

Gunnar Carlsson (6/26/17) Bedlewo: Local to global principles for persistent homology

One could say that the goal of topology is to formulate “local to global principles”. Such principles have not been discussed a great deal within the applied topology/persistent homology world. I will discuss some results and questions in this direction, including excision and Künneth form

From playlist Applied Topology in Będlewo 2017

Lecture 7: Hochschild homology in ∞-categories

In this video, we construct Hochschild homology in an arbitrary symmetric-monoidal ∞-category. The most important special case is the ∞-category of spectra, in which we get Topological Hochschild homology. Feel free to post comments and questions at our public forum at https://www.uni-mu

From playlist Topological Cyclic Homology

Omer Bobrowski: Random Simplicial Complexes, Lecture III

A simplicial complex is a collection of vertices, edges, triangles, tetrahedra and higher dimensional simplexes glued together. In other words, it is a higher-dimensional generalization of a graph. In recent years there has been a growing effort in developing the theory of random simplicia

From playlist Workshop: High dimensional spatial random systems

Lewis Mead (5/27/20): From large to infinite random simplicial complexes

Title: From large to infinite random simplicial complexes Abstract: The talk will introduce two general models of random simplicial complexes which extend the highly studied Erdos-Renyi model for random graphs. These models include the well known probabilistic models of random simplicial

From playlist AATRN 2020

Intuitive Persistence - Damiano - 2020

Intuitive Persistence This lecture introduces the persistent homology of a filtration of a simplicial complex. In each dimension inclusions of the filtration subcomplexes induce a sequence of maps on the homology vector spaces of the given dimension of the subcomplexes. Tracking cycles fr

From playlist Applied Topology - David Damiano - 2020

Gunnar Carlsson(9/6/16): Local to global principles in persistent homology

From playlist AATRN 2016

Lecture 14: The Definition of TC

In this video, we finally give the definition of topological cyclic homology. In fact, we will give two definitions: the first is abstract in terms of a mapping spectrum spectrum in cyclotomic spectra and then we unfold this to a concrete definition on terms of negative topological cyclic

From playlist Topological Cyclic Homology