In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold. Loosely speaking, a cycle is a closed submanifold, a boundary is a cycle which is also the boundary of a submanifold, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries. A homology class is thus represented by a cycle which is not the boundary of any submanifold: the cycle represents a hole, namely a hypothetical manifold whose boundary would be that cycle, but which is "not there". There are many different homology theories. A particular type of mathematical object, such as a topological space or a group, may have one or more associated homology theories. When the underlying object has a geometric interpretation as topological spaces do, the nth homology group represents behavior in dimension n. Most homology groups or modules may be formulated as derived functors on appropriate abelian categories, measuring the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category. (Wikipedia).
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 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
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
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 introduction to homology | Algebraic Topology | NJ Wildberger
We briefly describe the higher homotopy groups which extend the fundamental group to higher dimensions, trying to capture what it means for a space to have higher dimensional holes. Homology is a commutative theory which also deals with this issue, assigning to a space X a series of homolo
From playlist Algebraic Topology
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
An interesting homotopy (in fact, an ambient isotopy) of two surfaces.
From playlist Algebraic Topology
Cohomology in Homotopy Type Theory - Eric Finster
Eric Finster Ecole Polytechnique Federal de Lausanne; Member, School of Mathematics March 6, 2013 For more videos, visit http://video.ias.edu
From playlist Mathematics
Algebraic Topology - 11.3 - Homotopy Equivalence
We sketch why that the homotopy category is a category.
From playlist Algebraic Topology
Jie Wu (7/25/22): Topological Approaches to Graph Data
Abstract: In this talk, we will discuss some topological approaches to graph data beyond classical persistent homology, including path homology and $\delta$-homology introduced by S. T. Yau et al, and their generalizations such as hypergraph homology, weighted persistent homology, and twis
From playlist Applied Geometry for Data Sciences 2022
Secondary products in SUSY QFT by Tudor Dimofte
Program: Quantum Fields, Geometry and Representation Theory ORGANIZERS : Aswin Balasubramanian, Saurav Bhaumik, Indranil Biswas, Abhijit Gadde, Rajesh Gopakumar and Mahan Mj DATE & TIME : 16 July 2018 to 27 July 2018 VENUE : Madhava Lecture Hall, ICTS, Bangalore The power of symmetries
From playlist Quantum Fields, Geometry and Representation Theory
The rising sea: Grothendieck on simplicity and generality - Colin McLarty [2003]
Slides for this talk: https://drive.google.com/file/d/1yDmqhdcKo6-YpDpRdHh2hvuNirZVbcKr/view?usp=sharing Notes for this talk: https://drive.google.com/open?id=1p45B3Hh8WPRhdhQAd0Wq0MvmY0JYSnmc The History of Algebra in the Nineteenth and Twentieth Centuries April 21 - 25, 2003 Colin Mc
From playlist Mathematics
Henry Adams (6/4/20): Descriptors of Energy Landscapes using Topological Analysis (DELTA)
Title: Descriptors of Energy Landscapes using Topological Analysis (DELTA) Abstract: Many of the properties of a chemical system are described by its energy landscape, a real-valued function defined on a high-dimensional domain. I will explain how topology, and in particular persistent ho
From playlist DELTA (Descriptors of Energy Landscape by Topological Analysis), Webinar 2020
You Could Have Invented Homology, Part 1: Topology | Boarbarktree
The first video in my series "You Could Have Invented Homology" Become a patron: https://patreon.com/boarbarktree
From playlist You Could Have Invented Homology | Boarbarktree
JunJie Wee (7/27/22): Mathematical AI in Molecular Sciences
Abstract: With great accumulations in experimental data, computing power and learning models, artificial intelligence (AI) is making great advancements in molecular sciences. Recently, the breakthrough of AlphaFold 2 in protein folding herald a new era for AI-based molecular data analysis
From playlist Applied Geometry for Data Sciences 2022
Knots, three-manifolds and instantons – Peter Kronheimer & Tomasz Mrowka – ICM2018
Plenary Lecture 11 Knots, three-manifolds and instantons Peter Kronheimer & Tomasz Mrowka Abstract: Over the past four decades, input from geometry and analysis has been central to progress in the field of low-dimensional topology. This talk will focus on one aspect of these developments
From playlist Plenary Lectures
Ippei Obayashi (3/30/22): Stable volumes for persistent homology
Persistent homology is a powerful tool to characterize the shape of data quantitatively using topology. A persistence diagram (or barcode) is the output of persistent homology. The diagram is a scatter plot on the X-Y plane, and each point on the diagram called a birth-death pair correspon
From playlist AATRN 2022
Wojtek Chacholski, "Homological algebra and persistence"
Slides available at https://www.mat.uniroma2.it/Eventi/2022/Topoldata/Slides/chacholski.pdf The talk is part of the Workshop Topology of Data in Rome (15-16/09/2022) https://www.mat.uniroma2.it/Eventi/2022/Topoldata/topoldata.php The event was organized in partnership with the Romads Cen
From playlist Workshop: Topology of Data in Rome
Peter Bubenik - Lecture 1 - TDA: Summaries and Distances
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Peter Bubenik, University of Florida Title: TDA: Summaries and Distances Abstract: Topological Data Analysis (TDA) uses tools based on topology to address challenges in data science. In these talks I will focus on the part
From playlist 38th Annual Geometric Topology Workshop (Online), June 15-17, 2021
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