Homotopy theory | Algebraic topology
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space is denoted by . (Wikipedia).
Definition of a group Lesson 24
In this video we take our first look at the definition of a group. It is basically a set of elements and the operation defined on them. If this set of elements and the operation defined on them obey the properties of closure and associativity, and if one of the elements is the identity el
From playlist Abstract algebra
A group is (in a sense) the simplest structure in which we can do the familiar tasks associated with "algebra." First, in this video, we review the definition of a group.
From playlist Modern Algebra - Chapter 15 (groups)
Group Definition (expanded) - Abstract Algebra
The group is the most fundamental object you will study in abstract algebra. Groups generalize a wide variety of mathematical sets: the integers, symmetries of shapes, modular arithmetic, NxM matrices, and much more. After learning about groups in detail, you will then be ready to contin
From playlist Abstract Algebra
Jacob explains the fundamental concepts in group theory of what groups and subgroups are, and highlights a few examples of groups you may already know. Abelian groups are named in honor of Niels Henrik Abel (https://en.wikipedia.org/wiki/Niels_Henrik_Abel), who pioneered the subject of
From playlist Basics: Group Theory
Algebraic topology: Calculating the fundamental group
This lecture is part of an online course on algebraic topology. We calculate the fundamental group of several spaces, such as a ficure 8, or the complement of a circle in R^3, or the group GL3(R). For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EF
From playlist Algebraic topology
Symmetric Groups (Abstract Algebra)
Symmetric groups are some of the most essential types of finite groups. A symmetric group is the group of permutations on a set. The group of permutations on a set of n-elements is denoted S_n. Symmetric groups capture the history of abstract algebra, provide a wide range of examples in
From playlist Abstract Algebra
Now that we know what a quotient group is, let's take a look at an example to cement our understanding of the concepts involved.
From playlist Abstract algebra
This lecture is part of an online math course on group theory. We review free abelian groups, then construct free (non-abelian) groups, and show that they are given by the set of reduced words, and as a bonus find that they are residually finite.
From playlist Group theory
(Fundamental Group of an Elliptic Curve) = (Tate Module)
Here we talk about how the abelianization of the geometric algebraic fundmental group is really the Tate module.
From playlist Fundamental Groups
Sergio Zamora (1/20/23): The lower semi-continuity of \pi_1 and nilpotent structures in persistence
When a sequence of compact geodesic spaces X_i converges to a compact geodesic space X, under minimal assumptions there are surjective morphisms $\pi_1(X_i) \to \pi_1(X)$ for i large enough. In particular, a limit of simply connected spaces is simply connected. This is clearly not true for
From playlist Vietoris-Rips Seminar
Lie Groups and Lie Algebras: Lesson 39 - The Universal Covering Group
Lie Groups and Lie Algebras: Lesson 39 - The Universal Covering Group We are finally in position to understand the nature of the Universal Covering Group and its connection to all the Lie groups which share a single Lie algebra. This is a critical lecture! In this lecture we simply state
From playlist Lie Groups and Lie Algebras
Hyperbolicity and Fundamental groups (Lecture 1) Yohan Brunebarbe
PROGRAM : TOPICS IN BIRATIONAL GEOMETRY ORGANIZERS : Indranil Biswas and Mahan Mj DATE : 27 January 2020 to 31 January 2020 VENUE : Madhava Lecture Hall, ICTS Bangalore Birational geometry is one of the current research trends in fields of Algebraic Geometry and Analytic Geometry. It ca
From playlist Topics In Birational Geometry
On the algebraic fundamental group of surfaces of general type by Margarida Lopes
Algebraic Surfaces and Related Topics PROGRAM URL : http://www.icts.res.in/program/AS2015 DESCRIPTION : This is a joint program of ICTS with TIFR, Mumbai and KIAS, Seoul. The theory of surfaces has been the cradle to many powerful ideas in Algebraic Geometry. The problems in this area
From playlist Algebraic Surfaces and Related Topics
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
Philippe Eyssidieux: Examples of Kähler groups
Abstract : Malgré les succès de la théorie de Hodge non abélienne de Corlette-Simpson pour exclure que de nombreux groupes de présentation finie soient groupes fondamentaux de variétés projectives lisses (ou des groupes Kähleriens), les techniques de construction manquent. La construction
From playlist Analysis and its Applications
Dejan Govc (03/15/2023): Fundamental groups of small simplicial complexes
Title: Fundamental groups of small simplicial complexes Abstract: Minimal triangulations of manifolds have been widely studied, but many problems remain open. I will present a computation of the fundamental groups of simplicial complexes with at most 8 vertices and explain how it can be u
From playlist AATRN 2023
Lauren Ruth: "Von Neumann Equivalence"
Actions of Tensor Categories on C*-algebras 2021 "Von Neumann Equivalence" Lauren Ruth - Mercy College, Mathematics Abstract: We introduce a new equivalence relation on groups, which we call von Neumann equivalence, that is coarser than both measure equivalence and W*-equivalence. Our ge
From playlist Actions of Tensor Categories on C*-algebras 2021
Lie groups: Lie groups and Lie algebras
This lecture is part of an online graduate course on Lie groups. We discuss the relation between Lie groups and Lie algebras, and give several examples showing how they behave differently. Lie algebras turn out to correspond more closely to the simply connected Lie groups. We then explain
From playlist Lie groups
Abstract Algebra - 11.1 Fundamental Theorem of Finite Abelian Groups
We complete our study of Abstract Algebra in the topic of groups by studying the Fundamental Theorem of Finite Abelian Groups. This tells us that every finite abelian group is a direct product of cyclic groups of prime-power order. Video Chapters: Intro 0:00 Before the Fundamental Theorem
From playlist Abstract Algebra - Entire Course