Fiber bundles | Group actions (mathematics) | Differential geometry
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with 1. * An action of on , analogous to for a product space. 2. * A projection onto . For a product space, this is just the projection onto the first factor, . Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of . Likewise, there is not generally a projection onto generalizing the projection onto the second factor, that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a principal bundle is the frame bundle of a vector bundle , which consists of all ordered bases of the vector space attached to each point. The group in this case, is the general linear group, which acts on the right in the usual way: by changes of basis. Since there is no natural way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section. Principal bundles have important applications in topology and differential geometry and mathematical gauge theory. They have also found application in physics where they form part of the foundational framework of physical gauge theories. (Wikipedia).
The TRUTH about TENSORS, Part 9: Vector Bundles
In this video we define vector bundles in full abstraction, of which tangent bundles are a special case.
From playlist The TRUTH about TENSORS
What is a Manifold? Lesson 12: Fiber Bundles - Formal Description
This is a long lesson, but it is not full of rigorous proofs, it is just a formal definition. Please let me know where the exposition is unclear. I din't quite get through the idea of the structure group of a fiber bundle fully, but I introduced it. The examples in the next lesson will h
From playlist What is a Manifold?
Introduction to Fiber Bundles part 1: Definitions
We give the definition of a fiber bundle with fiber F, trivializations and transition maps. This is a really basic stuff that we use a lot. Here are the topics this sets up: *Associated Bundles/Principal Bundles *Reductions of Structure Groups *Steenrod's Theorem *Torsor structure on arith
From playlist Fiber bundles
Introduction to Fiber Bundles Part 4: Torsor Interlude
Torsors and Principal Homogeneous Spaces. What is the difference?
From playlist Fiber bundles
Introduction to Fiber Bundles Part 3: Associated Bundles and Amalgamated Products
This is an incomplete introduction here. The basic idea is that the associated principal bundle knows all. This should be obvious since all bundles with G-structure are classified by H^1(X,G) --- it turns out you can recover your original bundle from a principal bundle by taking "amalgamat
From playlist Fiber bundles
Introduction to the Principal Unit Normal Vector
Introduction to the Principal Unit Normal Vector
From playlist Calculus 3
The TRUTH about TENSORS, Part 8: Tangent bundles & vector fields
In this video, we discuss the definition of the tangent bundle of a manifold, which in turns inspires the more general definition of vector bundles, to be discussed in the next video. The notion of tangent bundle, further lets us formalize our intuitive notion of vector fields.
From playlist The TRUTH about TENSORS
The TRUTH about TENSORS, Part 10: Frames
What do the octonions have to do with spheres? Skip to the end of the video to find out!
From playlist The TRUTH about TENSORS
What Is A Tensor Lesson #1: Elementary vector spaces
We define a vector space and lay the foundation of a solid understanding of tensors.
From playlist What is a Tensor?
Equivariant principal bundle over toric variety by Arijit Dey
Higgs bundles URL: http://www.icts.res.in/program/hb2016 DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore DESCRIPTION: Higgs bundles arise as solutions to noncompact analog of the Yang-Mills equation. Hitchin showed that irreducible solutio
From playlist Higgs Bundles
Equivariant principal bundles on toric varieties- Part 1 by Mainak Poddar
DISCUSSION MEETING ANALYTIC AND ALGEBRAIC GEOMETRY DATE:19 March 2018 to 24 March 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore. Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions be
From playlist Analytic and Algebraic Geometry-2018
Minhyong Kim: Recent progress on the effective Mordell problem
SMRI Algebra and Geometry Online: Minhyong Kim (University of Warwick) Abstract: In 1983, Gerd Faltings proved the Mordell conjecture stating that curves of genus at least two have only finitely many rational points. This can be understood as the statement that most polynomial equations
From playlist SMRI Algebra and Geometry Online
Index Theory, survey - Stephan Stolz [2018]
TaG survey series These are short series of lectures focusing on a topic in geometry and topology. May_8_2018 Stephan Stolz - Index Theory https://www3.nd.edu/~math/rtg/tag.html (audio fixed)
From playlist Mathematics
Moduli Spaces of Principal 2-group Bundles and a Categorification of the Freed.. by Emily Cliff
Program Quantum Fields, Geometry and Representation Theory 2021 (ONLINE) ORGANIZERS: Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pandi
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Branched Holomorphic Cartan Geometries by Sorin Dumitrescu
DISCUSSION MEETING ANALYTIC AND ALGEBRAIC GEOMETRY DATE:19 March 2018 to 24 March 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore. Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions be
From playlist Analytic and Algebraic Geometry-2018
Lukasz Fidkowski - Symmetry Protected Topological phases, cobordism, and QCA - IPAM at UCLA
Recorded 02 September 2021. Lukasz Fidkowski of the University of Washington presents "Symmetry Protected Topological phases, cobordism, and QCA" at IPAM's Graduate Summer School: Mathematics of Topological Phases of Matter. Abstract: We give an overview of the cobordism classification of
From playlist Graduate Summer School 2021: Mathematics of Topological Phases of Matter
Introduction to Fiber Bundles Part 5.1: Steenrod's Theorem
This video is about how to reduce structure groups of fiber bundles.
From playlist Fiber bundles
What is a Manifold? Lesson 13: The tangent bundle - an illustration.
What is a Manifold? Lesson 13: The tangent bundle - an illustration. Here we have a close look at a complete example using the tangent bundle of the manifold S_1. Next lesson we look at the Mobius strip as a fiber bundle.
From playlist What is a Manifold?
Elmar Schrohe: Fourier integral operators on manifolds with boundary and ...
Full Title: Fourier integral operators on manifolds with boundary and the Atiyah-Weinstein index theorem The lecture was held within the framework of the Hausdorff Trimester Program Non-commutative Geometry and its Applications. (18.12.2014)
From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"