Homotopy theory | Vector bundles | Differential topology | Algebraic topology
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle. From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point fiber, that is, like a double cover. A special case of this is the orientable double cover of a differentiable manifold, where the corresponding line bundle is the determinant bundle of the tangent bundle (see below). The Möbius strip corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the unit interval as a fiber, or the real line. Complex line bundles are closely related to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres. In algebraic geometry, an invertible sheaf (i.e., locally free sheaf of rank one) is often called a line bundle. Every line bundle arises from a divisor with the following conditions (I) If X is reduced and irreducible scheme, then every line bundle comes from a divisor. (II) If X is projective scheme then the same statement holds. (Wikipedia).
Calculus 3: Line Integrals (4 of 44) What is a Line Integral? NOT TO BE CONFUSED WITH
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the line integral is NOT the length of a line (curve). Next video in the series can be seen at: https://youtu.be/yUHGDBYxGe0
From playlist CALCULUS 3 CH 6 LINE INTEGRALS
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
Physics Ch 67.1 Advanced E&M: Review Vectors (48 of TBD) What is a Line Integral?
Visit http://ilectureonline.com for more math and science lectures! To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn that a line integral is the integral of the dot product of a vector and an infinitesimal displacement along a path from
From playlist PHYSICS 67.1 ADVANCED E&M VECTORS & FIELDS
Free ebook http://tinyurl.com/EngMathYT How to integrate over 2 curves. This example discusses the additivity property of line integrals (sometimes called path integrals).
From playlist Engineering Mathematics
Classification of Line Bundles on Schemes
Here I talk about the classifying cohomology class associated to a line bundle on scheme. This is an element of first cohomology. If you can here looking for Chern classes let me know and I'll slap something together for you. You can see more about those in, say, the Riemann-Roch videos.
From playlist Deformation/Obstruction Theory of Line Bundles
Free ebook http://tinyurl.com/EngMathYT How to integrate over curves to produce a line integral (involving scalar valued functions). I discuss an example and geometric interpretation.
From playlist Engineering Mathematics
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?
Calculus 3: Line Integrals (1 of 44) What is a Line Integral?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain some of the 4 different types of line integral. Next video in the series can be seen at: https://youtu.be/fwv1B0vryJU
From playlist CALCULUS 3 CH 6 LINE INTEGRALS
Vector Bundles in Algebraic Geometry by Poorna Narayanan
ICTS In-house 2022 Organizers: Chandramouli, Omkar, Priyadarshi, Tuneer Date and Time: 20th to 22nd April, 2022 Venue: Ramanujan Hall inhouse@icts.res.in An exclusive three-day event to exchange ideas and research topics amongst members of ICTS.
From playlist ICTS In-house 2022
Marc Levine: Refined enumerative geometry (Lecture 4)
The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Lecture 4: Characteristic classes in Witt-cohomology Classical enumerative geometry relies heavily on the theory of Chern classes of vector bundles and the splitting principle, which
From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
Introduction to Line Integrals
Introduction to Line Integrals If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Thank you:)
From playlist Calculus 3
Hilbert Space Techniques in Complex Analysis and Geometry (Lecture 1) by Dror Varolin
PROGRAM CAUCHY-RIEMANN EQUATIONS IN HIGHER DIMENSIONS ORGANIZERS: Sivaguru, Diganta Borah and Debraj Chakrabarti DATE: 15 July 2019 to 02 August 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Complex analysis is one of the central areas of modern mathematics, and deals with holomo
From playlist Cauchy-Riemann Equations in Higher Dimensions 2019
Zhaoting Wei: Determinant line bundles and cohesive modules
Talk by Zhaoting Wei in Global Noncommutative Geometry Seminar (Americas) https://www.math.wustl.edu/~xtang/NCG-Seminar on December 16, 2020
From playlist Global Noncommutative Geometry Seminar (Americas)
Modular spectral covers and Hecke eigensheaves on ... (Lecture 3) by Tony Pantev
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
An introduction to spectral data for Higgs bundles.. by Laura Schaposnik
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
This lecture give an overview of Chern classes of nonsingular algebraic varieties. We first define the Chern class of a lline bundle by looking at the cycle of zeros of a section. Then we define Chern classes of higher rank vector bundles by looking at the line bundle O(1) over the corresp
From playlist Algebraic geometry: extra topics
An introduction to spectral data for Higgs bundles.. by Laura Schaposnik
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
Vector Bundles on Riemann Surfaces and Metric Graphs by Martin Ulirsch
PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is the study of
From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)
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
B. Berndtsson - The curvature of (higher) direct images
I will first discuss some earlier work on the curvature of direct images of adjoint line bundles under a smooth proper fibration, or more generally a surjective map and (maybe) some of its applications. Then I will present a general formula for the curvature of higher direct images. Th
From playlist Complex analytic and differential geometry - a conference in honor of Jean-Pierre Demailly - 6-9 juin 2017