K-theory

Twisted K-theory

In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory. More specifically, twisted K-theory with twist H is a particular variant of K-theory, in which the twist is given by an integral 3-dimensional cohomology class. It is special among the various twists that K-theory admits for two reasons. First, it admits a geometric formulation. This was provided in two steps; the first one was done in 1970 (Publ. Math. de l'IHÉS) by Peter Donovan and Max Karoubi; the second one in 1988 by Jonathan Rosenberg in Continuous-Trace Algebras from the Bundle Theoretic Point of View. In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theory, see K-theory (physics). In the broader context of K-theory, in each subject it has numerous isomorphic formulations and, in many cases, isomorphisms relating definitions in various subjects have been proven. It also has numerous deformations, for example, in abstract algebra K-theory may be twisted by any integral cohomology class. (Wikipedia).

Video thumbnail

What is Spin?

Spin in quantum mechanics is an incredibly interesting property. However, it can be very difficult to understand what exactly it is. In this video, we dispel some misconceptions about spin as well as answer some of the more frequently asked questions about spin. #physics #quantum

From playlist Quantum Mechanics

Video thumbnail

You've Heard of SPIN - But How Is it Encoded in the Math of Quantum Physics? Parth G

The concept of Spin is hard, but the mathematics is actually quite simple! In this video I wanted to take a look at how we build up our mathematical representation (or at least one of them) of quantum mechanical spin. To do this, we'll start by looking at the spin of an electron, and unde

From playlist Quantum Physics by Parth G

Video thumbnail

Math of the twisting somersault

Mathematical models can be used to obtain an understanding of the mechanics of twists during somersaults. The twisting somersault can be described by a formula, which factors in the airborne time of the diver, the time spent in various stages of the dive, the number of somersaults, the num

From playlist What is math used for?

Video thumbnail

Vector Calculus: Understanding Curl

Some formal and informal intuition regarding curl, a vector calculus concept.

From playlist Summer of Math Exposition Youtube Videos

Video thumbnail

What is the difference between rotating clockwise and counter clockwise

👉 Learn how to rotate a figure and different points about a fixed point. Most often that point or rotation will be the original but it is important to understand that it does not always have to be at the origin. When rotating it is also important to understand the direction that you will

From playlist Transformations

Video thumbnail

Determining clockwise vs counter clockwise rotations

👉 Learn how to rotate a figure and different points about a fixed point. Most often that point or rotation will be the original but it is important to understand that it does not always have to be at the origin. When rotating it is also important to understand the direction that you will

From playlist Transformations

Video thumbnail

Physics, Torque (1 of 13) An Explanation

Explains what torque is, the definition, how it is described and the metric units. Also presented are two examples of how to calculate the torque produced by a force. Torque is a turning force. It is a measure of how much force acting on an object that causes the object to rotate. The ob

From playlist Mechanics

Video thumbnail

General Relativity to Quantum gravity (Intro for dummies?)

Support me on Patreon: https://www.patreon.com/quahntasy A brief introduction to Newtonian Gravity(or Gravity for dummies) ,General Relativity and Quantum gravity (Loop Quantum gravity). The whole idea of general relativity is that matter influence spacetime curvature. Incorporating this i

From playlist Gravity

Video thumbnail

Cycloid

#Cycloid: A curve traced by a point on a circle rolling in a straight line. (A preview of this Sunday's video.)

From playlist Miscellaneous

Video thumbnail

Alex Fok, Equvariant twisted KK-theory of noncompact Lie groups

Global Noncommutative Geometry Seminar(Asia-Pacific), Oct. 25, 2021

From playlist Global Noncommutative Geometry Seminar (Asia and Pacific)

Video thumbnail

Teena Gerhardt - 3/3 Algebraic K-theory and Trace Methods

Algebraic K-theory is an invariant of rings and ring spectra which illustrates a fascinating interplay between algebra and topology. Defined using topological tools, this invariant has important applications to algebraic geometry, number theory, and geometric topology. One fruitful approac

From playlist Summer School 2020: Motivic, Equivariant and Non-commutative Homotopy Theory

Video thumbnail

Michael Groechenig - Complex K-theory of Dual Hitchin Systems

Let G and G’ be Langlands dual reductive groups (e.g. SL(n) and PGL(n)). According to a theorem by Donagi-Pantev, the generic fibres of the moduli spaces of G-Higgs bundles and G’-Higgs bundles are dual abelian varieties and are therefore derived-equivalent. It is an interesting open probl

From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory

Video thumbnail

Tudor Dimofte - 3d SUSY Gauge Theory and Quantum Groups at Roots of Unity

Topological twists of 3d N=4 gauge theories naturally give rise to non-semisimple 3d TQFT's. In mathematics, prototypical examples of the latter were constructed in the 90's (by Lyubashenko and others) from representation categories of small quantum groups at roots of unity; they were rece

From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory

Video thumbnail

Supersymmetric Ground States of 3rd N=4 Theories on a Riemann surface by Heeyeon Kim

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)

Video thumbnail

Marc Levine: Refined enumerative geometry (Lecture 1)

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Marc Levine: Refined enumerative geometry Abstract: Lecture 1: Milnor-Witt sheaves, motivic homotopy theory and Chow-Witt groups We review the Hoplins-Morel construction of the Miln

From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"

Video thumbnail

Ulrich Pennig: "Fell bundles, Dixmier-Douady theory and higher twists"

Actions of Tensor Categories on C*-algebras 2021 "Fell bundles, Dixmier-Douady theory and higher twists" Ulrich Pennig - Cardiff University, School of Mathematics Abstract: Classical Dixmier-Douady theory gives a full classification of C*-algebra bundles with compact operators as fibres

From playlist Actions of Tensor Categories on C*-algebras 2021

Video thumbnail

Parametrization of the 2-, 3-, 4-, and 5-Selmer Groups of Elliptic... (Lecture 2) by Arul Shankar

PROGRAM ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (HYBRID) ORGANIZERS: Ashay Burungale (CalTech/UT Austin, USA), Haruzo Hida (UCLA), Somnath Jha (IIT Kanpur) and Ye Tian (MCM, CAS) DATE: 08 August 2022 to 19 August 2022 VENUE: Ramanujan Lecture Hall and online The program pla

From playlist ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (2022)

Video thumbnail

Tensionless AdS/CFT (Lecture 2) by Matthias Gaberdiel

PROGRAM KAVLI ASIAN WINTER SCHOOL (KAWS) ON STRINGS, PARTICLES AND COSMOLOGY (ONLINE) ORGANIZERS Francesco Benini (SISSA, Italy), Bartek Czech (Tsinghua University, China), Dongmin Gang (Seoul National University, South Korea), Sungjay Lee (Korea Institute for Advanced Study, South Korea

From playlist Kavli Asian Winter School (KAWS) on Strings, Particles and Cosmology (ONLINE) - 2022

Video thumbnail

Intro to General Relativity Learning Playlist

In this course, we'll take you step-by-step to an intuitive understanding of the ideas behind Einstein's theory of general relativity.

From playlist Curved Spacetime in General Relativity

Video thumbnail

Calista Bernard - Applications of twisted homology operations for E_n-algebras

An E_n-algebra is a space equipped with a multiplication that is commutative up to homotopy. Such spaces arise naturally in geometric topology, number theory, and mathematical physics; some examples include classifying spaces of braid groups, spaces of long knots, and classifying spaces of

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

Related pages

Michael Atiyah | Operator theory | Hilbert space | Abstract algebra | Atiyah–Hirzebruch spectral sequence | Superstring theory | Classifying space | Projective unitary group | String theory | Wess–Zumino–Witten model | Eilenberg–MacLane space | Fredholm operator | Bundle gerbe | K-theory | K-theory (physics) | Massey product | Associated bundle | Algebraic topology