C*-algebras | K-theory

KK-theory

In mathematics, KK-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician in 1980. It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C*-algebras by Lawrence G. Brown, Ronald G. Douglas, and Peter Arthur Fillmore in 1977. In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of nuclear C*-algebras, as it was the key to the solutions of many problems in operator K-theory, such as, for instance, the mere calculation of K-groups. Furthermore, it was essential in the development of the Baum–Connes conjecture and plays a crucial role in noncommutative topology. KK-theory was followed by a series of similar bifunctor constructions such as the E-theory and the , most of them having more category-theoretic flavors, or concerning another class of algebras rather than that of the separable C*-algebras, or incorporating group actions. (Wikipedia).

Video thumbnail

Arthur Bartels: K-theory of group rings (Lecture 1)

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Arthur Bartels: K-theory of group rings The Farrell-Jones Conjecture predicts that the K-theory of group rings RG can be computed in terms of K-theory of group rings RV where V vari

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

Video thumbnail

Guoliang Yu: Quantitative operator K-theory and its applications

Abstract: In this lecture, I will give an introduction to quantitative operator K-theory and apply it to compute K-theory of operator algebras which naturally arise from geometry. I will discuss applications to asymptotic behavior of positive scalar curvature when the dimension of the mani

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

Video thumbnail

What IS Quantum Field Theory? (For Dummies?)

The framework in which quantum mechanics and special relativity are successfully reconciled is called quantum field theory. Heres how you can easily understand Quantum Field Theory. Support me on patreon so that i can keep on making videos! https://www.patreon.com/quantasy In theoretical

From playlist Quantum Field Theory

Video thumbnail

Clustering 1: monothetic vs. polythetic

Full lecture: http://bit.ly/K-means The aim of clustering is to partition a population into sub-groups (clusters). Clusters can be monothetic (where all cluster members share some common property) or polythetic (where all cluster members are similar to each other in some sense).

From playlist K-means Clustering

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

Arthur Bartels: K-theory of group rings (Lecture 3)

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. The Farrell-Jones Conjecture predicts that the K-theory of group rings RG can be computed in terms of K-theory of group rings RV where V varies over the collection of virtually cycli

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

Video thumbnail

Rufus Willett: Decomposable C*-algebras and the UCT

Talk by Rufus Willett in in Global Noncommutative Geometry Seminar (Americas) https://globalncgseminar.org/talks/tba-25/ on March 11, 2022.

From playlist Global Noncommutative Geometry Seminar (Americas)

Video thumbnail

Ralf Meyer: On the classification of group actions on C*-algebras up to equivariant KK-equivalence

Talk by Ralf Meyer in Global Noncommutative Geometry Seminar (Europe) http://www.noncommutativegeometry.nl/ncgseminar/ on November 10, 2020.

From playlist Global Noncommutative Geometry Seminar (Europe)

Video thumbnail

Quantum Theory - Full Documentary HD

Check: https://youtu.be/Hs_chZSNL9I The World of Quantum - Full Documentary HD http://www.advexon.com For more Scientific DOCUMENTARIES. Subscribe for more Videos... Quantum mechanics (QM -- also known as quantum physics, or quantum theory) is a branch of physics which deals with physica

From playlist TV Appearances

Video thumbnail

Jamie Gabe: A new approach to classifying nuclear C*-algebras

Talk in the global noncommutative geometry seminar (Europe), 9 February 2022

From playlist Global Noncommutative Geometry Seminar (Europe)

Video thumbnail

Koen van den Dungen: Localisations and the Kasparov product in unbounded KK-theory

Talk by Koen van den Dungen in Global Noncommutative Geometry Seminar (Europe) http://www.noncommutativegeometry.nl/ncgseminar/ on May 19, 2021

From playlist Global Noncommutative Geometry Seminar (Europe)

Video thumbnail

(ML 16.1) K-means clustering (part 1)

Introduction to the K-means algorithm for clustering.

From playlist Machine Learning

Video thumbnail

How to Cluster Data in MATLAB

Clustering is the process of grouping a set of data given a certain criterion. In this way it is possible to define subgroups of data, called clusters, that share common characteristics. Determining the internal structure of the data is important in exploratory data analysis, but is also u

From playlist “How To” with MATLAB and Simulink

Video thumbnail

Christopher Schafhauser: On the classification of nuclear simple C*-algebras, Lecture 4

Mini course of the conference YMC*A, August 2021, University of Münster. Abstract: A conjecture of George Elliott dating back to the early 1990’s asks if separable, simple, nuclear C*-algebras are determined up to isomorphism by their K-theoretic and tracial data. Restricting to purely i

From playlist YMC*A 2021

Video thumbnail

José Carrión: "The abstract approach to classifying C*-algebras"

Actions of Tensor Categories on C*-algebras 2021 Mini Course: "The abstract approach to classifying C*-algebras" José Carrión - Texas Christian University Institute for Pure and Applied Mathematics, UCLA January 21, 2021 For more information: https://www.ipam.ucla.edu/atc2021

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

Video thumbnail

Christopher Schafhauser: "Non-stable extension theory and the classification of C∗-algebras"

Actions of Tensor Categories on C*-algebras 2021 "Non-stable extension theory and the classification of C∗-algebras" Christopher Schafhauser - University of Nebraska-Lincoln Abstract: Over the last decade, much of the progress in the classification and regularity theory of simple, nuclea

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

Video thumbnail

The KL Divergence : Data Science Basics

understanding how to measure the difference between two distributions Proof that KL Divergence is non-negative : https://www.youtube.com/watch?v=LOwj7UxQwJ0&t=520s My Patreon : https://www.patreon.com/user?u=49277905 0:00 How to Learn Math 1:57 Motivation for P(x) / Q(x) 7:21 Motivation

From playlist Data Science Basics

Related pages

K-homology | Functor | Cap product | Noncommutative topology | Cup product | Nuclear C*-algebra | Mathematics | Operator K-theory | Separable space | Ideal (ring theory) | Fredholm module | Ronald G. Douglas | Baum–Connes conjecture | Category theory | C*-algebra | Category (mathematics) | Atiyah–Singer index theorem