相关论文: K-theory. An elementary introduction

These are notes from a talk at the 2010 Talbot Workshop on Twisted K-theory and Loop Groups. This particular talk is an overview of index theory from the point of view of topological K-theory. Assuming little background in analysis, but…

If we want mathematics education to be valuable to everybody, including those who do not pursue mathematics-related careers, we need to use mathematics as a training ground for certain ways of thinking. This dissertation focuses on one…

In a previous paper I gave a presentation for the Quillen higher algebraic K-groups of an exact category in terms of "acyclic binary multicomplexes". In this paper I take that presentation as a definition of the higher K-groups, generalize…

The goal of this book is to provide an introduction to the mathematical theory of Kinetically constrained models developed in the last twenty years, intended for both mathematicians and physicists.

We review various simple analytical theories for homopolymers within a unified framework. The common guideline of our approach is the Flory theory, and its various avatars, with the attempt of being reasonably self-contained. We expect this…

This is a series of lecture notes explaining topos theory and its application in physics.

The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain…

Using the Evans spectral sequence and its counter-part for real $K$-theory, we compute both the real and complex $K$-theory of several infinite families of $C^*$-algebras based on higher-rank graphs of rank $3$ and $4$. The higher-rank…

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented…

Quantum Computing is an exciting field that draws from information theory, computer science, mathematics, and quantum physics to process information in fundamentally new ways. There is an ongoing race to develop practical quantum computers…

This is a gentle introduction to Colombeau nonlinear generalized functions, a generalization of the concept of distributions such that distributions can freely be multiplied. It is intended to physicists and applied mathematicians who…

We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.

This is a graduate-level introduction to graph theory, corresponding to a quarter-long course. It covers simple graphs, multigraphs as well as their directed analogues, and more restrictive classes such as tournaments, trees and…

For T an abelian compact Lie group, we give a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology. In the appendix we give applications of this by extending results of Chang-Skjelbred and…

We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.

Deep learning algorithms have made incredible strides in the past decade, yet due to their complexity, the science of deep learning remains in its early stages. Being an experimentally driven field, it is natural to seek a theory of deep…

Quantum field theory has formed the conceptual framework of most of physics for more than sixty years. It incorporates a complete revision of our conception of the nature of matter and existence itself. Yet it is rarely taught, or even…

Higher twisted $K$-theory is an extension of twisted $K$-theory introduced by Ulrich Pennig which captures all of the homotopy-theoretic twists of topological $K$-theory in a geometric way. We give an overview of his formulation and key…

What would you teach if you had only one course to help students grasp the essence of computation and perhaps inspire a few of them to make computing a subject of further study? Assume they have the standard college prep background. This…

This expository paper is an introductory text on topological K-theory and the Atiyah-Singer index theorem, suitable for graduate students or advanced undegraduates already possessing a background in algebraic topology. The bulk of the…