Some aspects of algebraic topology: An analyst's perspective

A short survey on applications of algebraic geometry in topological data analysis.

This book gives a thorough introduction to topological data analysis (TDA), the application of algebraic topology to data science. Algebraic topology is traditionally a very specialized field of math, and most mathematicians have never been exposed to it, let alone data scientists, computer scientists, and analysts. I have three goals in writing this book. The first is to bring people up to speed who are missing a lot of the necessary background. I will describe the topics in...

These lecture notes (in Spanish) are based on a mini-course given by A. Osorno and M. Rivera at the First Colombian Geometry and Topology Meeting that took place at the Universidad Nacional de Colombia in July 2024 in Bogota. They are intended to be a guide for a first encounter with homotopy theory and simplicial methods - emphasizing intuition and important statements - accessible to students with basic knowledge of point set topology. Estas notas surgieron como parte de ...

This paper has been withdrawn by the author due a few mistakes in the paper.

This article, written in honor of Fritz Rohrlich, briefly surveys the role of topology in physics.

In this article we define a metric on C(S1; S1). Also, we give some density results in C(S^1; S^1).

This is an article on the interaction between topology and physics which will appear in 1998 in a book called: A History of Topology, edited by Ioan James and published by Elsevier-North Holland.

This note is a survey of Analysis on Metric spaces, in connection with the upcoming AMS Mathematics Research Communities program in June 2020.

The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are continuous with regard to it. We redefine synthetic topology in order to incorporate closed sets, and several generalizations are made. Real numbers are reconstructed (to suit the new background) as open Dedekind cuts. An extensive theory ...

This research addresses a new tool for data analysis known as Topological Data Analysis TDA It underlies an area of Mathematics known as Combinatorial Algebra or more recently Algebraic Topology which through making strong use of Computation Statistics Probability and Topology among other concepts extracts mathematical characteristics from a set of data that allow us associate create and infer general and quality information about them