Some aspects of algebraic topology: An analyst's perspective

These informal notes were prepared in connection with a lecture at a high school mathematics tournament, and provide an overview of some examples of metric spaces and a few of their basic properties.

We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.

We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of number systems.

Persistent homology is currently one of the more widely known tools from computational topology and topological data analysis. We present in this note a brief survey on the evolution of the subject. The goal is to highlight the main ideas, starting from the subject's computational inception more than 20 years ago, to the more modern categorical and representation-theoretic point of view.

In this exposition the space of at most 3-element subsets of the circle, first identified by Borsuk and Bott, is used as a motivation to introduce the readers to a variety of methods in (algebraic) topology.

These are lecture notes for a short course about spectral sequences that was held at M\'alaga, October 18--20 (2016), during the "Fifth Young Spanish Topologists Meeting". The approach was to illustrate the basic notions via fully computed examples arising from Algebraic Topology and Group Theory.

This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as the background notes to a series of seminars given at NTNU and subsequently at Sheffield. I am posting them in the hope that they will be useful to people wishing to know a little about the basics of the subject. I have tried to make them readable by anyone with a good grounding in finite dimensional differential topology. Any comments...

This is the opening article of the abstract book of conference "Set-Theoretic Topology and Topological Algebra" in honor of professor Alexander Arhangelskii on the occasion of his 80th birthday held in 2018 at Moscow State University.

This article is a short survey of the history of functional analysis for topologists.

The constructive approach to mathematics has the advantage that witnesses can be extracted from statements of existence and theorems can be unwound to give algorithms. Even better, constructive theorems can be interpreted in any topos, giving many different results for the price of one. On the other hand, you might have heard that fundamental results from topology such as Tychonoff's theorem or even the intermediate value theorem do not hold constructively, which can make the...