22,363 research outputs found
Algebraic Topology
The chapter provides an introduction to the basic concepts of Algebraic
Topology with an emphasis on motivation from applications in the physical
sciences. It finishes with a brief review of computational work in algebraic
topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook
\emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael
Grinfeld from the University of Strathclyd
Constructive Algebraic Topology
The classical ``computation'' methods in Algebraic Topology most often work
by means of highly infinite objects and in fact +are_not+ constructive. Typical
examples are shown to describe the nature of the problem. The Rubio-Sergeraert
solution for Constructive Algebraic Topology is recalled. This is not only a
theoretical solution: the concrete computer program +Kenzo+ has been written
down which precisely follows this method. This program has been used in various
cases, opening new research subjects and producing in several cases significant
results unreachable by hand. In particular the Kenzo program can compute the
first homotopy groups of a simply connected +arbitrary+ simplicial set.Comment: 24 pages, background paper for a plenary talk at the EACA Congress of
Tenerife, September 199
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