Historical Foundations of Topology: Concepts and Contexts

Historical Foundations of Topology: Concepts and Contexts

I am teaching the course Historical Foundations of Topology: Concepts and Contexts in fall 2025 at the University of Zurich. The course will be given in English.

• Lecture: Tuesdays, 8am–10am (Y27 H12)

• Material:   

• “History of Topology”, I. M. James, Elsevier BV (1999)

• “Handbook of the History of  General Topology”, Vol. 1 & 2, C. E. Aull and R. Lowen, Kluwer Academic Publishers (1998)

• “Analysis situs”, H. Poincaré, Journal de l’École Polytechnique, Vol. 1 No. 2, pp. 1–123 (1895)

• “A History of Algebraic and Differential Topology”, J. Dieudonné, Birkhäuser Basel (2009)

• Description of the course: This course explores the historical development of topology, tracing its transformation from intuitive geometric notions to a central field in modern mathematics. We will investigate key conceptual milestones such as e.g. the formalization of topological dimension theory, the evolution of the manifold concept, the emergence of homotopy theory, and the development of complexes from early simplicial constructions to CW-complexes. The course also covers the history of differential forms and their relationship to topology and geometry, with particular emphasis on the foundational contributions of Henri Poincaré, including his Analysis Situs and the birth of algebraic topology. Students will engage directly with both primary sources and secondary historical analyses to understand how topological thinking developed in its mathematical, philosophical, and scientific contexts.

A central feature of the course is student participation through guided research projects. Students will explore a selected topic suggested by the instructor and produce a research essay and presentation. The use of AI tools for research and idea development is encouraged, and guidance will be provided to support ethical and effective use. The proper work with LaTeX will be communicated and used throughout the course for assignments, promoting clear and professional mathematical cal typesetting. The research essays will be collected for a potential future textbook, giving students the opportunity to actively contribute to a possible published reference. 

This course is ideal for students in mathematics who want to dive a bit into the history of science and the development of modern mathematical ideas. It is also recommended to students who want to do the teaching diploma in mathematics.