From Computable Analysis to Computable Algebraic Topology

• Applicant: Professor Dr. Vasco Brattka
• Subject Area: Mathematics ; Theoretical Computer Science
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 554999067

The purpose of this project is the development of some aspects of computable algebraic topology within the framework of computable analysis. Computable analysis aims to study computability questions within analysis. A prototypical result yields, for instance, the classification of the computational content of the Brouwer fixed-point theorem. Results of this nature are most naturally proved with certain methods of algebraic topology, similarly as their classical counterparts. However, so far no systematic investigation of computability-theoretic questions within algebraic topology exists and we aim to approach such an investigation from the angle of applications within computable analysis. This requires, for one, to understand the nature of the topological spaces that can be handled in this context. Secondly, basic constructions in algebraic topology such as that of forming the fundamental group or more generally of homotopy and homology groups should be developed from a computability-theoretic perspective. Another interesting example with important applications is calculating the winding number of a curve. In another subproject we aim to classify the computational content of theorems related to algebraic topology. Such theorems are, on the one hand, theorems from the core of algebraic topology, such as the Seifert-van Kampen theorem that possibly allows the calculation of certain homotopy groups and, on the other hand, theorems that can be proved with the help of algebraic topology, such as fixed-point theorems. Finally, we also plan to address questions of computable category theory and algebra that occur naturally in this context. This project belongs to the domain of mathematical logic and its subfield of computability theory, and more specifically to computable analysis.

• DFG Programme: Research Grants