Applied Topology

Applied Topology

Topology arose historically as an abstract framework for geometrical problems in Poincaré’s studies of dynamical systems. It has gone through a tremendous internal development and linked up with most mathematical research areas. In recent years, we have witnessed an explosion of insights, techniques and tools from algebraic topology used to great advantage in examining computation problems in data analysis, dynamical systems, distributed networks and concurrency theory.

Topological methodology has been successfully applied to concurrency, the study of parallel computations. A theory of directed topological spaces – with direction reflecting the time flow – had to be developed: in general and, in depth, for particular geometric/combinatorial models for parallel programs. The topology of associated path spaces modelling executions is investigated theoretically and for the purpose of analyzing and verifying correctness of parallel computations. Algorithms for calculations of invariants of execution spaces are under development in collaboration with partners from France and Poland. Techniques from category theory help to organize and analyse directed spaces and associated path spaces.

Topics of recent and current interest include:

  • Algebraic topological invariants of loop spaces
  • Homological studies of string topology
  • Properties of spaces of directed paths
  • Simplicial models of execution spaces
  • Lattices of d-structures
  • Cross-fertilization with related areas in applied and computational algebraic topology (persistence, distributed networks etc.)

Members of the research group and their research interest:

  • Iver Ottosen’s work focuses on free loop spaces, in particular on homology computations and on Morse theoretical methods.
  • Lisbeth Fajstrup works on broadening the range of applications for directed spaces from the original concurrency models to more general settings. Also for those, the detection of deadlocks and an investigation of trace spaces are of central interest.
  • Martin Raussen works on geometric/combinatorial models for execution spaces corresponding to parallel programs, on invariants and properties for these and their variation under deformations.