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Topology and applications

Topology and applications

Historically, topology arose as an abstract framework for geometrical problems in Poincaré’s studies of dynamical systems. The subject has gone through a tremendous internal development and has linked up with most mathematical research areas. Topological methods originally developed for theoretical purposes alone have led to a wealth of insights, techniques and tools used to great advantage in handling problems in e.g. data analysis, dynamical systems, distributed networks and concurrency theory.

The topologists at Aalborg University have concentrated their research efforts on the analysis of spaces of paths in various frameworks: A particular motivation stems from models for parallel computation in concurrency theory and led us to a theory of directed spaces; directions reflect the flow of time, almost as in relativity theory. Tweaking methodology from “traditional” algebraic topology, we investigate the many schedules (directed paths) that the models give rise to. We analyse the topology of associated path spaces from a theoretical perspective and give contributions to the analysis and correctness verification of parallel computations.

Another field of study concerns the space of closed curves in a given base space. These loop spaces are investigated using methods from algebraic topology, differential geometry and homotopy theory.  If the base space is a Riemannian manifold, the topology of the loop space contains crucial information regarding closed geodesics on the manifold and groups of diffeomorphisms of the manifold.

A new topic relates to applications of control theory studying the existence of observable sets and searching their connection with control sets for linear control systems on Lie groups. 

The group cooperates with partners from France and Poland on establishing algorithms calculating invariants of such schedule spaces. Techniques from category theory help organizing and analysing all spaces of schedules that a directed space gives rise to. Moreover, we collaborate with researchers in France, Germany, Poland, the US and Mexico working on mathematical underpinnings of distributed (hybrid) systems.

Over the years, the group has established a growing network comprising researchers within applied topology, among others through chairmanship in an ESF research networking programme on Applied and Computational Algebraic Topology and in a recent programme at the Hausdorff Research Institute in Bonn under the same name; furthermore by establishing the new Journal of Applied and Computational Topology and cooperating in its Scientific Board. Well-established contacts to researchers in topological data analysis and in the emerging area of neuro-topology are particularly noteworthy.

Members

  • Lisbeth Fajstrup studies properties of categories of directed spaces. She is interested in connections to the well-established categorical framework for (undirected) algebraic topology. Moreover, she studies symmetry properties of models of concurrent programs.
  • Bedia Akyar Møller studies a categorical description of prismatic sets (extensions of simplicial sets) and the simplicial structure of planar rooted trees.  She is also interested in applications of soft theory to control theory.
  • Iver Ottosen’s work focuses on free loop spaces, in particular on their homology groups and stable homotopy types.    
  • Martin Raussen studies topological and combinatorial models for schedule spaces corresponding to parallel programs. He analyses their properties using topological invariants and studies their variation depending on start and end conditions - and under deformations.