Computation and Reconfiguration in Low-Dimensional Topological Spaces

Motivtion

Low-dimensional topological structures are pervasive both in pure mathematics and more applied and natural settings: knots, curves, surfaces and embedded graphs occur as DNA strands, trajectories, meshes, maps and many other examples. This ubiquity leads to very similar objects and questions being studied independently in different communities: from geometric topology, graph algorithms, computational geometry and topology to graph drawing. In all these groups, there is a strong need and interest to develop efficient algorithms that harness the structure of low-dimensional spaces.

The goal of this Dagstuhl Seminar is to bring together researchers from different communities who are working on low-dimensional topological spaces, in order to foster collaborations and synergies. Indeed, while the mathematical study of these objects has a rich and old history, the study of their algorithmic properties is still in its infancy, and new questions and problems keep coming from theoretical computer science or more applied fields, yielding a fresh and renewed perspective on computation in topological spaces.

This seminar is a follow-up to the Dagstuhl Seminars 17072: “Applications of Topology to the Analysis of 1-Dimensional Objects” and 19352: "Computation in Low-Dimensional Geometry and Topology". The first previous seminar focused on the analysis of one-dimensional objects, and the second one widened this to low-dimensional objects and included the evolution of topological objects over time. While these topics are still very current and will not be excluded from the discussions, for the third iteration, we plan to give a new impetus to the seminar by placing a particular emphasis on the topics related to reconfiguration. How can one structure be changed into another? How far apart are two structures? Such questions lie at the heart of various geometric problems like computing the Fréchet distance as a way to quantify curve similarity, or morphing between two versions of a common graph. In many cases, the combinatorics and the geometry of a reconfiguration space also emerged as important objects of study: examples are associahedra and the flip graph of triangulations or the curve complex in geometric topology.

The main emphasis of the seminar will be to develop collaborations by working together on open problems. To foster communication, we will start with a few longer talks given by speakers that span the various areas, with the goal of quickly developing a strong set of open problems for groups of participants to work on.