TOP
Suche auf der Schloss Dagstuhl Webseite
Sie suchen nach Informationen auf den Webseiten der einzelnen Seminare? - Dann:
Nicht fündig geworden? - Einige unserer Dienste laufen auf separaten Webseiten mit jeweils eigener Suche. Bitte beachten Sie folgende Liste:
Schloss Dagstuhl - LZI - Logo
Schloss Dagstuhl Services
Seminare
Innerhalb dieser Seite:
Externe Seiten:
  • DOOR (zum Registrieren eines Dagstuhl Aufenthaltes)
  • DOSA (zum Beantragen künftiger Dagstuhl Seminare oder Dagstuhl Perspektiven Workshops)
Publishing
Innerhalb dieser Seite:
Externe Seiten:
dblp
Innerhalb dieser Seite:
Externe Seiten:
  • die Informatik-Bibliographiedatenbank dblp


Dagstuhl-Seminar 22062

Computation and Reconfiguration in Low-Dimensional Topological Spaces

( 06. Feb – 11. Feb, 2022 )


Permalink
Bitte benutzen Sie folgende Kurz-Url zum Verlinken dieser Seite: https://www.dagstuhl.de/22062

Organisatoren

Kontakt

Gemeinsame Dokumente



Programm

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.


Summary

This seminar was proposed as a followup 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 goal of these seminars was to bring together researchers from different communities who are working on low-dimensional topological spaces (curves, embedded graphs, knots, surfaces, three-manifolds), 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 and more applied fields, yielding a fresh and renewed perspective on computation in topological spaces.

The success of previous seminars demonstrated that research in low-dimensional topology is very active and fruitful, and also that there was a strong demand for a new seminar gathering researchers from the various involved communities, namely geometric topology and knot theory, computational geometry and topology, all the way to graph drawing and trajectory analysis.

For this iteration we placed a particular emphasis on topics related to geometric and topological 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 such as 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 include associahedra, the flip graphs of triangulations, and the curve complexes in geometric topology.

The seminar started with four overview talks given by researchers in geometric topology, computational geometry, topological dynamics, and graph drawing to motivate and propose open problems that would fit the diverse backgrounds of participants and the specific focus on reconfiguration chosen for this year's workshop. This was followed by an open problem session where we gathered fifteen open problems, some of which were circulated in advance of the meeting. The remainder of the week was spent actively working on solving these problems in small groups.

The Covid pandemic prevented many participants from attending the seminar physically, and the entirety of the seminar took place in a hybrid setting, with most working groups featuring both online and physical participants. In order to coordinate the progress, we used Coauthor, a tool designed for by Erik Demaine (MIT), which greatly facilitated the collaborations, and also allowed participants to have a record of the work when the seminar concluded. We also held two daily progress report meetings, allowing people to share progress and allow people to switch groups. In addition to the traditional hike, a virtual social meeting was held on Gather.town to foster interactions between the online and the physical participants.

We now briefly describe the problems that have been worked on, with a more in-depth survey of the problems and the progress being done being featured farther down in this Dagstuhl Report. Some more open problems that have been proposed but not worked on are also listed at the end of the document.

Two groups worked on questions pertaining to reconfiguring curves in the plane and on surfaces. The group 4.1 investigated problems inspired by nonograms, where one aims at introducing switches at intersections of curves in the plane to remove so-called popular faces. The group 4.5 looked at the reconfiguration graph obtained under the action of local moves on minimal closed (multi-)curves on surfaces, and whether such multi-curves could be realized as the set of geodesics of some hyperbolic metric on the surface.

A different flavor of surfaces was studied by the group 4.4, who investigated how square-tiled surfaces could be transformed under the action of shears of cylinder blocks.

The working group 4.2 studied the longstanding problem of the computational complexity of evaluation the rotation distance between elimination trees in graphs. A different flip graph, namely the one of order-k Delaunay triangulations was the topic of study of group 4.7.

Finally, two groups worked on motion of discrete objects in different contexts. The group 4.3 initiated a generalization of the classical theory of morphings of planar graph when one allows the morph to go through a third dimension. The group 4.6 investigated Turning machines, which is a simple model of molecular robot aiming to fold into specific shapes.

All in all, the seminar fostered a highly collaborative research environment by allowing researchers from very diverse backgrounds to work together on precise problems. While the hybrid setting proved to be a significant challenge, the quality of the equipment at Dagstuhl and the online tools that were used provided a practical way for all the participants to interact and to make progress on problems related to reconfiguration in geometric and topological settings.

Copyright Maike Buchin, Arnaud de Mesmay, Anna Lubiw, and Saul Schleimer

Teilnehmer
Vor Ort
Remote:

Verwandte Seminare
  • Dagstuhl-Seminar 17072: Applications of Topology to the Analysis of 1-Dimensional Objects (2017-02-12 - 2017-02-17) (Details)
  • Dagstuhl-Seminar 19352: Computation in Low-Dimensional Geometry and Topology (2019-08-25 - 2019-08-30) (Details)

Klassifikation
  • Computational Complexity
  • Computational Geometry
  • Data Structures and Algorithms

Schlagworte
  • Curve
  • Graph
  • Surface
  • Geometric Topology
  • Reconfiguration