https://www.dagstuhl.de/19352

### August 25 – 30 , 2019, Dagstuhl Seminar 19352

# Computation in Low-Dimensional Geometry and Topology

## Organizers

Maarten Löffler (Utrecht University, NL)

Anna Lubiw (University of Waterloo, CA)

Saul Schleimer (University of Warwick – Coventry, GB)

Erin Moriarty Wolf Chambers (St. Louis University, US)

## For support, please contact

Dagmar Hofmann for administrative matters

Michael Gerke for scientific matters

## Documents

List of Participants

Shared Documents

Dagstuhl Seminar Schedule [pdf]

## Motivation

One-dimensional structures embedded in higher-dimensional spaces are ubiquitous in both the natural and artificial worlds: DNA strands, migration paths, planetary orbits, rocket trajectories, robot motion planning, chip design, and much more. These are studied in different areas of mathematics and computer science, under many names: knots, curves, paths, traces, trajectories, graphs, and so on. However, researchers in many areas are just beginning to apply algorithmic techniques to find efficient algorithms for these structures, especially when more fundamental mathematical results are necessary tools.

Our Dagstuhl Seminar 19352 "Computation in Low-Dimensional Geometry and Topology" is a follow-up to "Applications of Topology to the Analysis of One-Dimensional Objects" (Dagstuhl Seminar 17072). Our overall goal is to identify connections between, and promote new research collaborations in, computational topology and computational geometry.

In this seminar, we will continue our study of one-dimensional objects, but now adding a time dimension. That is, we will consider these objects as they move from a given position towards an optimal position. Examples include:

- classical algorithms on trajectories like the Frechet distance as a way to formalize a distance measure as a curve changes;
- morphing between two versions of a common graph, which again tracks a higher dimensional space that corresponds to movement of a one-dimensional object;
- drawing and manipulating objects in three-manifolds, such as graphs, curves, or surfaces; and
- perhaps the simplest problem posed (in different ways) in all these areas, "how does one draw and morph a nice curve on a nice surface?"

In each of these examples, there is a core problem from one area that will be of interest to researchers in other areas.

In addition, we will invite participants and speakers with more of an emphasis on code development. In this way some of the groups can begin to focus on implementation issues that span these areas. While theoretical development of tools is key, there is also a great need for practical algorithms that can handle large datasets, from detecting knots in proteins to clustering trajectories and map matching.

Despite increasing collaborations between these areas, there is still a significant divide between the communities. To foster communication, we will have 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. The emphasis on code development reflects our growing faith in the implementability and usefulness of solutions to the core problems of these fields.

**Motivation text license**

Creative Commons BY 3.0 DE

Maarten Löffler, Anna Lubiw, Saul Schleimer, and Erin Moriarty Wolf Chambers

## Related Dagstuhl Seminar

## Classification

- Data Structures / Algorithms / Complexity

## Keywords

- Mathematics
- Topology
- Trajectories
- Graph drawing
- Knots