12.02.17 - 17.02.17, Seminar 17072

Applications of Topology to the Analysis of 1-Dimensional Objects

Diese Seminarbeschreibung wurde vor dem Seminar auf unseren Webseiten veröffentlicht und bei der Einladung zum Seminar verwendet.

Motivation

One-dimensional objects embedded in higher-dimensional spaces are one of the most natural phenomena we encounter: ranging from DNA strands to roads to planetary orbits, they occur at all granularities throughout the sciences. One-dimensional objects are studied under different names in different areas of mathematics and computer science (knots, curves, paths, traces, trajectories). In mathematics, 1-dimensional objects are historically well-studied; however, many application areas demand algorithms that deal with 1-dimensional objects, and so this remains a rich topic of study in computer science.

The goal of this Dagstuhl Seminar is to identify connections and seed new research collaborations along the spectrum from knot theory and topology, to computational topology and computational geometry, all the way to graph drawing. The focus will be on 1-dimensional objects embedded in 2- and 3-dimensional spaces, as this is both the most fundamental setting in many applications, as well as the setting where the discrepancy between generic mathematical theory and potential algorithmic solutions is most apparent. This novel combination of areas along a wide spectrum from pure mathematics to computer science has unique potential to provide new fundamental insights for 1-dimensional objects.

Seminar Approach The seminar will consist of a mix of talks and collaborative work on open problems. Longer survey talks will introduce the spectrum of different areas, and shorter talks will give participants the opportunity to present related work. A large portion of the time will be devoted to solving open problems and defining cross-cutting research directions. These will be conducted in several work groups. Topics of study include:

  • Applying computational topology to curve analysis and graph drawing. Applications in this area are in great demand, especially given the rise of massive amounts of data through GIS systems, map analysis, and many other application areas. There are many algorithmically interesting questions that can benefit from the rich mathematical history of related concepts in topology. Homotopy, for example, is notoriously difficult, as even deciding if two curves are homotopic is undecidable in a generic 2-complex; however, many application areas provide restrictions on the inputs that make computation more accessible.
  • Computational and algorithmic knot theory. Practical algorithms are showing their potential through experimentation and computer-assisted proofs, and we are now seeing key breakthroughs in our understanding of the complex relationships between knot theory and complexity theory. Early interactions between mathematicians and computer scientists in these areas have proven extremely fruitful, and as these interactions deepen it is hoped that major unsolved problems in the field will come within reach.

License
Creative Commons BY 3.0 Unported license
Benjamin Burton, Maarten Löffler, Carola Wenk, and Erin Moriarty Wolf Chambers