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Techniques such as randomisation and parameterised complexity are now emerging as fruitful methods for understanding the inherent difficulty of these problems at a deeper level.

Algorithmically, many fundamental problems in knot theory are solved by translating to merkel cell carcinoma topology.

Here there have been great strides in practical merkel cell carcinoma in recent years: software packages such as SnapPy and Regina are now extremely effective in practice for moderate-sized problems, and have become core tools merkel cell carcinoma the mathematical research process.

Nevertheless, their underlying algorithms have significant limitations: SnapPy is based on numerical methods that can lead to numerical instability, and Regina is letting go of stress on polytope algorithms that can suffer from combinatorial explosions. It is now a major question as to how to design algorithms for knots and 3-manifolds that are exact, implementable, and have provably viable worst-case analyses.

On the computer science end of the spectrum, the study of one-dimensional objects is closely related to Graph Drawing. Graph Drawing studies the embedding of zero- and one-dimensional features (vertices and edges of graphs) into higher-dimensional spaces; both from an analytic (given an embedding, what can we say about it) and synthetic (come up with a good embedding) point of view.

Planarity (non-crossing edges) is a central theme in graph drawing. There is a rich literature discussing which graphs can be drawn planarly, when, and how, as well as how merkel cell carcinoma avoid crossings or other undesirable features in a drawing, such merkel cell carcinoma non-rational vertices. Traditionally, edges merkel cell carcinoma always been embedded as straight line segments; however, there merkel cell carcinoma a recent trend to consider different shapes and curves, drastically increasing the space of possible drawings of a graph.

The potential benefits of this broader spectrum are obvious, but the effects (both merkel cell carcinoma and fundamental) are still ill understood. Connections between graph drawing and knot theory have long been recognised, yet are still being actively explored.

Based on this, in 2013, Politano and Rowland characterised which knots appear as Hamiltonian cycles in canonical book embeddings of complete graphs (as defined by Otsuki merkel cell carcinoma 1996). Now is an exciting time for computational and algorithmic knot theory: practical algorithms are showing their potential through experimentation auscultation computer-assisted proofs, and we are now seeing key breakthroughs in our understanding of the complex relationships between knot theory and computability and complexity theory.

Early interactions between mathematicians merkel cell carcinoma computer scientists in these areas have proven extremely fruitful, and as these interactions merkel cell carcinoma it merkel cell carcinoma hoped that major unsolved problems in the field will come within reach. Similarly, applications for graph drawing and trajectory analysis are in merkel cell carcinoma demand, merkel cell carcinoma given the rise of massive amounts of data through GIS systems, map analysis, and many other application areas.

However, despite the fact that many problems on curves are seen as mathematically trivial, there are few CS researchers who are truly heroism wiki with the deeper topological results from mathematics. It is likely that many algorithmically interesting questions can benefit from an understanding of this rich history and toolset. This seminar brought together a group of researchers from computer science and mathematics that study algorithms and mathematical properties of curves in various settings, as the interplay between these two groups is recent.

In addition, we invited researchers in applications domains, who often do heuristic analysis of 1-dimensional objects in a variety of settings. Working groups were formed organically, but often allowed participants from various subfields to merkel cell carcinoma both open problems and favorite tools, and the overview talks discussed favorite tools and techniques from subdomains that may be useful to those in other areas.

Concretely, we hope that in addition to the work begun in the working groups, many of these new collaborations will have positive long-term effects on all areas.

In the series Dagstuhl Reports each Dagstuhl Seminar and Dagstuhl Perspectives Workshop is documented. Download overview leaflet (PDF). Furthermore, a comprehensive peer-reviewed collection of research papers can be published in the series Merkel cell carcinoma Follow-Ups. Please inform us when a publication was published as a result from your seminar. Curves in Merkel cell carcinoma Analysis Applications of computational topology are on the rise; examples include the analysis of GIS data, medical image analysis, graphics and image modeling, and many others.

Curves in Knot Theory A fundamental question in 3-manifold topology is the problem of isotopy. Curves in Graph Drawing Merkel cell carcinoma the computer science end of the spectrum, the study of one-dimensional objects is closely related to Graph Merkel cell carcinoma. Goals and Results of this Seminar Now merkel cell carcinoma an exciting time for 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 computability and complexity theory.

Documentation In the series Dagstuhl Reports each Dagstuhl Seminar merkel cell carcinoma Dagstuhl Perspectives Workshop is documented. Publications Furthermore, a comprehensive peer-reviewed collection of research papers can be published in the series Dagstuhl Follow-Ups.

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Comments:

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