Crossing Numbers: Theory and Applications (18w5029)

Arriving in Banff, Alberta Sunday, October 21 and departing Friday October 26, 2018

Organizers

(Universidad Autónoma de San Luis Potosí)

(University of South Carolina)

Bojan Mohar (Simon Fraser University)

Objectives

General objectives.

\item{\em Interaction of researchers with a variety of backgrounds and interests}

The community of researchers working on crossing numbers has grown and broadened greatly in the last few years. We have recently witnessed a flurry of activity both in the parametric, structural, and algorithmical aspects of the theory. We have seen a series of successful applications of geometric techniques to the topological setting; we have gained a lot of insight from tackling structural problems with an eye of obtaining improved approximation algorithms; and we have also continued to see the fruitful application of techniques from other branches of mathematics (such as semidefinite programming and flag algebras) to crossing numbers. One of the foremost objectives of this workshop is to bring together all the researchers that have played a major role in this impressive run, so that we can enrich our understanding and horizons learning from each other's experience and background.

\itemExposing younger researchers to the breadth of this thriving field

The area of crossing numbers has quickly established itself as an important, mainstream part of Topological Graph Theory. Most of the cornerstone results in the field have been proved in the last couple of decades. This remarkable progress has involved researchers from many parts of mathematics and computer science, from people coming from the Graph Minors tradition of Structural Graph Theory, to computer scientists with an interest in the explicit implementation of graph drawing and visualization algorithms.

Since this discipline brings together people from such diverse backgrounds and interests, most younger researchers and graduate students interested in the field almost never have the opportunity to interact with all the protagonists in the field in the same event. Several meetings on the general field of Topological Graph Theory take place each year; the same is true about Discrete Geometry or (the much broader field of) Algorithms. A central objective of this workshop is to expose junior researchers to the many facets of crossing numbers, as seen and told by their most active players.

\itemSetting an agenda

Bringing together researchers from the variety of areas that nurture the field of crossing numbers is not only an ideal environment for proving new results; it is also a great opportunity to come up with new questions, and to redefine the main goals and paradigms of the area. This branch of Topological Graph Theory has benefited enormously from the interaction of graph theorists, computer scientists, and people with background and interests in mathematical programming, extremal combinatorics, and discrete geometry, among others. An intensive, weeklong interaction promises to be a very fruitful ground to set an ambitious agenda for the area for years to come.

Relevance, importance, and timeliness.

We have been witnessing in the last few years the coming of age of Crossing Numbers as a mainstream part of Topological Graph Theory. Some important conjectures and questions have been settled~[2page,m1,m2}; important applications to other branches of mathematics have been unveiled~\cite{ap,ap2,tao}; and different perspectives have been applied to old problems that have allowed to apply techniques and results from areas such as mathematical programming~\cite{dekp,dps1,dps2}, extremal combinatorics~\cite{norinstalksiam}, and discrete geometry~\cite{cyl,kyncl].

We seem to face a unique opportunity to bring together both the major veterans in the field, as well as the new, younger protagonists that have developed an interest in the area unveiling the close relationship of crossing numbers to other parts of mathematics. Several problems that have given a direction to this field have been successfully tackled, and the time seems ripe to get together and set a fresh agenda for years to come.

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