at the

American Institute of Mathematics, Palo Alto, California

organized by

Jozsef Balogh, Silvia Fernandez-Merchant, and Gelasio Salazar

This workshop, sponsored by AIM and the NSF, will be devoted to tackling several long-standing open problems in the field of crossing numbers. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. This field can be traced back to a question raised by Paul Turán in 1944, which in modern terminology asks for the crossing number of the complete bipartite graph. Since the appearence of Turán's question (which remains open to this day), many fundamental crossing number questions have been tackled, both for their theoretical interest and for their applications to other branches of mathematics. Another tantalizingly open question asks for the crossing number of the complete graph. As Richter and Thomassen showed, this is closely related to Turán's original question. Several variants of these two questions (including their rectilinear versions, in which edges are required to be drawn as straight line segments) are of great interest of their own. In particular, the rectilinear crossing number of the complete graph is of fundamental importance, since settling this problem would solve the Four Point Problem posed by J.J. Sylvester in 1864.

The main topics for the workshop are:

- Zarankiewicz's Crossing Number Conjecture (Turán's Brickyard Problem), including its weaker variants: rectilinear crossing number and 2-page crossing number.
- The Harary-Hill Conjecture on the crossing number of the complete graph.
- The rectilinear crossing number of the complete graph.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email *workshops@aimath.org*

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