Extremal Problems in Combinatorial Geometry (18w5058)

Arriving in Banff, Alberta Sunday, February 4 and departing Friday February 9, 2018


Jozsef Solymosi (University of British Columbia)

(École Polytechnique Fédérale de Lausanne)

(University of California - San Diego)


Problems in combinatorial geometry are often simply described, and usually involve finite set of points, lines, convex sets, and other geometric objects in Euclidean space. Extremal problems in the field asks how large or small a finite set of geometric objects can be under certain restrictions. For example, given a set of n points in the plane, how often can the unit distance occur among them, or what is the size of the largest subset in convex position? Many questions in the field are very natural and worth studying for their own sake, while others are fueled by the more recent development of computational geometry. Over the past 6 years, in particular, combinatorial geometry has seen tremendous growth, and numerous unexpected connections to other fields of mathematics are being discovered. Two notable examples using algebraic geometry are the works of Guth and Katz, who solved the Erdos distinct distances problem, and of Green and Tao, who solved the long-standing conjecture of Dirac and Motzkin on the number of ordinary lines.

This workshop will focus on several areas in combinatorial geometry, such as incidence geometry, intersection graphs, graph drawing, Erdos-Szekeres-type theorems, and combinatorial number theory. One of the major goals of this workshop is to study the interplay between algebraic and combinatorial methods used in the field. Among the main themes that the workshop will cover are 1) The polynomial method and its applications in incidence geometry and combinatorial number theory, 2) The container method and its applications in discrete and computational geometry, 3) The theory of semi-algebraic hypergraphs: regularity lemmas, VC-dimension arguments, and applications.

With all of the recent developments and exciting breakthrough results in the field, it is important now to bring together prominent experts in the field from all over the world, such as J. Balogh (Urbana), I. Barany (Budapest), J. Fox (Palo Alto), L. Guth (Boston), M. Sharir (Tel Aviv), and G. Tardos (Budapest), as well as very promising young researchers such as B. Bukh (Pittsburgh (US)), O. Raz (Princeton), and J. Zahl (Vancouver). We believe it is very important to bring together and foster interactions between senior and junior researchers in combinatorial geometry. We are planning to schedule a number of key lectures by international experts, which will survey the state-of-the-art of several long-standing open problems in the field. Other participants will have the opportunity to give a 30 minute talk to present their work. Our workshop schedule will provide ample time and opportunity for participants to interact and engage in mathematical discussion.