הרצאה פומבית בביה"ס למדעי המחשב - Extremal Problems in Combinatorial Geometry: Algebraic Approaches

Abstract:

Combinatorial Geometry is a field that studies combinatorial problems that have some geometric aspect. It was pioneered and developed by Paul Erdos, starting at the beginning of the 20th century. While such problems are often easy to state, some of them are very difficult, have a deep underlying theory, and remain (or have remained) open for many decades.

A significant class of geometric questions lead to polynomial relations of the form F(x,y,z)=0, for some trivariate polynomial F. In this talk I will review results concerning the number of zeros of a trivariate polynomial F on an n x n x n Cartesian product, and several extensions and variants of this problem. I will then show how these results can be regarded as a general tool suitable for problems in combinatorial geometry.

The talk is based on work developed as part of my PhD studies under the supervision of Prof. Micha Sharir. Part of the contributions that will be presented are a result of collaboration with Micha Sharir, Jozsef Solymosi, Frank de Zeeuw, Oliver Roche-Newton, Ilya Shkredov and others.