These results were often quoted during the last century, and were generally considered as basic and as solid fundaments on which to build other results. However, during the last year both were shown to be false--even though in a subtle way! (A polite way of saying this is that their statements require "reinterpretation".) This has opened a whole lot of exciting questions and directions of investigation, and a main aim of the course will be to present as many of these as possible.
Here is a short description of the topic of configurations for those who are new to it. A configuration is a collection of points and a collection of lines, together with a list of incidences. In a geometric sense "incidence" can be interpreted as "line" containing "point"; in combinatorial interpretation "lines" are sets consisting of "points" incident with them. Like many other simple concepts about elementary-geometric entities, configuration of points and lines are related in intriguing and nontrivial ways to many branches of mathematics. In particular, these ramifications go from pure combinatorics and graph theory to algebraic geometry of cubic curves, from symmetry and group theory to problems about trigonometric polynomials, from Diophantine equations to questions of computer implementation of construction and enumeration. I expect to give at least a taste of most of these aspects, with information about the currently known facts (published and unpublished). A particularly intriguing collection of facts deals with the interplay of geometric continuity with combinatorial discreteness.
Prerequisites: The course has no particular prerequisites, besides some mathematical maturity.