Tutte Colloquium - Patricia Klein

Title: Combinatorial models in enumerative geometry

Abstract: How many circles are tangent to three given circles in the plane?  How many lines lie on a smooth cubic surface? How many lines intersect four given lines in 3-space? These are classical questions in enumerative geometry, a field at least as old as Apollonius of Perga, to whom the question about tangent circles is attributed in the third century BCE.  It is not hard to see that the answers to these questions may depend on the choice of circles, surface, or lines, respectively. It is harder to see that, in each case, there is a "typical" answer. 

In the millennia since Apollonius, the subject has advanced. With a great deal of effort, especially in the wake of the work of Hermann Schubert around the turn of the 20th century, mathematicians made rigorous the notion of a "typical" answer and also made rigorous certain simplifying strategies Schubert had suggested.  Indeed, making Schubert's arguments rigorous was the topic of Hilbert's 15th problem. The simplifications Schubert had suggested entail, roughly speaking, sliding or deforming the geometric objects to be studied while preserving the total number of whatever it is one wants to count. These strategies are what are now called degeneration techniques. In this talk, we will describe some modern questions in enumerative geometry, explain how these are studied via degeneration techniques, and give examples of the kinds of combinatorics that can be attached to enumerative geometry questions when degeneration techniques are applied.