Georgia Institute of Technology College of Sciences

Elliptic surfaces are some of the most well-behaved families of smooth, simply-connected four-manifolds from the geometric and analytic perspective. Many of their smooth invariants are easily computable and they carry a fibration structure which makes it possible to modify them by various surgical operations. However, elliptic surfaces have large Euler characteristics which means even their simplest handle-decompositions appear to be quite complicated. In this seminar, we will learn how to draw several different handle diagrams of elliptic surfaces which show explicitly many of their nice properties. This will allow us to see many useful properties of elliptic surfaces combinatorially, and gives insight into the constructions of their exotic smooth structures.