MATH 7520 - Berstein Seminar: Four Manifolds (with a Symplectic Bias)

Tara Holm, spring 2016.

The world of four manifolds is well-explored, and yet many phenomena remain mysterious. In the symplectic category, four dimensions is where we know that there are abundant examples, and at the same time many analytic and topological techniques have been established. More generally, four dimensions is where there remain a number of open questions: which topological four manifolds are smoothable? We will take a tour of four manifolds. The semester will begin with a broad overview, and we will proceed to choose some topics and papers of particular interest to the participants, to examine and dig deeper.

Participants will be expected to read papers and present material to the class. Prerequisite background includes some basic differential geometry; a bit of algebraic topology would be helpful. We will discuss and decide together the best way to document our investigations.