Mathematics 790-03, Fall 2012

Knots and three-dimensional topology

Time: Tuesdays and Thursdays, 1:25 - 2:40
Dates: November 6 through December 6
Location: TBA

This minicourse will occupy the last third of the semester (I'm away on November 1). Math 790-03 is the new number for the former Math 378.

In recent years, there has been an explosion of activity in the study of the topology of three-manifolds. Results such as the Poincare conjecture/theorem have greatly expanded our understanding of classification questions, but the subject does not end there. Much recent excitement has centered on new invariants derived from gauge theory and symplectic geometry.

In this minicourse, we will cover some basic topics in low-dimensional topology and present a quick discussion of some recent developments in the field. We will approach the subject through a discussion of knots, Heegaard splittings of three-manifolds, and Dehn surgery, culminating in the result that all compact orientable three-manifolds are obtained from the three-sphere by Dehn surgery on some link. More advanced topics might include the Berge conjecture, Khovanov homology, or Heegaard Floer homology, depending on interest.

Not much background will be assumed. Some basic familiarity with algebraic topology will be helpful, but a willingness to work with a plethora of three-dimensional diagrams is probably more important.