Georgia Institute of Technology College of Sciences

The pants complex of a surface has as its 0-cells the pants decompositions of a surface and as its 1-cells some elementary moves relating two pants decompositions; the 2-cells are disks glued along certain cycles in the 1-skeleton of the complex. In "Pants Decompositions of Surfaces," Hatcher proves that this complex is contractible.

 During this interactive talk, we will aim to understand the structure of the pants complex and some of the important tools that Hatcher uses in his proof of contractibility.

Every knot in S^3 bounds a PL (piecewise-linear) disk in the four ball. But this is no longer true for knots in other three manifolds, as demonstrated first by Akbulut, who constructed a knot which does not bound any PL disk in a specific contractible four manifold. Then Levine showed that there exist knots that do not bound a PL disk in any homology four ball. What happens if we relax the condition of bounding PL disk to bounding a PL surface with some given genus?

In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory.

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world.

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world.

The classical braid groups can be viewed from many different angles and admit generalizations in just as many directions. Surface braid groups are a topological generalization of the braid groups that have close connections with mapping class groups of surfaces. In this talk we review a recent result on minimal nonabelian finite quotients of braid groups. In considering the analogous problem for surface braid groups, we construct nilpotent nonabelian quotients by generalizing the Heisenberg group.

We will describe an elegant construction of potential counterexamples to the Smooth 4-Dimensional Poincaré Conjecture whose input is a fibered, homotopy-ribbon knot in the 3-sphere. The construction also produces links that are potential counterexamples to the Generalized Property R Conjecture, as well as balanced presentations of the trivial group that are potential counterexamples to the Andrews-Curtis Conjecture.

When do commuting homeomorphisms of S^2 have a common fixed point? Christian Bonatti gave the first sufficient condition: Commuting diffeomorphisms sufficiently close to the identity in Diff^+(S^2) always admit a common fixed point. In this talk we present a result of Michael Handel that extends Bonatti's condition to a much larger class of commuting homeomorphisms. If time permits, we survey results for higher genus surfaces due to Michael Handel and Morris Hirsch, and connections to certain compact foliated 4-manifolds.