Georgia Institute of Technology College of Sciences

 I will discuss the orderability of the fundamental groups of knot complements including known results, a useful technique using some ideas of Baumslag, and some interesting questions that have recently arisen from this study.

Heegaard Floer homology gives a powerful invariant of closed 3-manifolds. It is always computable in the purely combinatorial sense, but usually computing it needs a very hard work. However, for certain graph 3-manifolds, its minus-flavored Heegaard Floer homology can be easily computed in terms of lattice homology, due to Nemethi. I plan to give the basic definitions and constructions of Heegaard Floer theory and lattice homology, as well as the isomorphism between those two objects.

A question going back to Serre asks which groups arise as fundamental groups of smooth, complex projective varieties, or more generally, compact Kaehler manifolds.  The most basic examples of these are surface groups, arising as fundamental groups of 1-dimensional projective varieties.  We will survey known examples and restrictions on such groups and explain the special role surface groups play in their classification. Finally, we connect this circle of ideas to more general questions about surface bundles and mapping class groups. 

The field of complex dynamics melds a number of disciplines, including complex analysis, geometry and topology. I will focus on the influences from the latter, introducing some important concepts and questions in complex dynamics, with an emphasis on how the concepts tie into and can be enhanced by a topological viewpoint.