Georgia Institute of Technology College of Sciences

The study of representations of fundamental groups of surfaces into Lie groups is captured by the character variety. One main tool to study character varieties are Higgs bundles, a complex geometric tool. They fail to see the mapping class group symmetry. I will present an alternative approach which replaces Higgs bundles by so-called higher complex structures, given in terms of commuting nilpotent matrices. The resulting theory has many similarities to the non-abelian Hodge theory. Joint with Georgios Kydonakis and Charlie Reid.

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. These Heisenberg quotients do not arise as quotients of the braid 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. We will then turn our attention to generalized square knots (connected sums of torus knots with their mirrors), which provide a setting where the potential counterexamples mentioned above can be explicitly understood. Here, we show that the constructed 4-manifolds are diffeomorphic to the 4-sphere; but the potential counterexamples to the other conjectures persist. In particular, we present a new, large family of geometrically motivated balanced presentations of the trivial group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy and isotopy rel-boundary. This talk is based on joint work with Alex Zupan.

The fundamental groups of knot complements have lots of finite quotients. We give a criterion for a hyperbolic knot in the three-sphere to be distinguished (up to isotopy and mirroring) from every other knot in the three-sphere by the set of finite quotients of its fundamental group, and we use this criterion as well as recent work of Baldwin-Sivek to show that there are infinitely many hyperbolic knots distinguished (up to isotopy and mirroring) by finite quotients.