Georgia Institute of Technology College of Sciences

Symplectic geometry has become very important for our understanding of 4-manifolds and exotic structures on them. In this talk, I hope to: 

On a given Riemann surface, we may talk about various projective structures that are compatible with the given complex structure. These structures are equivalent to what are known as $sl_2$ opers, or in terms of Faltings’ paper that we mainly follow, as permissible connections. Then there are several interesting questions we may ask about the monodromies of these objects, and the one we focus on today is whether the monodromy group is conjugate into $PSL_2(\mathbb R)$ in $PSL_2(\mathbb C)$ (“the real monodromy property”).

The Conway knot C is an 11-crossing knot that bounds a locally flat disk in B^4. After many decades of unknown status, C was proven to not bound a smoothly embedded disk by Piccirillo. In this talk I will show that in fact, no sum of Conway knots bounds a smoothly embedded disk. This is joint work with Marco Golla (and independent of work of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.)

The dual Garside left canonical form is used to solve the conjugacy problem in braid theory, introduced by Xu (3-braids), Kang-Ko-Lee (4-braids) and Birman-Ko-Lee (n-braids). The fractional Dehn twist coefficient (FDTC) is a measurement of the boundary twist of a surface homeomorphism, introduced by Honda-Kazez-Matic. In contact geometry, it is a powerful tool to detect tightness/overtwistedness of a given contact structure. In this talk, I will give applications of the left canonical form to the computation of the FDTC of 4-braids.

Knot concordance, in both the smooth and topological categories, is a well-studied equivalence relation on knots. We will introduce the smooth, topological, and algebraic concordance groups, and discuss several invariants that help answer questions about the structures of these groups and the relationships between them. We will also discuss the ways facts about knot concordance can be used to construct exotic pairs of 4-manifolds.