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Series: Geometry Topology Seminar

We introduce a new surgery operation for contact manifolds called the Liouville connect sum. This operation -- which includes Weinstein handle attachment as a special case -- is designed to study the relationship between contact topology and symplectomorphism groups established by work of Giroux and Thurston-Winkelnkemper. The Liouville connect sum is used to generalize results of Baker-Etnyre-Van Horn-Morris and Baldwin on the existence of "monodromy multiplication cobordisms" as well as results of Seidel regarding squares of symplectic Dehn twists.

Series: Geometry Topology Seminar

Given any smooth manifold, there is a canonical symplectic structure on its cotangent bundle. A long standing idea of Arnol'd suggests that the symplectic topology of the cotangent bundle should contain a great deal of information about the smooth topology of its base. As a contrast, I show that when X is an open 4-manifold, this symplectic structure on T^*X does not depend on the choice of smooth structure on X. I will also discuss the particular cases of smooth structures on R^4 and once-punctured compact 4-manifolds.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

We give a simple generating set for the following three closely related groups: the hypereliptic Torelli group, the kernel of the integral Burau representation, and the fundamental group of the branch locus of the period mapping. Our theorem confirms a conjecture of Hain. This is joint work with Tara Brendle and Andy Putman.

Series: Geometry Topology Seminar

I will explain how to construct a 4-variable knot invariant which expresses a recursion for the colored HOMFLY polynomial of a knot, and its implications on (a) asymptotics (b) the SL2 character variety of the knot (c) mirror symmetry.

Series: Geometry Topology Seminar

In this talk we are going to present a theorem that can be seen as related to S. Smale's theorem on the topology of the space of Legendrian loops. The framework will be slightly different and the space of Legendrian curves will be replaced by a smaller space $C_{\beta}$, that appears to be convenient in some variational problems in contact form geometry. We will also talk about the applications and the possible extensions of this result. This is a joint work with V. Martino.

Series: Geometry Topology Seminar

We discuss two concepts of low-dimensional topology in higher dimensions: near-symplectic manifolds and overtwisted contact structures. We present a generalization of near-symplectic 4-manifolds to dimension 6. By near-symplectic, we understand a closed 2-form that is symplectic outside a small submanifold where it degenerates. This approach uses some singular mappings called generalized broken Lefschetz fibrations. An application of this setting appears in contact topology. We find that a contact 5-manifold, which appears naturally in this context, is PS-overtwisted. This property can be detected in a rather simple way.

Series: Geometry Topology Seminar

We classify the Legendrian torus knots in S1XS2 with tight contact structure up to isotopy. This is a joint work with Feifei Chen and Fan Ding.

Series: Geometry Topology Seminar

We study some finite quotients of the A_n Milnor fibre which coincide with the Stein surfaces that appear in Fintushel and Stern's rational blowdown construction. We show that these Stein surfaces have no exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the A_n Milnor fibre coming from homological mirror symmetry. On the contrary, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori that we studied in the A_n Milnor fibre. We conclude that these Stein surfaces have non-vanishing symplectic cohomology. This is joint work with M. Maydanskiy.

Series: Geometry Topology Seminar

We will give an overview of open book foliation method by emphasizing the aspect that it is a generalization of Birman-Menasco's braid foliation theory. We explain how surfaces in open book reflects topology and (contact) geometry of underlying 3-manifolds, and will give several applications. This talk is based on joint work with Keiko Kawamuro.