Georgia Institute of Technology College of Sciences

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.

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.

In this talk we introduce the notions of the contact angle and of the holomorphic angle for immersed surfaces in $S^{2n+1}$. We deduce formulas for the Laplacian and for the Gaussian curvature, and we will classify minimal surfaces in $S^5$ with the two angles constant. This classification gives a 2-parameter family of minimal flat tori of $S^5$. Also, we will give an alternative proof of the classification of minimal Legendrian surfaces in $S^5$ with constant Gaussian curvature. Finally, we will show some remarks and generalizations of this classification.

By a classical result of Eliashberg, contact manifolds in dimension 3 come in two flavors: tight (rigid) and overtwisted (flexible). Characterized by the presence of an "overtwisted disk", the overtwisted contact structures form a class where isotopy and homotopy classifications are equivalent.In higher dimensions, a class of flexible contact structures is yet to be found. However, some attempts to generalize the notion of an overtwisted disk have been made. One such object is a "plastikstufe" introduced by Niederkruger following some ideas of Gromov. We show that under certain conditions, non-isotopic contact structures become isotopic after connect-summing with a contact sphere containing a plastikstufe. This is a small step towards finding flexibility in higher dimensions. (Joint with E. Murphy, K. Niederkruger, and A. Stipsicz.)