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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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

The fractional Dehn twist coefficient (FDTC), defined by Honda-Kazez-Matic, is an invariant of mapping classes. In this talk we study properties of FDTC by using open book foliation method, then obtain results in geometry and contact geometry of the open-book-manifold of a mapping class. This is joint work with Tetsuya Ito.

Series: Geometry Topology Seminar

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.)

Series: Geometry Topology Seminar

The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces. Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined. The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams. The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo. We show that these two previously defined invariant agree. Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.

Series: Geometry Topology Seminar

It is a theorem of Bass, Lazard, and Serre, and, independently, Mennicke, that the special linear group SL(n,Z) enjoys the congruence subgroup property when n is at least 3. This property is most quickly described by saying that the profinite completion of the special linear group injects into the special linear group of the profinite completion of Z. There is a natural analog of this property for mapping class groups of surfaces. Namely, one may ask if the profinite completion of the mapping class group embeds in the outer automorphism group of the profinite completion of the surface group. M. Boggi has a program to establish this property for mapping class groups, which couches things in geometric terms, reducing the conjecture to determining the homotopy type of a certain space. I'll discuss what's known, and what's needed to continue his attack.