Planar contact manifolds have been intensively studied to understand several aspects of 3-dimensional contact geometry. In this talk, we define "iterated planar contact manifolds", a higher-dimensional analog of planar contact manifolds, by using topological tools such as "open book decompositions" and "Lefschetz fibrations”. We provide some history on existing low-dimensional results regarding Reeb dynamics, symplectic fillings/caps of contact manifolds and explain some generalization of those results to higher dimensions via iterated planar structure. This is partly based on joint work in progress with J. Etnyre and B. Ozbagci.
Seminars and Colloquia by Series
Monday, April 30, 2018 - 14:00 , Location: Skiles 006 , Michael Harrison , Lehigh University , email@example.com , Organizer: Mohammad Ghomi
The h-principle is a powerful tool in differential topology which is used to study spaces of functionswith certain distinguished properties (immersions, submersions, k-mersions, embeddings, free maps, etc.). Iwill discuss some examples of the h-principle and give a neat proof of a special case of the Smale-HirschTheorem, using the "removal of singularities" h-principle technique due to Eliashberg and Gromov. Finally, I willdefine and discuss totally convex immersions and discuss some h-principle statements in this context.
Monday, April 23, 2018 - 14:00 , Location: Skiles 006 , Hong Van Le , Institute of Mathematics CAS, Praha, Czech Republic , firstname.lastname@example.org , Organizer: Thang Le
Novikov homology was introduced by Novikov in the early 1980s motivated by problems in hydrodynamics. The Novikov inequalities in the Novikov homology theory give lower bounds for the number of critical points of a Morse closed 1-form on a compact differentiable manifold M. In the first part of my talk I shall survey the Novikov homology theory in finite dimensional setting and its further developments in infinite dimensional setting with applications in the theory of symplectic fixed points and Lagrangian intersection/embedding problems. In the second part of my talk I shall report on my recent joint work with Jean-Francois Barraud and Agnes Gadbled on construction of the Novikov fundamental group associated to a cohomology class of a closed 1-form on M and its application to obtaining new lower bounds for the number of critical points of a Morse 1-form.
Monday, April 16, 2018 - 15:30 , Location: Skiles 005 , Yu Pan , MIT , Organizer: Caitlin Leverson
Augmentations and exact Lagrangian fillings are closely related. However, not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we show that all the augmentations come from possibly immersed exact Lagrangian fillings. In particular, let ∑ be an immersed exact Lagrangian filling of a Legendrian knot in $J^1(M)$ and suppose it can be lifted to an embedded Legendrian L in J^1(R \times M). For any augmentation of L, we associate an induced augmentation of the Legendrian knot, whose homotopy class only depends on the compactly supported Legendrian isotopy type of L and the homotopy class of its augmentation of L. This is a joint work with Dan Rutherford.
Monday, April 16, 2018 - 14:00 , Location: Skiles 006 , Ken Baker , University of Miami , Organizer: Caitlin Leverson
Based on the known examples, it had been conjectured that all L-space knots in S3 are strongly invertible. We show this conjecture is false by constructing large families of asymmetric hyperbolic knots in S3 that admit a non-trivial surgery to the double branched cover of an alternating link. The construction easily adapts to produce such knots in any lens space, including S1xS2. This is joint work with John Luecke.
Monday, April 9, 2018 - 14:00 , Location: Skiles 006 , Bahar Acu , Northwestern University , Organizer: John Etnyre
Monday, April 2, 2018 - 14:00 , Location: Skiles 006 , Linh Truong , Columbia University , Organizer: Jennifer Hom
Heegaard Floer homology has proven to be a useful tool in the study of knot concordance. Ozsvath and Szabo first constructed the tau invariant using the hat version of Heegaard Floer homology and showed it provides a lower bound on the slice genus. Later, Hom and Wu constructed a concordance invariant using the plus version of Heegaard Floer homology; this provides an even better lower-bound on the slice genus. In this talk, I discuss a sequence of concordance invariants that are derived from the truncated version of Heegaard Floer homology. These truncated Floer concordance invariants generalize the Ozsvath-Szabo and Hom-Wu invariants.
Monday, March 26, 2018 - 14:30 , Location: Room 304 , Bob Gompf and Sergei Gukov , UT Austin and Cal Tech , Organizer: Caitlin Leverson
For oriented manifolds of dimension at least 4 that are simply connected at infinity, it is known that end summing (the noncompact analogue of boundary summing) is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. We will examine how and when uniqueness fails. There are examples in various categories (homotopy, TOP, PL and DIFF) of nonuniqueness that cannot be detected in a weaker category. In contrast, we will present a group-theoretic condition that guarantees uniqueness. As an application, the monoid of smooth manifolds homeomorphic to R^4 acts on the set of smoothings of any noncompact 4-manifold. (This work is joint with Jack Calcut.)
Monday, February 19, 2018 - 15:30 , Location: Skiles 005 , Greg Kuperberg , UC Davis , Organizer: Caitlin Leverson
Now that the geometrization conjecture has been proven, and the virtual Haken conjecture has been proven, what is left in 3-manifold topology? One remaining topic is the computational complexity of geometric topology problems. How difficult is it to distinguish the unknot? Or 3-manifolds from each other? The right approach to these questions is not just to consider quantitative complexity, i.e., how much work they take for a computer; but also qualitative complexity, whether there are efficient algorithms with one or another kind of help. I will discuss various results on this theme, such as that knottedness and unknottedness are both in NP; and I will discuss high-dimensional questions for context.