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
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.
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.
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.