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Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

Although the Alexander polynomial does not satisfy an unoriented skein relation, Manolescu (2007) showed that there exists an unoriented skein exact triangle for knot Floer homology. In this talk, we will describe some developments in this direction since then, including a combinatorial proof using grid homology and extensions to the Petkova-Vertesi tangle Floer homology (joint work with Ina Petkova) and Zarev's bordered sutured Floer homology (joint work with Shea Vela-Vick).