Geometry and Topology Seminar
Monday, December 10, 2018 at 3:00pm to 5:00pm
Thomas Kragh (Uppsala University)
Title: Stable homotopy theory in symplectic geometry
Abstract: I will go through some of the ideas and consequences in applying stable homotopy theory to represent Floer homology. If time permits I will discus how it relates to: Algebraic K-theory of spaces, generating families and/or micro-local sheaf theory.
Professor Anar Akhmedov (University of Minnesota and Simons Fellow at Harvard University)
Title: Title: Genus two Lefschetz fibrations via lantern substitutions
Abstract: Lefschetz fibrations play very important role in 4-manifold topology. It was shown by Donaldson that, perhaps after some blow-ups, any closed symplectic 4-manifold admit a Lefschetz fibration over two sphere. Conversely, Gompf showed that the total space of a Lefschetz fibration admits a symplectic structure, provided the fibers are non-trivial in homology, generalizing an earlier work of Thurston. In this talk, we will construct a family of genus two Lefschetz fibrations X(n) over two sphere by applying the lantern substitutions to the twisted fiber sums of Matsumoto's genus two Lefschetz fibration. We will compute the fundamental group of X(n) and show that it is isomorphic to the trivial group if n =-3, -1, Z if n=-2, and finite cyclic group Z_|n+2| for all integers n not equal to -3,-2,-1. We also show that the total spaces of these Lefschetz fibrations are symplectically minimal and have the symplectic Kodaira dimension equal to 2. We will also discuss very recent result on indecomposable minimal genus two Lefschetz fibrations. These are joint works with Naoyuki Monden.
Thomas Kragh 3:00-4:00pm
Anar Akhmedov 4:00-5:00PM