Title: The nearby Lagrangian conjecture and its relation to algebraic K-theory
Speaker: Kiyoshi Igusa (Brandeis University)
Abstract: Around 1975, using algebraic K-theory of spaces, F. Waldhausen predicted, as a special case of a more general theory, that there are exotic smooth disk bundles over 4k-spheres. These are smooth disk bundles which are trivial as topological bundles but nontrivial as smooth bundles. This was a difficult result requiring the work of many mathematicians.
Also in 1975, A. Hatcher proposed a simple construction of these exotic bundles using G/O. Bökstedt proved that Hatcher's idea works, i.e., that Hatcher's construction gives a rational homotopy equivalence from G/O to the stablised pseudoisotopy space of a point. Later I gave another proof using parametrized Morse theory.
In 2018, T. Kragh showed that the homotopy fiber of this map, which he called the "Hatcher-Waldhausen map" is the Eliashberg-Gromov space which might be an obstruction to the validity of the "nearby Lagrangian conjecture". Kragh's construction uses parametrized Morse theory to obtain "generating functions" for Lagrangian manifolds.
In this talk I will tell this story in reverse order starting with generating functions for Lagrangians, the Eliashberg-Gromov construction and Kragh's argument which, running in reverse, will reproduce Hatcher's construction. The equivariant version of Hatcher construction (joint work with Goodwillie and Ohrt) will give some interesting examples in symplectic topology (joint work with Álvarez-Gavela).
Bio-sketch: Kiyoshi Igusa earned his PhD at Princeton University in 1979 with Allen Hatcher and went on to join Brandeis University, where he is currently a professor. He was a recipient of a Sloan Fellowship in 1981, and was an invited speaker at ICM Kyoto in 1990. He became a Fellow of the AMS in 2013. His work is in representation of quivers, cluster algebras, and more broadly in combinatorial and algebraic topology, and he has major publications around stability theorems for isotopy and higher torsion.
Colloquium URL: https://web.math.osu.edu/colloquium/
