Geometry and Topology Seminar 2019-2020
The seminar will be held in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
|January 21||Mohammed Abouzaid (Clay Institute & MIT)||Yong-Geun|
|February 4||Laurentiu Maxim (UW-Madison)||local|
|March 4||David Massey (Northeastern)||Maxim|
|March 11||Danny Calegari (Cal Tech)||Yong-Geun|
|March 23, Wed, Room: TBA||Joerg Schuermann (University of Muenster, Germany)||Maxim|
|May 6||Alex Suciu (Northeastern)||Maxim|
|May 13||Alvaro Pelayo (IAS)||Yong-Geun|
Mohammed Abouzaid (Clay Institute & MIT)
A plethora of exotic Stein manifolds
In real dimensions greater than 4, I will explain how a smooth manifold underlying an affine variety admits uncountably many distinct (Wein)stein structures, of which countably many have finite type, and which are distinguished by their symplectic cohomology groups. Starting with a Lefschetz fibration on such a variety, I shall per- form an explicit sequence of appropriate surgeries, keeping track of the changes to the Fukaya category and hence, by understanding open-closed maps, obtain descriptions of symplectic cohomology af- ter surgery. (joint work with P. Seidel)
Laurentiu Maxim (UW-Madison)
Intersection Space Cohomology and Hypersurface Singularities
A recent homotopy-theoretic procedure due to Banagl assigns to a certain singular space a cell complex, its intersection space, whose rational cohomology possesses Poincare duality. This yields a new cohomology theory for singular spaces, which has a richer internal algebraic structure than intersection cohomology (e.g., it has cup products), and which addresses certain questions in type II string theory related to massless D-branes arising during a Calabi-Yau conifold transition.
While intersection cohomology is stable under small resolutions, in recent joint work with Markus Banagl we proved that the new theory is often stable under smooth deformations of hypersurface singularities. When this is the case, we showed that the rational cohomology of the intersection space can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface.
David Massey (Northeastern)
Lê Numbers and the Topology of Non-isolated Hypersurface Singularities
The results of Milnor from his now-classic 1968 work "Singular Points of Complex Hypersurfaces" are particularly strong when the singular points are isolated. One of the most striking subsequent results in this area, was the 1976 result of Lê and Ramanujam, in which the h-Cobordism Theorem was used to prove that constant Milnor number implies constant topological-type, for families of isolated hypersurfaces.
In this talk, I will discuss the Lê cycles and Lê numbers of a singular hypersurface, and the results which seem to indicate that they are the "correct" generalization of the Milnor number for non-isolated hypersurface singularities.
Danny Calegari (Cal Tech)
Alex Suciu (Northeastern)
Joerg Schuermann (Muenster)
Alvaro Pelayo (IAS)
Symplectic Dynamics of integrable Hamiltonian systems
I will start with a review the basic notions of Hamiltonian/symplectic vector field and of Hamiltonian/symplectic group action, and the classical structure theorems of Kostant, Atiyah, Guillemin-Sternberg and Delzant on Hamiltonian torus actions. Then I will state a structure theorem for general symplectic torus actions, and give an idea of its proof. In the second part of the talk I will introduce new symplectic invariants of completely integrable Hamiltonian systems in low dimensions, and explain how these invariants determine, up to isomorphisms, the so called "semitoric systems". Semitoric systems are Hamiltonian systems which lie somewhere between the more rigid toric systems and the usually complicated general integrable systems. Semitoric systems form a fundamental class of integrable systems, commonly found in simple physical models such as the coupled spin-oscillator, the Lagrange top and the spherical pendulum. Parts of this talk are based on joint work with with Johannes J. Duistermaat and San Vu Ngoc.