Princeton University

Applications of additive combinatorics to Diophantine equations

Alexei Skorobogatov
Imperial College London
April 10, 2014
The work of Green, Tao and Ziegler can be used to prove existence and approximation properties for rational solutions of the Diophantine equations that describe representations of a product of norm forms by a product of linear polynomials. One can also prove that the Brauer-Manin obstruction precisely describes the closure of rational points in the adelic points for pencils of conics and quadrics over \(\mathbb Q\) when the degenerate fibres are all defined over \(\mathbb Q\).

Minimal Discrepancy of Isolated Singularities and Reeb Orbits

Mark McLean
Stony Brook University
April 4, 2014
Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry.

Princeton/IAS Symplectic Geometry Seminar

Keon Choi
University of California, Berkeley
March 7, 2014
Embedded contact homology is an invariant of a contact three-manifold, which is recently shown to be isomorphic to Heegaard Floer homology and Seiberg-Witten Floer homology. However, ECH chain complex depends on the contact form on the manifold and the almost complex structure on its symplectization. This fact can be used to extract symplectic geometric information (e.g. ECH capacities) but explicit computation of the chain complexes has been carried out only on a few cases.

Contact invariants in sutured monopole and instanton homology

Steven Sivek
University of Warwick
March 5, 2014
Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of Juhász in Heegaard Floer homology. In this talk, I will construct an invariant of a contact structure on a 3-manifold with boundary as an element of the associated sutured monopole homology group. I will discuss several interesting properties of this invariant, including gluing maps and an exact triangle associated to bypass attachment, and explain how this construction leads to an invariant in the sutured version of instanton Floer homology as well.

On Floer cohomology and non-archimedian geometry

Mohammed Abouzaid
Columbia University
February 14, 2014
Ideas of Kontsevich-Soibelman and Fukaya indicate that there is a natural rigid analytic space (the mirror) associated to a symplectic manifold equipped with a Lagrangian torus fibration. I will explain a construction which associates to a Lagrangian submanifold a sheaf on this space, and explain how this should be the mirror functor.

Cylindrical contact homology as a well-defined homology?

Joanna Nelson
Institute for Advanced Study; Member, School of Mathematics
February 7, 2014
In this talk I will explain how the heuristic arguments sketched in literature since 1999 fail to define a homology theory. These issues will be made clear with concrete examples and we will explore what stronger conditions are necessary to develop a theory without the use of virtual chains or polyfolds in 3 dimensions. It turns out that this can be accomplished by placing strong conditions on the growth rates of the indices of Reeb orbits.

Complex analytic vanishing cycles for formal schemes

Vladimir Berkovich
Weizmann Institute of Sciences; Member, School of Mathematics
December 12, 2013
Let \(R={\cal O}_{{\bf C},0}\) be the ring of power series convergent in a neighborhood of zero in the complex plane. Every scheme \(\cal X\) of finite type over \(R\) defines a complex analytic space \({\cal X}^h\) over an open disc \(D\) of small radius with center at zero. The preimage of the punctured disc \(D^\ast=D\backslash\{0\}\) is denoted by \({\cal X}^h_\eta\), and the preimage of zero coincides with the analytification \({\cal X}_s^h\) of the closed fiber \({\cal X}_s\) of \(\cal X\).