Gysin sequences and cohomology ring of symplectic fillings

Zhengyi Zhou
Member, School of Mathematics
March 4, 2019

It is conjectured that contact manifolds admitting flexible fillings have unique exact fillings. In this talk, I will show that exact fillings (with vanishing first Chern class) of a flexibly fillable contact (2n-1)-manifold share the same product structure on cohomology if one of the multipliers is of even degree smaller than n-1. The main argument uses Gysin sequences from symplectic cohomology twisted by sphere bundles.

Local and global expansion of graphs

Yuval Peled
New York University
March 4, 2019

The emerging theory of High-Dimensional Expansion suggests a number of inherently different notions to quantify expansion of simplicial complexes. We will talk about the notion of local spectral expansion, that plays a key role in recent advances in PCP theory, coding theory and counting complexity. Our focus is on bounded-degree complexes, where the problems can be stated in a graph-theoretic language: 

Global well-posedness and scattering for the radially symmetric cubic wave equation with a critical Sobolev norm

Benjamin Dodson
Johns Hopkins University; von Neumann Fellow, School of Mathematics
February 28, 2019

In this talk we discuss the cubic wave equation in three dimensions. In three dimensions the critical Sobolev exponent is 1/2. There is no known conserved quantity that controls this norm. We prove unconditional global well-posedness for radial initial data in the critical Sobolev space.

Geodesic nets: examples and open problems.

Alexander Nabutovsky
University of Toronto; Member, School of Mathematics
February 26, 2019

Geodesic nets on Riemannian manifolds is a natural generalization of geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere. 

In this talk I am going to survey some results and open questions (old
and new) about geodesic nets on Riemannian manifolds. Many of these

Strongly log concave polynomials, high dimensional simplicial complexes, and an FPRAS for counting Bases of Matroids

Shayan Oveis Gharan
University of Washington
February 25, 2019

A matroid is an abstract combinatorial object which generalizes the notions of spanning trees,
and linearly independent sets of vectors. I will talk about an efficient algorithm based on the Markov Chain Monte Carlo technique
to approximately count the number of bases of any given matroid. 

The proof is based on a new connections between high dimensional simplicial complexes, and a new class
of multivariate polynomials called completely log-concave polynomials. In particular, we exploit a fundamental fact from our

Positive geometries

Thomas Lam
University of Michigan; von Neumann Fellow, School of Mathematics
February 25, 2019

Positive geometries are real semialgebraic sets inside complex varieties characterized by the existence of a meromorphic top-form called the canonical form. The defining property of positive geometries and their canonical forms is that the residue structure of the canonical form matches the boundary structure of the positive geometry. A key example of a positive geometry is a projective polytope. 

Higher symplectic capacities

Kyler Siegel
Columbia University
February 25, 2019
I will describe a new family of symplectic capacities defined using rational symplectic field theory.
These capacities are defined in every dimension and give state of the art obstructions for various "stabilized" symplectic embedding problems such as one ellipsoid into another. They can also be described via symplectic cohomology and are related to counting pseudoholomorphic curves with tangency conditions. I will explain the basic idea of the construction and then give some computations, structural results, and applications.

Deep Learning: Alchemy or Science?

See agenda
February 22, 2019

Deep learning has led to rapid progress in open problems of artificial intelligence—recognizing images, playing Go, driving cars, automating translation between languages—and has triggered a new gold rush in the tech sector. But some scientists raise worries about slippage in scientific practices and rigor, likening the process to “alchemy.” How accurate is this perception? And what should the field do to combine rapid innovation with solid science and engineering?