## On the number of ordinary lines determined by sets in complex space

Shubhangi Saraf
Rutgers University
December 5, 2016
Consider a set of $n$ points in $\mathbb R^d$. The classical theorem of Sylvester-Gallai says that, if the points are not all collinear then there must be a line through exactly two of the points. Let us call such a line an "ordinary line". In a recent result, Green and Tao were able to give optimal linear lower bounds (roughly $n/2$) on the number of ordinary lines determined $n$ non-collinear points in $\mathbb R^d$. In this talk we will consider the analog over the complex numbers.

## A spectral gap in $\mathrm{SL}^2(\mathbb R)$ and applications: expansion, Furstenberg measures and the Anderson-Bernoulli model

Jean Bourgain
IBM von Neumann Professor, School of Mathematics
November 30, 2016

## $C^0$ Hamiltonian dynamics and a counterexample to the Arnold conjecture

After introducing Hamiltonian homeomorphisms and recalling some of their properties, I will focus on fixed point theory for this class of homeomorphisms. The main goal of this talk is to present the outlines of a $C^0$ counterexample to the Arnold conjecture in dimensions four and higher. This is joint work with Lev Buhovsky and Vincent Humiliere.
## Rectification and the Floer complex: quantizing Lagrangians in $T^*N$
We shall give a construction of the quantized sheaf of a Lagrangian submanifold in $T^*N$ and explain a number of features and applications.