The classical Sylvester-Gallai theorem states the following: Given a finite set of points in the 2-dimensional Euclidean plane, not all collinear, there must exist a line containing exactly 2 points (referred to as an ordinary line). In a recent result, Green and Tao were able to give optimal lower bounds on the number of ordinary lines for large finite point sets.

In this talk we will consider the situation over the complex numbers. While the Sylvester-Gallai theorem as stated is false in the complex plane, Kelly's theorem states that if a finite point set in 3-dimensional complex space is not contained in a plane, then there must exist an ordinary line. Using techniques developed for bounding ranks of design matrices, we will show that either such a point set must determine at least 3n/2 ordinary lines or at least n-1 of the points are contained in a plane. (Joint work with Z. Dvir, S. Saraf and C.Wolf.)

Geometric analysis in differential geometry is a powerful tool in Riemannian geometry. It has been used to solve many problems in Riemannian geometry. In the field of several complex variables, it was not the most popular weapon to attack questions. One of the reasons is that many problems in the several complex variables relates to some types of differential equations of complex-valued functions which is currently not well understood. In this talk, we consider problems in the Diederich-Forn\ae ss index with a viewpoint of geometric analysis and see what we obtain. This is a joint work with Krantz and Peloso.

Some prominent conjectures in the theory of C*-algebras ask whether or not every C*-algebra of a particular form embeds into an ultrapower of a particular C*-algebra.  For example, the Kirchberg Embedding Problem asks whether every C*-algebra embeds into an ultrapower of the Cuntz algebra O_2.  In this series of lectures, we show how techniques from model theory, most notably model-theoretic forcing, can be used to give nontrivial reformulations of these conjectures.  We will start from scratch, assuming no knowledge of C*-algebras nor model theory.

The Toda lattice, beyond being a completely integrable dynamical system, has many important properties.  Classically, the Toda flow is seen as acting on a specific class of bi-infinite Jacobi matrices.  Depending on the boundary conditions imposed for finite matrices, it is well known that the flow can be used as an eigenvalue algorithm. It was noticed by P. Deift, G. Menon and C. Pfrang that the fluctuations in the time it takes to compute eigenvalues of a random symmetric matrix with the Toda, QR and matrix sign algorithms are universal. In this talk, I will present a proof of such universality for the Toda algorithm.  I will also discuss empirical and rigorous results for other algorithms from numerical analysis.  This is joint work with P. Deift.