# Algebraic Geometry

## Ben Green: the Sylvester-Gallai Theorem

These pictures are of a lecture given by Ben Green on the Sylvester-Gallai Theorem

## On the Sylvester-Gallai Theorem

The Sylvester-Gallai Theorem states that, given any set P of n points in the plane not all on one line, there is at least one line through precisely two points of P. Such a line is called an ordinary line. How many ordinary lines must there be? The Sylvester-Gallai Theorem says that there must be at least one but, in recent joint work with T. Tao, we have shown that there must be at least n/2 if n is even and at least 3n/4 - C if n is odd, provided that n is sufficiently large. These results are sharp

Photos of this event are also available.

## Expanders, Group Theory, Arithmetic Geometry, Cryptography and Much More

This is a lecture given on the occasion of the launch of the PIMS CRG in "L-functions and Number Theory".

The theory of expander graphs is undergoing intensive development. It finds more and more applications to diverse areas of mathematics. In this talk, aimed at a general audience, I will introduce the concept of expander graphs and discuss some interesting connections to arithmetic geometry, group theory and cryptography, including some very recent breakthroughs.

## Frozen Boundaries and Log Fronts

In this talk, based on joint work with Richard Kenyon and Grisha Mikhalkin, Andrei Okounkov discusses a binary operation on plane curves which

- generalizes classical duality for plane curves and
- arises naturally in probabilistic context,

namely as a facet boundary in certain random surface models.

## Hyperplane Arrangements and Applications

Some photos from the Hyperplane Arrangements and Applications conference which took place at UBC Vancouver, August 8-12. This conference was held in honour of Hiroaki Terao.

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