Let $A(n)$ be the minimum area of convex lattice $n$-gons.
(Here lattice is the usual lattice of integer points in $R^2$.)
G. E. Andrews proved in 1963 that $A(n)>cn^3$ for a suitable positive $c$.
We show here that $\\lim A(n)/n^3$ exists, and explain what the shape of
the minimizing convex lattice $n$-gon is. This is joint work with
Norihide Tokushige.

A set of the form $C\\cap\\bf{Z}^d$, where $C\\subseteq R^d$ is convex
and $Z^d$ denotes the integer lattice, is called a {\\it convex
lattice set}. I will explain that the Helly number
of $d$-dimensional convex lattice sets is $2^d$.
However, the {\\it fractional Helly number\\/} is only $d+1$:
For every $d$ and every $\\alpha\\in (0,1]$ there exists $\\beta>0$
such that whenever $F_1,\\ldots,F_n$ are convex lattice sets in $\\bf{Z}^d$
such that $\\bigcap _{i\\in I} F_i\\neq\\emptyset$
for at least $\\alpha{n\\choose d+1}$ index sets $I\\subseteq\\{1,2,\\ldots,n\\}$
of size $d+1$, then there exists a (lattice) point common to
at least $\\beta n$ of the $F_i$. This implies a $(p,d+1)$-theorem
for every $p\\geq d+1$; that is, if $H$ is a finite family
of convex lattice sets in $\\bf{Z}^d$ such that among every $p$ sets of $H$,
some $d+1$ intersect, then $H$ has a transversal of size
bounded by a function of $d$ and $p$. This is joint work with J.
Matousek.

We will show how number theory can be used to make communication secure.

Refreshments will be served!

For a sequence A of integers, S(A) denotes the collection of partial sums
of A. About forty years ago, Erdos and Folkman made the following
conjecture: Let A be an infinite sequence of integers with density at
least $Cn^{1/2}$ (i.e., A contains at least $Cn^{1/2}$ numbers between $1$ and n
for every larger n), then S(A) contains an infinite arithmetic
progression. Partial results have been obtained by Erdos (1962), Folkman
(1966), Hegyvari (2000), Luczak-Schoen (2000). Together with Szemeredi,
we have recently proved the conjecture. In this talk, I plan to survey
this development.