Frits Beukers: Hall's conjecture      

Hall's conjecture states that there is a positive real number $C$ such that
for every pair of positive integers $x,y$ we have either $x^3=y^2$
or $|x^3-y^2|>C\sqrt{x}$. In this lecture we discuss the pro's and
con's for this conjecture together with a beautiful idea of Elkies
to carry out numerical verifications.          

Phong Nguyen: The two faces of lattices in cryptology

Lattices are regular arrangements of points in n-dimensional sapce.
Since the appearance of the celebrated Lenstra-Lenstra-Lovasz
lattice basis reduction algorithm twenty years ago, lattices have had
surprising applications in cryptology.
Until recently, the applications of lattices to cryptology
were only negative, as lattices were used to break
various cryptographic schemes.
Paradoxically, several positive cryptographic applications of lattices
have emerged in the past five years: there now exist public-key
cryptosystems based on the hardness of lattice problems, and lattices
play a crucial role in a few security proofs.
We survey a few examples of the two faces of lattices in cryptology. 

Aart Blokhuis: Button Madness

Button Madness is a solitaire game, played on a 4 by 4 array of buttons,
that are either red or green. Pushing a button changes its color,
and that of its four neighbours. A button on the edge is considered to be
a neighbour of the corresponding button on the opposite edge.
The object of the game is, starting from an arbitrary
initial position, to reach the situation where every button is green.
Experience shows
that this can always be achieved.
The worst case is the situation where every button
is red. In this case every button has to be pushed.

For a 5 by 5 board the situation
is different. In this case not every pattern can be solved.
Let is call the number
$n$ {\em mad} if it is not possible to solve every pattern on a $n\times
n$-board.
It can be verified that the numbers
$n=3,\,5,\,6,\,9,\,10,\,12,\,15,\,17,\,18,\,20$
are the mad numbers $n\le 20$.      
In the talk we will give some equivalent criteria to determine whether a given
number $n$ is mad. In this way we are able to determine all mad numbers
$n<100000$,
and give infinite sequences of mad numbers, and of non-mad numbers.