Arithmetic of K3 surfaces (08w5083)

The Banff International Research Station will host the workshop: “Arithmetic of K3 surfaces” next week, November 30 - December 5, 2008.

Arguably the most famous theorem of mathematics is Pythagoras'
theorem. Most people remember it from high school as "a-squared plus
b-squared equals c-squared." Here c is the hypotenuse of a
right-angled triangle, while a and b are the other two sides. Although
this theorem allows the hypotenuse of any right-angled triangle to be
computed, most exercises about this theorem make use of some special
triangles where all sides happen to be whole numbers. The smallest
example is the triangle with sides (3,4,5), but other commonly used
examples are (5,12,13), or (7,24,25), or (8,15,17). These triangles
correspond to the points (3/5,4/5), (5/13,12/13), (7/25,24/25), and
(8/17,15/17) on the unit circle given by x^2 + y^2 = 1. One can prove
that there are in fact infinitely many of these special triangles,
corresponding to infinitely many points with rational coordinates on
the unit circle. Such points are called rational points.

Some mathematicians, often described as number theorists, do not care
much about the triangles, but they are interested in the solutions in
whole numbers of the equation a^2 + b^2 = c^2. These (infinitely many)
solutions have been understood for millennia now. Only a little over
a decade ago, Fermat's Last Theorem was proved, which states that if
we replace the exponent 2 by any other whole number greater than 2,
then all solutions to the equation are trivial in the sense that one
of the variables a, b, and c is equal to zero. Increasing the
exponents in the equation makes the problem significantly harder.
Another way to increase the complexity of the problem is by increasing
the number of variables. Instead of looking for rational points on the
unit circle, we then look for rational points on objects of dimension
2 or greater. These varieties, as they are called, can be classified
and on some of them the rational points are very well understood.
On many varieties, however, they are not. A K3 surface is an example
of such a variety. For example, for some years it was not known
whether the K3 surface given by the equation x^4 + 2y^4 - 4z^4 = 1
has any solutions other than x = 1 or x = -1 and y = z = 0. In fact
there are others, such as x = 1484801/1157520, y = 1203120/1157520,
z = 1169407/1157520, and the solutions that are obtained by taking the
negative of some of the coordinates. It is still not known, however,
whether there are any more solutions. In the last five years the rate
of progress on the understanding of rational points on K3 surfaces
has increased dramatically. This is the first international workshop
to join the forces of all the mathematicians involved in this process.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologí­a (CONACYT).