Title: A curious diophantine problem
Abstract: The origin of the problem was in trying to prove a
non-trivial bound for the triple correlation in a certain physical
system.
The problem can be studied using
1) Combinatorics
2) Diophantine Approximation
3) Arithmetic Geometry
In this talk I will discuss the approach using combinatorics.
(work in progress with Bourgain)
Tusday
June 7
15:00-16:00
Gert-Martin Greuel (Universität
Kaiserslautern)
Title: Asymptotic Bounds for Singularities on Algebraic Curves
Abstract: We study families of projective algebraic curves with
prescribed
number and types of singularities: their existence, smoothness and
irreducibility. Substantial progress on these classical problems has
been made only in the last decade. Bezout's theorem or the genus
formula show that the sum of the Milnor numbers of the singularities
is bounded by a quadratic function in the degree of the curves. We
show that, with respect to existence, this necessary bound is
asymptotically proper. That is, up to some constant, it is also
sufficient for the existence of a curve with prescribed singularity
types if the degree goes to infinity. We discuss further proper and in
some cases even optimal asymptotic bounds for the existence,
T-smoothness, and irreducibility of such families and report on open
problems and conjectures.