current general 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001	March 24 Utrecht, Room K11 (basement) 11:30-12:30, 13:30-14:30 E. Friedman (U. of Chile), Lehmer's Conjecture and Relative Regulators Abstract. Smyth showed in 1971 that if a polynomial (assumed monic, irreducible, with integer coefficients and of degree at least 2) is not palindromic, then its roots cannot be too close to the unit circle, in a certain precise sense. (A polynomial is palindromic if the sequence of coefficients reads the same backwards or forwards.) Lehmer's conjecture states that Smyth's conclusion should hold even for palindromic polynomials, as long as they are not cyclotomic. The simplest open case of Lehmer's conjecture is that of Salem numbers, i.e. of a palindromic polynomial having only one root outside the unit circle. Such a root α is called a Salem number. It is an algebraic unit which generates a number field L=Q(α), endowed with a subfield K=Q(α+α-1) with [L:K]=2 and NormL/K(α)=1. Thus α is a ``relative unit'' of the extension L/K. I will describe regulators of relative units and a conjectural lower bound for them which would imply the Salem number case of Lehmer's conjecture. I will sketch a proof of this conjecture in the high-rank case. Unfortunately the Salem number case is the rank-one case. I will end by presenting a formula which might pave the way for a proof of the low-rank case. This is joint work with N.-P. Skoruppa, published some 6 years ago, but still unfinished in the low-rank case. 14:45-15:30, 15:45-16:30 F. Beukers (Utrecht), Lower bounds for heights of algebraic points Abstract. We consider the algebraic points on an algebraic variety defined over the algebraic closure of Q and their absolute normalised logarithmic heights. In many cases it turns out that one can give uniform non-trivial lower bounds for these heights. We start with a remarkable elementary result by Zhang-Zagier and then proceed to discuss some wide generalisations, with applications to diophantine equations.