James Lewis  University of Alberta
Title: An Archimedean height pairing on the equivalence relation defining
Bloch's higher Chow groups
Abstract: The existence of a height pairing on
the equivalence relation defining Bloch's higher Chow
groups is a surprising consequence of some
recent joint work by myself and Xi Chen on
a nontrivial $K_1$class on a selfproduct
of a general $K3$ surface. I will explain how this
pairing comes about.

Patrick Brosnan University of British Columbia Title: The zero locus of an admissible normal function
Abstract:
Let S be an algebraic variety and H a variation of pure Hodge structure on S of odd weight. Then H induces over S a family J(H) of compact complex tori whose fibers are called Griffiths intermediate Jacobians. Admissible normal functions are certain sections of J(H) which are wellbehaved at infinity. Since J(H) is an analytic variety, it is obvious that the locus where a normal function is zero is analytic in S. However, it is far from clear that it is algebraic because, in general, the Griffiths intermediate Jacobian is not an algebraic object and, therefore, normal functions cannot be thought of as exactly algebraic objects.
I will discuss joint work with Gregory Pearlstein in which we prove that, in fact, the zero locus of an admissible normal function over an algebaic base is algebraic.

S.Mahanta  Johns Hopkins University Title: F_1geometry and Galois extensions Abstract: The motivation for developing a geometry over F_1 (the field with one element) arises, on the one hand, from Tits' early work on Chevalley group schemes and, on the other, from some considerations in arithmetic geometry. After a brief survey of some of the approaches towards F_1 geometry I shall discuss a notion of finite Galois extensions over F_1 and, if time permits, outline a possible construction of the Witt vectors.

G.V.Ravindra University of Missouri, St. Louis Title: Curves and bundles on threefolds
Abstract: The NoetherLefschetz theorem says that for a "very general"
hypersurface X of degree at least 4 in P^3, any curve C in X occurs as
an intersection of X and S where S is another surface in P^3.
Motivated by this, Griffiths and Harris asked whether any curve C in
X, where X is now a general hypersurface of degree at least 6 in P^4,
is an intersection of X with a surface S in P^4. C. Voisin showed that
there exists curves which are not of this form. One would still be
interested to know whether there is indeed a "generalised
NoetherLefschetz theorem" in this situation.
We will explain how arithmetically CohenMacaulay (ACM) curves and
bundles provide an answer in this direction.
1) In the first part, we shall show that ACM curves provide examples
of curves which are not intersections as above and show that Voisin
examples can be understood as special cases.
2) We will then sketch a proof of the following characterisation of
complete intersection curves on a general smooth projective
hypersurface of dimension three and degree at least six:
" a curve in such a hypersurface is a complete intersection if and
only if it is arithmetically Gorenstein (i.e. it is the zero locus of
a nonzero section of a rank two bundle with vanishing intermediate
cohomology). "
Apart from the obvious motivation for such a theorem, we shall show
a) how this theorem can be thought of as a generalisation of the
classical NoetherLefschetz theorem for curves on two dimensional
hypersurfaces.
b) that this implies the following interesting fact: a "general"
homogeneous polynomial in five variables of degree at least six cannot
be obtained as the Pfaffian (square root of the determinant) of an
even sized "minimal" skewsymmetric matrix with homogeneous polynomial
entries.
c) Relate it to Horrocks' criteria for a vector bundle on projective
space to be split, and finally
d) show that it can be viewed as a verification of a (strengthening of
a) conjecture of BuchweitzGreuelSchreyer in commutative algebra and
algebraic geometry.
This is based on joint work with N.Mohan Kumar and A.P.Rao in St.
Louis and I.Biswas and J.Biswas in India.
