xii CONFERENCE TALKS
we prove that the moduli space of rank n, relatively semi-stable vector bundles is
corepresented by the relative Douady space of length n and relative dimension 0
subspaces of the relative Jacobian.
The signature operator on Riemannian pseudomanifolds
We consider an oriented Riemannian manifold which can be compactiﬁed by
adjoining a smooth compact oriented Riemannian manifold, B, of codimension at
least two, such that a neighbourhood of the singular stratum is given by a family
of metric cones. We show that there is a natural self-adjoint extension for the
Dirac operator on smooth compactly supported diﬀerential forms with discrete
spectrum, and we determine the condition of essential self-adjointness. We describe
the boundary conditions analytically and construct a good parametrix which leads
to the asymptotic expansion of the associated heat trace. We also give a new proof
of the local formula for the
The lost Riemann-Roch index problem
I will describe recent results of joint work with C. Consani. We determined the
real counting function N(q), (q ∈ (1, ∞)) for the hypothetical ”curve” C = Spec Z
over F1, whose corresponding zeta function is the complete Riemann zeta function.
Then, we develop a theory of functorial F1-schemes which reconciles the previous
attempts by C. Soul´e and A. Deitmar. Our construction ﬁts with the geometry
of monoids of K. Kato, is no longer limited to toric varieties and covers the case
of Chevalley groups. Finally we show, using the monoid of ad`ele classes over an
arbitrary global ﬁeld, how to apply our functorial theory of Mo-schemes to interpret
conceptually the spectral realization of zeros of L-functions. I will end the lecture
by a speculation concerning a Riemann-Roch index problem which is so far lost in
associated with integral domains
We associate canonically a C∗-algebra with every (countable) integral domain.
We describe diﬀerent realizations of this algebra which are used to analyze its
structure and K-theory.
The Baum-Connes conjecture and parametrization of group
Associated to any connected Lie group G is its so-called contraction of G to a
maximal compact subgroup. This is a smooth family of Lie groups, and a conse-
quence of the Baum-Connes conjecture is that the reduced duals of all the groups
in the family are the same, at least at the level of K-theory. A rather surprising
development from the last several years is that in key cases the duals are actually
the same at the level of sets. I shall report on recent eﬀorts, both algebraic and
geometric, to understand this phenomenon better.