Abstract of Talks Beginning 10/4/00 <p>

		by W. L. Baily, Jr.    <p>

In these talks we consider moduli problems connected with,
or possibly connected with, some arithmetic quotients of
certain bounded symmetric domains. The investigation
of moduli problems possibly connected with a certain "nice"
arithmetic quotient of the 27-dimensional exceptional domain
associated with the real exceptional Lie group E_{7(-25)}
is the central object of interest.  It is at the top of
(or of highest dimension among) the list of the four
irreducible bounded symmetric tube domains of real rank  3.
The other three such domains are the Siegel upper half-space
of real rank 3 and dimension 6, the tube domain over the cone
of positive definite complex 3X3 hermitian matrices of
dimension 9, and the tube domain over the cone of positive
definite hermitian quaternion 3X3 matrices of dimension 15.
Corresponding to these four symmetric tube domains there
are four so-called Severi varieties of dimensions 2, 4,
8, and 16, realized explicitly as the Veronese surface in
P^5, the product P^2 X P^2 imbedded by the Segre mapping
in  P^8, the Grassmannian  G(2,6) contained in P^{14}
via the Plucker imbedding, and the Cayley projective
plane contained in  P^{26} via a certain imbedding described
in terms of the irreducible idempotents in an exceptional
Jordan algebra of dimension 27.  These two sets of four
objects appear to be related in some way, but a completely
clear explanation of the relationship seems to be unknown. <p>

Anyway, the 6-dimensional Siegel upper half-space
is connected in a well-known way with the moduli of
principally polarized Abelian varieties of dimension 3
and thus, via Torelli's theorem, with the moduli of
non-hyperelliptic curves of genus 3; the 9-dimensional
tube domain over the cone of positive complex hermitian
3X3 matrices is connected with the moduli of principally
polarized Abelian varieties with complex multiplication
of Weil-type by  i and also, somewhat surprisingly, via
a tube domain of type IV of dimension 9, with the moduli
of nodal Enriques surfaces; the 15-dimensional tube domain
over the cone of positive quaternion hermitian 3X3 matrices
is connected with the moduli of principally polarized
Abelian varieties of dimension 12 having complex
multiplicationby the Hurwitz ring of quaternion integers
and now, conjecturally (which is where part of the
mystery begins) with the moduli of certain Fano  4-folds.
At the next, and top, stage, the 27-dimensional exceptional
Jordan domain, one can so far only conjecture the existence
of something parallel to the earlier cases.  This is
the main and, so far, quite unexplored area.