The investigator and his collaborator have established a connection between the Castelnuovo Theory of subvarieties in projective space, and the geometric Schottky problem, i.e. that of identifying Jacobians among all principally polarized abelian varieties based on geometric properties of the polarization. This includes in particular a relationship with the celebrated Trisecant Conjecture of Welters. They are now approaching directly the Trisecant Conjecture, using these connections, as well as homological $M$-regularity methods. They are also interested in developing a "higher Castelnuovo-Schottky theory", based on deeper similarities between subvarieties in projective space and those of abelian varieties. In this respect, one of the main points they are interested in is a conjecture of Debarre on which are the subvarieties of principally polarized abelian varieties representing minimal classes.
The Trisecant Conjecture, or the Strange Duality Conjecture, are among the most prominent conjectures in the respective directions (which can be called roughly speaking the abelian and non-abelian theory of theta functions). The proved statements, or even significant progress towards them, would have a large number of consequences, as documented in numerous places in the literature. Further development of the asymptotic methods has the potential to provide new insight into the geometry of every smooth projective variety, the main objects of study in algebraic geometry. All parts of the project would further the knowledge in the field, would have a broad range of applications, and will create interaction with people of different backgrounds. Many of the problems in this proposal are of interest to researchers in adjacent fields, like complex analysis (theta functions), complex analytic geometry (transcendental methods in algebraic geometry) and conformal field theory (conformal blocks), and will lead to interactions with some of them.
