Abstract

The aim of this program is to explore central questions related to moduli spaces, algebraic cycles, and enumerative geometry. These are subjects at the very heart of geometry, with origins in the Italian and German schools of algebraic geometry from the 19th century. But they are also closely connected with modern developments such as the mathematics of string theory and the theory of motives. The notions of cycles and moduli are intertwined in many subtle ways, often by means of various cohomology theories for algebraic varieties, or the geometry of period domains. One example is Deligne-s theorem about Hodge cycles on abelian varieties, which is perhaps the strongest known result in the direction of the Hodge conjecture, and whose proof uses the moduli theory of abelian varieties. As another example, the recent proof by Charles (and Lieblich-Maulik-Snowden) of the Tate conjecture for K3 surfaces over finite fields passes via the moduli theory of K3 surfaces. We expect some specific focus areas of the program to be tautological classes, logarithmic geometry, moduli spaces of sheaves, hyperkähler manifolds, and rationality questions. Participation in the program is by invitation only.