COMS

The geometry of moduli spaces has been widely investigated and it is certainly one of the main subjects in Algebraic Geometry. In particular, moduli spaces of curves gave a major boost to the theory due to their many connections with Theoretical Physics and counting invariants. However, to count invariants it is very useful to apply intersection theory techniques that require the space to be proper, and moduli spaces are usually not proper. The problem of compactifying them is not trivial. Geometric Invariant Theory (GIT) may be a way to tackle the problem, but the limit points can be hard to describe geometrically.

Nowadays, new notions of stability and new techniques related to the minimal model program have been combined with GIT to provide new points of view to attack the problem and, more generally, to study the birational geometry of compactified moduli spaces. This has brought back many important insights again about counting invariants and mirror symmetry and has also been used to study other moduli problems, like the moduli of coherent sheaves or the moduli spaces of higher dimensional varieties.

The lecturers have been chosen among the leading experts of the area. Following the usual policy of the Advanced Schools at CRM, informal notes will be delivered by the speakers to the participants at the beginning of the school.

In its simplest form, Hodge theory is the study of periods - integrals of algebraic differential forms which arise in the study of complex geometry, number theory and physics. Its difficulty and richness arises in part from the non-algebraicity of these integrals. According to the beautiful conjectures of Hodge, Bloch and Beilinson, what algebraic structure they have should be explained by (generalized) algebraic cycles. There has been much recent progress on these conjectures and on classifying spaces for periods, as well as their asymptotic and arithmetic; this conference will bring together leading scholars and students of these topics.

Given that our desire is to create a number of new collaborations, we would aim at a small number of high-quality talks per day with ample breaks between each. In addition to bringing together leading scholars on the different aspects of the asymptotic, symmetries, and arithmetic of periods together for a research conference, the organizers will also arrange for a prior workshop for graduate students and recent Ph.D.'s on major themes of the conference. We expect this expository part to run approximately 4 days, with lectures by the organizers and some of the invited speakers.

We welcome participation from both theoretical mathematical areas and application areas not on this list which fall under this broadly interpreted notion of algebraic geometry and its applications.