The investigator will work on several problems in higher dimensional algebraic geometry. In a joint project with Alexeev, Hassett, and Koll'ar he plans to complete the proof of existence of coarse complete moduli spaces of stable log surfaces, an analog of the moduli space of stable curves. In another project the investigator is going to work on boundedness, rigidity and hyperbolicity of subvarieties of moduli stacks of canonically polarized smooth projective varieties. These questions evolved from a landmark conjecture of Shafarevich, and its solution by Arakelov and Parshin, which played an important role in Faltings' proof of the Mordell Conjecture. Part of this project is joint work with Kebekus. In a third project the investigator and Hacon are going to study the impact of the existence of nowhere vanishing differential forms on the geometry of the underlying variety. Their goal is to prove several outstanding conjectures in the area. In a fourth project the investigator and Araujo will work toward proving a conjecture of Beauville giving a new characterization of projective spaces and quadric hypersurfaces.
This research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one that blossomed to the point where it has solved problems that have stood for centuries. Originally, and still in its simplest form it treats figures defined in the plane by polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. A central problem in algebraic geometry is the classification of all geometric objects. In turn, an important part of classification theory is the theory of moduli. The latter's core idea is that one does not only want to understand these objects, but also understand the way they can be deformed. Moduli spaces play a very important role in theoretical physics. Studying curves on moduli spaces provides information on how an object is changing in space-time. One of the focuses of this project is on compact moduli spaces. Those are extensions of moduli spaces in general and they give additional information about singular deformations, ones that are essentially different from others. Other goals of the project involve a better understanding of certain higher dimensional varieties.
