CAREER: Theory of Moduli

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The investigator studies problems in algebraic geometry related to families of higher dimensional varieties. The main goal of the project is to find good definitions of compact moduli functors, with special regard to the moduli theory of surfaces. As part of the moduli theory project special efforts are devoted to reflexive sheaves and their behavior with respect to morphisms in an effort to develop a theory that includes many results for reflexive sheaves of rank 1 that are similar to results already known for line bundles. This is very important in order to develop a moduli theory of singular varieties, which in turn is essential for geometrically meaningful compact moduli spaces. Another problem the investigator is studying is generalizations of Shafarevich's conjecture and its applications to the case of higher dimensional bases, in particular the problem of boundedness and rigidity for families of varieties of general type. 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. This project focuses 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. Presently the existence of compact moduli spaces is known for low dimensional problems. The investigator is studying the higher dimensional case. As part of the project, but somewhat independent of the above, the investigator and his graduate students are running a website that works as a forum for graduate students and young researchers in algebraic geometry. Users of this website can pose questions that are likely to be known to experts but not available in the literature. The main goal of this project is to open a new venue for a larger number of students to access the knowledge of experts in the field helping to achieve a more equal opportunity environment. In addition, the collected database provides references to results that are too "small" to occupy an entire article, but too important to ignore, as well as a collection of potential research problems for graduate students.