This research project 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. 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 foci of this project is on compact moduli spaces, which give additional information about singular deformations, ones that are essentially different from others. The investigator is also involved in promoting mathematics to high school students; he and fellow directors of the Summer Institute for Mathematics at the University of Washington make special efforts to involve women at all levels of the program, from students through teaching assistants to instructors, to reinforce their leadership roles in mathematics.
This project concerns several topics in higher dimensional algebraic geometry, especially moduli theory and singularities. The overarching theme of the research is centered on compact moduli spaces of stable log varieties, an important area that is still in the developmental stage. In particular, even the correct moduli functor needs to be identified, and most of the research is motivated by understanding the basic properties of these moduli spaces. This involves understanding the singularities that can occur on stable log varieties. Important classes of singularities in this regard are that of rational singularities and other singularities of the minimal model program. The investigator will work on advancing our currently very limited understanding of these singularities in arbitrary characteristic. The project also aims to develop cohomological methods to deal with rational pairs and thrifty resolutions in arbitrary characteristic. In particular, the investigator will study logarithmic versions of Hodge cohomology and of Grothendieck's fundamental class. Also stemming from the moduli project, the investigator plans to study properties of Du Bois singularities and Du Bois pairs. The main objectives of this part of the project are to develop a definition of these singularities that makes sense in arbitrary characteristic and to prove a subadjunction theorem for rational and Du Bois pairs.
