Moduli theory and singularities

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The investigator will work on several problems in higher dimensional algebraic geometry, especially moduli theory and singularities. In several projects joint with Kollár, the PI plans to work on various problems related to the existence of a coarse moduli space of stable log varieties, an analog of the moduli space of stable pointed curves. These include the study of rational pairs and thrifty resolutions and their connections with Du Bois singularities and other singularities of the minimal model program. In another project, also motivated by the moduli project, jointly with Patakfalvi the PI will work on proving a logarithmic version of Kollár's Ampleness Lemma and use it to prove the projectivity of the moduli space of stable log varieties. The PI will also continue his work on the refined Viehweg conjecture regarding subvarieties of moduli stacks of canonically polarized smooth projective varieties. This conjecture 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. This project is joint work with Kebekus. 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.