Higher Dimensional Algebraic Geometry

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The investigator will work on several problems in higher dimensionalalgebraic geometry. In a joint project with Alexeev, Hassett, andKoll\'ar he plans to complete the proof of existence of coarsecomplete moduli spaces of stable log surfaces, an analog of the modulispace of stable curves. In another project the investigator is goingto work on boundedness, rigidity and hyperbolicity of subvarieties ofmoduli stacks of canonically polarized smooth projective varieties. These questions evolved from a landmark conjecture of Shafarevich, andits solution by Arakelov and Parshin, which played an important rolein Faltings' proof of the Mordell Conjecture. Part of this project isjoint work with Kebekus. In a third project the investigator andHacon are going to study the impact of the existence of nowherevanishing differential forms on the geometry of the underlyingvariety. Their goal is to prove several outstanding conjectures in thearea. In a fourth project the investigator and Araujo will worktoward proving a conjecture of Beauville giving a new characterizationof projective spaces and quadric hypersurfaces.This research is in the field of algebraic geometry, one of the oldestparts of modern mathematics, but one that blossomed to the point whereit has solved problems that have stood for centuries. Originally, andstill in its simplest form it treats figures defined in the plane bypolynomials. Today, the field uses methods not only from algebra, butalso from analysis and topology, and conversely it is extensively usedin those fields. Moreover it has proved itself useful in fields asdiverse as physics, theoretical computer science, cryptography, codingtheory and robotics. A central problem in algebraic geometry is theclassification of all geometric objects. In turn, an important part ofclassification theory is the theory of moduli. The latter's core ideais that one does not only want to understand these objects, but alsounderstand the way they can be deformed. Moduli spaces play a veryimportant role in theoretical physics. Studying curves on modulispaces provides information on how an object is changing inspace-time. One of the focuses of this project is on compact modulispaces. Those are extensions of moduli spaces in general and they giveadditional information about singular deformations, ones that areessentially different from others. Other goals of the project involvea better understanding of certain higher dimensional varieties.