Compactification of Certain Moduli Spaces, and Some Finiteness Problems in Arithmetic Geometry

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This award supports the research of Professor D. Abramovich in the field of algebraic geometry. Abramovich will continue his study of the Gieseker-Morrison compactification of the moduli space of vector bundles of rank 2 on semistable curves, with the goals of identifying the vector bundles corresponding to points on the boundary, extending the construction to higher ranks and comparing with other compactifications. He will also continue his work on Lang's conjecture in characteristic p, and will study integral and torsion points on elliptic curves and integral points on algebraic stacks. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; 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.