Arakelov Theory, Motives and Special Values

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Professor Gillet will be studying a number of problems related to Arakelov Theory, Motives, and special values of zeta and L-functions. He plans to show that there is a specialization map for motives of varieties over discretely valued fields; this would for example describe the relationship between the motive of a variety defined over the rational numbers and the associated, possibly singular, varieties over finite fields. He plans to study whether one can do arithmetic intersection theory on singular varieties, and improve our understanding of the height pairing on algebraic cycles which are homologically trivial. He also intends to study questions relating heights and similar integrals to special values of Dirichlet series. This includes understanding the relationship between heights of conics and the eigenvalues of Heun's equation, which is a classical differential equation arising from problems in physics.The overall goal of these areas of research is to improve our understanding of Diophantine equations, that is, understanding the properties of the set of integer solutions of equations with integer coefficients. Diophantine equations have applications in particular to problems in cryptography and coding theory.