Mathematical Sciences: Diophantine Approximations of Algebraic Points

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This award supports the research in Diophantine approximations of Professor Paul Vojta of the University of California at Berkeley. Dr. Vojta's project is to extend his already-productive analogy between Nevanlinna Theory and Diophantine approximation theory, so as to get information on algebraic points of bounded degree on varieties. In addition, he hopes to generalize to varieties of higher dimension his independent proof of the Mordell-Faltings Theorem that arose from applying his viewpoint to Dyson's sharpening of the Thue-Siegel Theorem. This is research in the field of arithmetic algebraic geometry, a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory started with the whole numbers and such questions as divisibility of one whole number by another. These two subjects, seemingly so far apart, have in fact influenced each other from the earliest times, but in the past quarter century the mutual influence has increased greatly. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics, and is having corresponding influence beyond its own borders.