Diophantine problems: new and old perspectives

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Number theory, one of the oldest branches of mathematics, has enjoyed a renaissance in recent years. On one hand, the remarkable proof of Fermat's Last Theorem by Andrew Wiles has reaffirmed Number theory's position as a central one in modern mathematics. On the other, numerous and striking applications to such diverse problems as data encryption and signal transmission have demonstrated the utility of the field, both for industrial and governmental purposes.One of the fundamental problems in Number  theory is that of determining the number and distribution of integral or rational points upon curves of positive genus. In the project at hand, the investigator studies a number of related questions in the subfield of Diophantine equations, including some stemming from work of Wiles. The techniques employed allow the complete resolution of a number of classic problems in this area. These equations are connected to the theory of error-correcting codes which, in turn, are of crucial importance to, for example, data processing from satellites, cd and dvd players, cell phones and smart cards.