CAREER: Undecidability in Number Theory and Applications of Arithmetic Geometry

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The investigator will study several questions related to diophantine equations. One focus is on generalizations of Hilbert's Tenth Problem. Hilbert's Tenth Problem over the rational numbers and over number fields in general is still open while undecidability is known for several subrings of the rational numbers. One goal is to investigate for which subrings of number fields the problem is undecidable. Another is to extend the currently known undecidability results for function fields. Research on Hilbert's Tenth Problem has led to new areas of interaction between number theory, arithmetic geometry and logic. These will be further explored in this project. Another focus is the study of computational problems in arithmetic geometry that have applications to cryptography. One goal is to construct curves of small genus that are suitable for cryptographic applications. Another goal is to generalize the quantum algorithms for number fields to function fields.The investigator proposes several research projects that involve studying the solutions to multivariable polynomial equations. Looking for solutions to such equations over the integers or rational numbers is one of the fundamental problems in number theory. It has a long history that goes back to ancient Greece. For the first project the investigator will study the fundamental question of whether it is possible to find a procedure that determines whether an arbitrary multivariable polynomial equation has a solution in a given number system. The second project focuses on computational aspects of certain special classes of equations that have applications to cryptography. This area of mathematics is very well-suited for motivating young students to study mathematics. The investigator will teach middle school and high school girls about cryptography and its mathematical background. There will be two yearly workshops which will involve hands-on experiments in the computer lab. There will also be professional development workshops for local mathematics teachers.