Number Theory and Geometry

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This project concerns research in number theory and geometry. These areas of mathematics have applications to computer science and to cyber security, which are areas of emphasis in the project. In addition to the research the PI will continue to- work with undergraduate and graduate students and postdoctoral researchers, to introduce them to ideas and methods of use in their own research; - write expository articles and notes for journals and for online mathematical sites; - maintain and develop web pages that feature interaction between sophisticated research mathematics and elementary topics and puzzles; - present expository mathematical lectures, some aimed at high school students or non-math majors in college, often featuring mathematical aspects of music or chess. This research project addresses fundamental theoretical and computational questions in arithmetic geometry: the behavior of elliptic curves and higher varieties (especially K3 surfaces, Calabi-Yau threefolds, and their moduli varieties) over finite fields, function fields, and number fields. It also explores the construction and application of generalized weight enumerators of error-correcting codes and related problems on spherical codes and analogous combinatorial structures. Better understanding of spherical and error-correcting codes may increase the efficiency of analog and digital communication that relies on such structures.