Groups and Arithmetic Geometry

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The principal investigator's research explores the connections between three of the central concepts in algebra: groups, fields, and varieties. Groups are symmetry types; for instance, all bilaterally symmetric animals share a symmetry group, which is different from the symmetry group of a starfish. Fields are systems of numbers which can be added, subtracted, multiplied, or divided; for instance, the usual real numbers form a field, but the rational numbers, which can be expressed as fractions, form a smaller field of particular interest to number theorists. Varieties are systems of simultaneous equations in a number of variables. The PI is trying to understand the deep connections between these three concepts. For instance, simple groups, the building blocks for all finite groups, can almost always be expressed essentially as the points of a variety over a finite field. Varieties determine fields whose symmetry groups have been of interest in mathematics for more than 200 years, when Gauss gave a compass and straightedge construction for the regular 17-gon. This project also provides research training opportunities for graduate students working with the PI on these questions.More specifically, this project involves using group theory to describe the images of Galois representations arising either from varieties or from automorphic forms. Problems of this kind can often be approached via by what might be termed the "inverse problem" in invariant theory, recognizing an algebraic group from data about its representation category. Group theory and algebraic geometry can be applied together to analyze the Galois action on Mordell-Weil groups of abelian varieties over Galois extensions. In a different direction, the project involves using algebraic geometry, alone or in combination with character-theoretic methods, to solve problems about finite simple groups. In particular, these tools can be applied to investigate word maps, defined by elements in free groups or related objects such as surface groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.