Groups and Arithmetic

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This project concerns the deep connections between two apparently very different areas of algebra. The first is group theory, which is the formal study of symmetry. The second is algebraic number theory, the study of numbers which can be constructed by algebraic processes, like taking square roots (as opposed to numbers like e and pi which arise only through limit processes belonging to calculus). The main link between these two fields is provided by algebraic geometry, the geometric study of systems of polynomial equations. A key theme in the proposed work is monodromy, which encapsulates the symmetries revealed by a varying system as the variable follows a closed loop. Monodromy problems arise in many guises, in both pure and applied mathematics. For instance some of the techniques under study in this project have been used to determine which kinds of computations can be carried out by different kinds of quantum computer. Algebraic number theory has found important practical applications and especially plays a key role in the development of modern cryptosystems.The theme of this project is the reciprocal relationship between group theory and algebraic number theory or arithmetic algebraic geometry. This includes using group theory as a tool, for instance in analyzing images of l-adic Galois representations, or Mordell-Weil groups of abelian varieties over Galois extensions of the rationals. It also includes studying groups, especially discrete linear groups (including finite groups), using methods from number theory and algebraic geometry, including the circle method, etale cohomology, and deformation theory.