in algebraic and arithmetic geometry. In particular, he studies generalizations of the conjectures of Shafarevich and Abhyankar on Galois groups over varieties; lifting problems for covers, from finite to mixed characteristic; and Galois extensions of the one-variable function field over the field of rational numbers. Goals include the development of Galois theory in higher dimensions, understanding good reduction of covers, and computing fields of moduli of covers. Achieving these goals would increase the understanding of the geometry of surfaces and the link between number theory and topology. Methods include formal patching, embedding problems, cohomology, birational transformations, and group theory.
This project relates aspects of algebra and geometry in a way that makes it possible to achieve research results that could not be obtained using either of these two fields alone. The geometric spaces being studied can be defined algebraically using equations, and the project makes use of both the algebraic and geometric aspects of these spaces. A particular focus of the project concerns the symmetries of these spaces, which in turn correspond to symmetries involving the algebraic functions on these spaces. The project studies what types of symmetries can occur, which types of extensions of symmetry are possible, and what types of algebraic numbers are needed in order to obtain a space with a given type of symmetry.
