William Fulton Distinguished University Professor of Mathematics
Karen Smith's research lies at the interface of commutative algebra and algebraic geometry. Algebraic geometry is the study of geometric shapes which are defined by polynomial equations; commutative algebra is the study of the rings of polynomial functions on such geometric objects. Specifically, one focus of Smith's research the use of prime characteristic methods to prove results about complex projective varieties. For example, the singularities of varieties can be measured in various ways using reduction to characteristic p and then iteration of the Frobenius map, and Smith was one of the leaders in unravelling the connections with rational singularities and other singularities in birational geometry. Similarly, global properties of projective varieties, such as the ways in which they can embed in different projective spaces, can be understood by studying the splitting properties of the Frobenius map, and Smith was a leader in establishing connections with positivity. Karen Smith also has been involved with the development of asymptotic multiplier ideals (in char 0) and their characteristic p analog, test ideals, and invariants such a jumping numbers derived from them.