Research Statement Daniel

Overview. My work is in the area of commutative algebra, and is motivated by the connections between algebra and geometry. Commutative algebra is the study of commutative rings (e.g. polynomial rings over fields and their quotients) and modules over these rings, while algebraic geometry is the study of finitely many polynomial equations in finitely many unknowns. The set of all solutions to such a system of equations gives rise to a geometric object, often called a variety. Basic examples of a variety include lines (e.g., solutions to the equation y = x) and parabolas (e.g., solutions to the equation y = x2), though a variety described by many equations in many unknowns can be quite complicated. Commutative algebra and algebraic geometry are intimately related, as many geometric properties of an algebraic variety X can be measured by the ring theoretic properties of the ring of functions on X. For example, the geometric notion of smoothness is equivalent to the algebraic notion of regularity. In a very broad sense, both commutative algebra and algebraic geometry deal with the theory of equations; as such, both are closely related to many areas of math, including differential geometry, representation theory, and number theory. Furthermore, the fields of computational commutative algebra and algebraic geometry form the basis for many “real world” applications. These applications are quite diverse, and include contributions to epidemiology, biology, computer science, and coding theory. My research is in the subfield of positive characteristic commutative algebra. For nearly forty years, mathematicians have used the Frobenius (or pth-power) map to investigate phenomena in commutative algebra, algebraic geometry, representation theory, and number theory. Notable applications include the key role of “Frobenius purity” (or F -purity, for short) in the proof of the well-known Hochster-Roberts theorem (which states that rings of invariants are Cohen-Macaulay) [HR74]. Another important application appears in the work of the Indian school of researchers studying algebraic groups, who introduced the concept of “Frobenius splittings,” and used it to deduce important vanishing theorems for Schubert varieties [MR85]. Today, the notions of F -purity and Frobenius splittings are understood to be very closely related [Smi00a]. Furthermore, emerging connections with the singularities of hypersurfaces over C (typically defined via L2-methods, or via resolution of singularities) appearing in the so-called minimal model program have led to a renewed interest in the theory of F -purity (and some of its natural generalizations). My research aims to shed light on the connection; in particular, I seek to understand the connection between certain invariants of singularities defined using Frobenius in characteristic p, and invariants defined using L2-methods (and more generally, resolution of singularities) over C.