Singularities and numerical invariants defined via the Frobenius endomorphism are an important part of the study of Commutative Algebra and Algebraic Geometry in positive characteristic. To that end, the program proposed by the PI will focus on the F-signature and other related so-called F-invariants, including Hilbert-Kunz multiplicity and test ideals. Central to the investigations of the PI is the interaction with geometric methods in characteristic zero stemming from complex algebraic geometry. One of the main objectives of the program is to better describe the geometry and broader connections of F-signature, as well as its many generalizations. The PI aims to approach various problems related to a number of long standing open questions in the field, including the equivalence of weak versus strong F-regularity and the direct summand conjecture. Furthermore, the PI plans to build upon recent work describing test ideals via regular alterations in exploring the local and global geometry of algebraic varieties in positive characteristic.
Commutative Algebra and Algebraic Geometry are among the oldest and yet most active disciplines in mathematics. The fields have strong ties to such diverse areas as complex analysis, topology, and number theory, and are used in a wide variety of applied settings. Applications range from error-correcting codes in computer science and genomics to control theory and modeling in engineering. These fields seek to understand geometric objects (algebraic varieties) given locally as the solutions to polynomial equations. For instance, a plane curve is the zero set of a polynomial in two variables (such as the cusp y^2 = x^3). The richness and simplicity of polynomial equations make algebraic varieties fascinating objects of study. The particular questions the PI proposes to study will hopefully lead to a deeper understanding of the varieties and singularities in positive characteristic, i.e. over number systems having the property that a prime number vanishes. In particular, these systems include the finite fields at the heart of essentially all electronic computation.
