Computing and Interpreting Frobenius Invariants

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Commutative algebra and algebraic geometry are among the oldest and yet most active disciplines in mathematics. The fields have strong ties to diverse areas of mathematics, and are also used in a wide variety of applied settings. This research project will explore important open questions in these fields, aiming for a deeper understanding of the singularities of varieties over positive characteristic number systems. These systems, those where a prime number vanishes, include the finite fields at the heart of essentially all electronic computation. While carrying out this research program, the investigator will also organize research seminars, support and supervise graduate student research, and conduct activities, including an annual symposium, to promote undergraduate research. Explicitly, the investigator plans to continue study of singularities of and invariants defined via the Frobenius map in positive characteristic commutative algebra. In particular, the research will focus on the circle of ideas surrounding F-signature, Hilbert-Kunz multiplicity, and test ideals, in two very different directions. First, in a fixed characteristic, the project aims to expand upon recent progress in computing these invariants, and to approach some long standing important open questions about them, such as the equivalence of weak and strong F-regularity. Second, the research aims to show these invariants have limiting values under reduction to positive characteristic. Here, it is hoped that the limits as the characteristic tends towards infinity have simpler interpretations and values, and can be related to geometric measures of singularities for complex algebraic varieties. The interaction with geometric methods in characteristic zero stemming from complex algebraic geometry is central to the project, and one of the main objectives of the work is to better describe the geometry and broader connections of F-invariants.