Motivic Invariants, Old and New

Algebraic geometry is the study of shapes, called varieties, that form the solutions of polynomial equations. These shapes arise naturally in physics, robotics, computer vision, statistics, and many other areas of science. Homotopy theory is the study of shapes up to deformation. Deformations provide a degree of flexibility to the study of varieties, which makes some difficult problems more tractable. A measurement that is not affected by deformations is called an invariant. Such measurements provide crucial information about varieties. This project will develop new invariants for varieties and offer novel reformulations of known, important invariants. A key feature of the invariants that will be developed is their validity in any number system, rather than just in the real or complex numbers. This will establish new connections between algebraic geometry, number theory, and low-dimensional topology. This project includes mathematical training opportunities for students across the country. There are three primary goals to the project. The first is to understand the motivic Euler characteristic, which is a quadratic form whose rank is the compactly supported Euler characteristic, of Hilbert schemes of K3 surfaces. This will involve a characterization of the Hasse--Witt invariants of the motivic Euler characteristic in terms of the bad reduction of the scheme. Other arithmetic aspects of these motivic Euler characteristics will also be studied. The second goal is explain the behavior of unstable local degrees in motivic homotopy theory under field extensions. In contrast to the stable case, the transfer map of the unstable local degree fails to be a homomorphism. The PI will construct an obstruction in Milnor--Witt K-theory to the homomorphicity of the transfer map. The payoff of the second goal is a toolkit for certain problems in enumerative geometry. The third goal of the project is to relate Levine's quadratic Donaldson--Thomas invariants, which are algebraically defined, to the Casson invariant in low-dimensional topology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.