In addition to the PI Levine, the members of the research group supported by the project are: Dhyan Aranha, Ran Azouri, Chirantan Chowdhury, Alessandro D'Angelo, Andrés Jaramillo Puentes, Fangzhou Jin, Andrei Konovalov, Sabrina Pauli and Herman Rohrbach.
This project consisted of three main parts:
I: Developing mathematical tools and methods using motivic homotopy theory.
II: Applying these methods to key problems in enumerative geometry.
III: Developing an arithmetic enumerative geometry.
Under (I) we find papers by Jin (et al.) and by Levine (et al.) dealing with questions on the structure of the motivic stable homotopy category of schemes. Papers by Aranha (with Pstrągowski), Aranha-Chowdhury , Chowdhury, and D’Angelo-Chowdhury establish extensions of aspects of the motivic stable homotopy category of schemes, including the six-functor formalism, to algebraic stacks. This forms the foundation for both a study of the equivariant theory and localization methods mentioned below, and gives the background for the extension of the main ideas of this project to areas where stacks are essential, such as a quadratic Gromov-Witten theory. Papers by Rohrbach (et al.) study Hermitian K-theory and equivariant Hermitian K-theory.
Papers by Azouri, D’Angelo, Jin (et al.) and Levine build a theory of quadratic enumerative invariants, constructing a quadratic enumerative geometry, a calculus of quadratic characteristic classes, and develop a quadratic version of virtual fundamental classes, forming the core tools for the project.
Papers by Aranha (et al.), D’Angelo and Levine build a theory of localization of quadratic invariants, one of the main achievements of this project, giving an effective method for computations.
Papers by Jaramillo Puentes and Pauli pioneer a link between quadratic enumerative geometry and tropical geometry that goes far beyond what was envisioned in the Project Description. This has yielded far-reaching computations of the quadratic invariants constructed here and elsewhere.
Under (II), papers by Levine (et al.) gives explicit computations of quadratic Euler characteristics. A paper by Pauli (et al.) computes quadratic invariants of self-maps of the projective line. A paper by Levine-Pauli uses Levine's localization method to give a quadratic count of twisted cubics on hypersurfaces and complete intersections. Two papers by Levine (with Kass, Solomon, Wickelgren) construct a quadratic refinement of the count of rational curves on a del Pezzo surface, a quadratic refinement of one of the main motivating problems of Gromov-Witten theory. A paper by Azouri proves a quadratic version of a classical conductor formula relating Euler characteristics in a degenerating family. Work of Levine (with Ananyevskiy) gives an algebraic analog of classical results about existence of nowhere vanishing vector fields on a compact manifold. A paper by Rohrbach (with Pajwani, Viergever) begins to answer a basic question about the quadratic Euler characteristic of symmetric powers. A paper by Levine (with Viergever) gives the first complete computation of the quadratic Donaldson-Thomas invariants for a smooth projective threefold.
The papers of Jaramillo Puentes and Pauli on quadratic-tropical methods have yielded impressive computational results for the quadratic invariants counting rational curves constructed in the two papers by Kass-Levine-Solomon-Wickelgren.
In the category (III) we find the the identification of the signature of the quadratic curve-counting invariant defined in the work of Kass-Levine-Solomon-Wickelgren with the classical Welschinger invariant. The conductor formulas in papers by Azouri and Levine-Pepin Lehalleur-Srinivas recover earlier results in the real case. The curve counts of Levine-Pauli recovers earlier computations for real curves.
The works were disseminated through lectures in seminars and conferences throughout the international mathematics community. Levine was lead organiser of a 3 week PCMI workshop on Motivic Homotopy Theory (July 7-27, Park City, Utah), a large portion of which was devoted to topics in quadratic enumerative geometry.
