Periodic Reporting for period 4 - QUADAG (Quadratic refinements in algebraic geometry)
Período documentado: 2024-03-01 hasta 2025-02-28
At its root, algebra is the science of solving equations. Enumerative geometry is a branch of mathematics that is concerned with counting the number of solutions of equations that arise from geometric situations, rather than finding the explicit solutions themselves. In many situations, this number of solutions remains constant under a change in the input parameters of the problem, which points to an intrinsic underlying aspect of the original geometric problem. As an elementary example, the number of solutions in the complex numbers of a degree 5 equation is always at most 5, and if counted with the proper "multiplicity” is always equal to 5.
The choice of number system for finding solutions to the geometric problem often has a large effect on the number of solutions and can even affect the invariance-of-parameter phenomenon mentioned above. The main thrust of the project QUADAG is to replace the simple integer count of solutions with an object known as a quadratic form. In contrast with a simple numerical count, the quadratic form can itself reflect the difference in the number system under consideration, and thus gives a way to unify the disparate counting results that one finds when working over different number systems, such as the rational, real or complex numbers.
Supporting this central objective is the development of effective methods of computing these quadratic invariants, applying these tools to make explicit computations in interesting geometric situations and giving conceptual interpretations of the resulting computations. Although a direct application of this work to pressing societal issues is difficult to see, we hope that, as a part of the overall quest for truth and knowledge that is represented by modern mathematics, this project will make a small contribution to society as a whole.
The choice of number system for finding solutions to the geometric problem often has a large effect on the number of solutions and can even affect the invariance-of-parameter phenomenon mentioned above. The main thrust of the project QUADAG is to replace the simple integer count of solutions with an object known as a quadratic form. In contrast with a simple numerical count, the quadratic form can itself reflect the difference in the number system under consideration, and thus gives a way to unify the disparate counting results that one finds when working over different number systems, such as the rational, real or complex numbers.
Supporting this central objective is the development of effective methods of computing these quadratic invariants, applying these tools to make explicit computations in interesting geometric situations and giving conceptual interpretations of the resulting computations. Although a direct application of this work to pressing societal issues is difficult to see, we hope that, as a part of the overall quest for truth and knowledge that is represented by modern mathematics, this project will make a small contribution to society as a whole.
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.
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.
1. The development of a calculus of quadratic characteristic classes, with applications to giving explicit formulas for quadratic Euler classes and quadratic Euler characteristics, including a quadratic Riemann-Hurwitz formula and quadratic virtual fundamental classes.
2. The introduction and further development of the method of normalizer localization for quadratic invariants, leading to a quadratic Atiyah-Bott localization theorem, a quadratic Bott residue formula and a localization formula for computing quadratic degrees of virtual fundamental classes.
3. Quadratic refinement of counts of rational curves on del Pezzo surfaces, unifying the classical counts over the complex numbers with the Welschinger count over the reals, and introducing new arithmetic information.
4. The introduction and application of tropical methods to the computation of quadratic enumerative invariants, including the new examples of the invariants described in (3).
5. A quadratic count of twisted cubics on smooth hypersurfaces.
6. The first complete computation of the quadratic Donaldson-Thomas invariants for a smooth threefold, namely, for (P^1)^3.
7. Quadratically enriched conductor formulas and the development of explicit algebraic computations of quadratic Euler characteristics of smooth hypersurfaces.
8. Contributions to the development of the A^1-homotopy theory of algebraic stacks, paving the way for a quadratic enumerative geometry of moduli stacks.
2. The introduction and further development of the method of normalizer localization for quadratic invariants, leading to a quadratic Atiyah-Bott localization theorem, a quadratic Bott residue formula and a localization formula for computing quadratic degrees of virtual fundamental classes.
3. Quadratic refinement of counts of rational curves on del Pezzo surfaces, unifying the classical counts over the complex numbers with the Welschinger count over the reals, and introducing new arithmetic information.
4. The introduction and application of tropical methods to the computation of quadratic enumerative invariants, including the new examples of the invariants described in (3).
5. A quadratic count of twisted cubics on smooth hypersurfaces.
6. The first complete computation of the quadratic Donaldson-Thomas invariants for a smooth threefold, namely, for (P^1)^3.
7. Quadratically enriched conductor formulas and the development of explicit algebraic computations of quadratic Euler characteristics of smooth hypersurfaces.
8. Contributions to the development of the A^1-homotopy theory of algebraic stacks, paving the way for a quadratic enumerative geometry of moduli stacks.