Advances in quadratic enumerative geometry

• Applicant: Professor Dr. Marc Levine
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 568776200

This project is set in the emerging field of quadratic enumerative geometry. The main idea is to refine the classical numerical invariants and numerical counts for geometric problems to yield quadratic forms, that capture interesting arithmetic information about the original geometric problem. We plan advances in this area in five directions: extending quadratic degree maps to DM stacks, studying and computing equivariant versions of quadratic Euler characteristics, improving localisation methods to enable new computations of quadratic Donaldson-Thomas invariants, relating the quadratic invariants to invariants constructed via methods of motivic nearby cycles, and extending degeneration methods to the quadratic setting. This will greatly expand the available toolkit for computing and analysing quadratic enumerative invariants, and will allow us to make many new computations in basic geometric settings.

• DFG Programme: Research Grants