Curve Enumeration via Quadratic Localization and Mirror Symmetry

• Applicant: Dr. Felipe Espreafico Guelerman Ramos
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 581779604

In the 1980s and 1990s, physicists predicted that counts of rational curves on a Calabi–Yau threefold, can be computed from period integrals on another Calabi–Yau threefold. Such duality between Calabi-Yau threefolds became known as Mirror Symmetry. These ideas revolutionised the field, allowing many previously unknown invariants to be computed. In a different direction, the recent field of motivic homotopy theory provides a new universal approach to questions in enumerative geometry, producing invariants that can be evaluated at any base field (not only real or complex numbers). However, it remains unexplored whether techniques from Mirror Symmetry can be used in the motivic framework to enable computations that are unachievable now. This project aims to extend classical computations to the motivic setting and make techniques from Mirror Symmetry available for these new universal counts. This contributes to the development of a new, algebraic, point of view of Mirror Symmetry. The main examples considered are the counts of rational curves of fixed degree on a quintic threefold, the first and most famous example in the field, where such universal/motivic counts only exist in low degree.

• DFG Programme: Position