Mirror symmetry at genus one

Research Project, 2022 – 2025

The study of mirror symmetry is a central theme in modern algebraic geometry. It was conceived by string theorists some 40 years ago via mirror pairs of six dimensional geometries appearing in dual physical models.
Since then, this mainly conjectural framework has been eagerly adopted by mathematicians, and the interplay between the two fields has led to many fruitful crosspollinations. A particular feature is the precise relation between curve counting, a classic but difficult topic, and other functions on the mirror. This project focuses on better understanding both the geometric and analytic theoretical properties of the mathematical aspects of natural objects appearing in mirror symmetry. More precisely, the current project uses a specific form of the heat equation; this partial differential equation was developed 200 years ago to describe how heat flows from warmer to colder areas. Analytic torsion is a fundamental invariant which is built up on the basis of various energy levels described by the equation, and is conjectured to be related to genus one curve counting in mirror symmetry. While the questions can be formulated, the techniques are not here yet, and a major portion of the project concerns developing the mathematical formulations and the theory for these purposes. Along the way, and adapted to the specific approach taken, one expects previously hidden geometric and number-theoretical properties to reveal themselves.