Title: Exploring the landscape of Fano varieties (slides)

Speaker: Alexander Kasprzyk (Nottingham)

Time: 12:15-12:50 (GMT+0), 2021.08.12

Abstract:

Fano varieties are fundamental objects in algebraic geometry: they form the "atomic pieces" of geometry. Although we know that there are finitely many deformation families of Fano varieties in each dimension, despite over eighty years of work their classification eludes us. Recent advances, drawing on ideas from mirror symmetry, allow us to begin systematically mapping the landscape of Fano varieties. Mirror symmetry suggests that geometric objects come in "mirror pairs": in this setting, the mirror partner to a Fano variety is a Laurent polynomial. These mirrors satisfy many beautiful combinatorial properties, and are accessible to systematic classification. With the help of large-scale computation, we have started populating the classifications of Fano varieties in dimensions 3 and 4, finding hundreds of new Fano varieties in the process. Furthermore, techniques from data science and Machine Learning have exposed deep, previously unsuspected structure in this data. Although the mathematics required to explain these structures may take several decades to develop, this approach is allowing us to start exploring the fascinating landscape of Fano varieties today.