The Manin conjecture was proposed in the late 1980s by Yuri Manin and his collaborators. It predicts an asymptotic formula for the counting function of rational points on a class of varieties called Fano varieties. Over the past forty years, this conjecture has been the subject of intensive study and has become one of the more interdisciplinary areas in mathematics, involving arithmetic geometry, Arakelov geometry, analytic number theory, automorphic forms, ergodic theory, and birational geometry.
One can also formulate the Manin conjecture over global function fields, such as the function field of a curve over a finite field. In this setting, the conjecture can be reformulated as a problem of counting $\mathbb{F}_q$-points on moduli spaces of curves on Fano varieties. Batyrev and later Ellenberg–Venkatesh proposed to study this counting problem using the homological stability of the relevant moduli spaces.
In this talk, I will present our proof of the Manin conjecture for quartic del Pezzo surfaces, which relies on a homological sieve method—an abstraction of the inclusion–exclusion principle that we have developed. This is joint work with Das, Lehmann, and Tosteson, with additional contributions from Sawin and Shusterman.

16:15 - tea