Numerical periods in effective algebraic geometry (with a focus on quartic surfaces), Workshop „Calabi-Yau Motives“, Mainz.
Séminaire de géométrie et algèbre effectives, IRMAR, Rennes.
“A numerical transcendental method in algebraic geometry”. It comes with a proof-of-concept database of quartic surfaces quarticdb.
Diagonales de fractions rationnelles et sommes binomiales, Institut Camille Jordan, Lyon.
Generalized Hermite reduction, creative telescoping and definite integration of differentially finite functions, presented at ISSAC 2018, New York City.
Conférence Algèbre, arithmétique et combinatoire des équations différentielles et aux différences, CIRM, Marseille.
2018
w/ E. Sertöz
In “A transcendental method in algebraic geometry”, Griffiths emphasized the role of certain multivariate integrals, known as periods, “to construct a continuous invariant of arbitrary smooth projective varieties”. Periods often determine the projective variety completely and therefore its algebraic invariants. Translating periods into discrete algebraic invariants is a difficult problem, exemplified by the long standing Hodge conjecture which describes how periods determine the algebraic cycles within a projective variety.
Recent progress in computer algebra makes it possible to compute periods with high precision and put transcendental methods into practice. We focus mainly on algebraic surfaces and give a numerical method to compute Picard groups. As an application, we count smooth rational curves on quartic surfaces using the Picard group.
A K3 surface containing 133056 smooth rational curves of degree 4 generating a Picard group of rank 19.