Geometry, Arithmetic and Differential Equations of Periods (GADEPs)
Geometry, Arithmetic and Differential Equations of Periods (GADEPs)
Fridays 10:30 (Rio de Janeiro's time), 21:30 (Beijing Time)
Organizers: Hossein Movasati, Younes Nikdelan
Invited Organizer: Jin Cao (03/2023-12/2023)
Past invited organizers: Tiago Fonseca (02/2021-11/2021), Roberto Villaflor (03/2022-06/2022)
M. Picard a donné à ces integrales le nom de périodes; je ne saurais l'en blâmer puisque cette dénomination lui a permis d'exprimer dans un langage plus concis les intéressants résultats auxquels il est parvenu. Mais je crois qu'il serait fâcheux qu'elle s'introduisit définitivement dans la science et qu'elle serait propre à engendrer de nombreuses confusions, H. Poincaré's remarks on the name period used for integrals, Acta Mathematica, 1887 page 323.
This seminar is about all confusions around periods (a bad but already established name). The talks are available in the YouTube Channel. And the detailed info. of Zoom is as below:
Zoom ID: 559 700 6085
Passcode: BIMSA
Schedule
2023-04-14 Fri
Title: Tropical Hodge conjecture for Abelian varieties.
Speaker: Ilia Zharkov (KSU, USA)
Zoom: 559 700 6085
Password: BIMSA
Abstract:
The classical Hodge conjecture states that for a smooth projective variety any rational (p,p)-class can be represented by an algebraic cycle. The first non-trivial case for abelian varieties is so called abelian varieties of Weil type of dimension 4 which possess primitive (2,2)-classes.
I will describe the tropical analog of the conjecture in this case and an idea of M. Kontsevich from around 2010 of how one can try to disprove it.
2023-04-07 Fri
Title: Absolute Hodge classes and the Mumford-Tate conjecture for hyperkahler manifolds
Speaker: Andrey Soldatenkov (IMPA, Brazil)
Zoom: 559 700 6085
Password: BIMSA
Abstract:
The Hodge conjecture implies that all Hodge cycles on a smooth complex projective variety are absolute, i.e. they remain Hodge if one conjugates the variety by an automorphism of the field of complex numbers. It was shown by Deligne that all Hodge cycles on abelian varieties are absolute, although the Hodge conjecture for abelian varieties remains open. I will present my recent results about Hodge cycles on compact hyperkahler manifolds, showing that all Hodge cycles on all known examples of such manifolds are absolute. I will also discuss the related results on the Mumford-Tate conjecture for hyperkahler manifolds.
2023-03-24 Fri
Title: Quasi-modularity of Hodge cycles
Speaker: François Greer (Michigan State University, USA)
Zoom: 559 700 6085
Password: BIMSA
Abstract:
Period spaces contain Hodge cycles, whose cohomology classes form the coefficients of certain modular forms, by work of Kudla and Millson. I will explain how this phenomenon survives when we pass to a toroidal compactification in the case of K3 type Hodge structures, and then give some geometric applications. This work is joint with Phil Engel and Salim Tayou.
2023-03-17 Fri
Title: On attractor points on the moduli space of Calabi-Yau threefolds
Speaker: Emanuel Scheidegger
Zoom: 559 700 6085
Password: BIMSA
Abstract:
We briefly review the origin in physics of attractor points on the moduli space of Calabi-Yau threefolds. We turn to their mathematical interpretation as special cases of Hodge loci. This leads to fascinating conjectures on the modularity of the Calabi-Yau threefolds at these points in terms of their periods and L-functions. For hypergeometric one-parameter families of Calabi-Yau threefolds, these conjectures can be verified at least numerically to very high precision.
2023-02-17 Fri
Title: Nodes and the Hodge conjecture
Speaker: Richard Thomas (Imperial College, UK)
Zoom: 559 700 6085
Password: BIMSA
Abstract: I'll summarise my old paper giving a simple inductive approach to the Hodge conjecture. The upshot is that -- given a middle dimensional (p,p) Hodge class -- instead of finding a codimension-p variety carrying it, we only need to find a codimension 1 hyperplane containing it.
Past Talks
https://w3.impa.br/~hossein/GADEPs/GADEPs.html
