# Seminars Calendar

**Wednesday July 6, 2022 - 14:00 - 2AB40 - David Smith (Padova) **

**A model-theoretic proof of Hilbert's Nullstellensatz **

Abstract: Model theory, a sub-field of mathematical logic, studies relationships between formal language expressing statements and mathematical structure. With comparatively less formal rigour than other sub-topics of logic, model theory has been known to have ties to algebraic and diophantine geometry. The purpose of this talk is to give a brief introduction to model theory whilst illustrating how model-theoretic techniques could be used to prove algebro-geometric results

**Thursday June 29, 2022 - 11:00 - 1BC45 - Michele Fornea (Columbia) **

**Plectic Jacobians and Hodge theory **

Abstract: Gehrmann, Guitart, Masdeu and myself recently proposed, and gave evidence for, plectic generalizations of Stark-Heegner points. The construction is p-adic, cohomological, and unfortunately lacking a satisfying geometric interpretation. Nevertheless, we formulated precise conjectures on the algebraicity of plectic points and their significance for the arithmetic of higher rank elliptic curves. In this talk I will report on work in progress on the Archimedean side of the story where geometry has a prominent role: I will describe a collection of complex tori — called plectic Jacobians — associated with the plectic Hodge structure appearing in the middle degree cohomology of Hilbert modular varieties. Interestingly, the Oda-Yoshida conjecture can be used to prove that plectic Jacobians are modular abelian varieties defined over \bar{Q}. Moreover, the existence of exotic Abel-Jacobi morphisms (mapping zero-cycles to plectic Jacobians) further highlights the arithmetic appeal of the construction.

**Thursday June 21, 2022 - 14:00 - room 2AB40 and on Zoom - Anna Barbieri (University of Padova) **

**On quiver categories associated to quadratic differentials **

Abstract: In a paper in 2015, Bridgeland and Smith identified some moduli spaces of quadratic differentials with simple zeroes on a Riemann surface with some spaces of stability conditions on certain categories. This identification passes through associating a quiver Q and a triangulated category D(Q) to a triangulation of a marked bordered surface with boundaries defined by a quadratic differential. I will review this correspondence and discuss how the picture changes when quadratic differentials with zeroes of arbitrary order are considered. This is part of a joint work with M. Moeller and J. So.

**Thursday June 9, 2022 - 12:15 - room 2AB40 - Antonio Cauchi (Concordia University, Montreal) **

**Quaternionic diagonal cycles and instances of the Birch and Swinnerton-Dyer conjecture for elliptic curves **

Abstract: In the early nineties, Kato’s Euler system of Beilinson elements and the theory of Heegner points revolutionised the arithmetic of (modular) elliptic curves over the rationals. For instance, the former led Kato to proving instances of the Birch and Swinnerton-Dyer conjecture for twists of elliptic curves over Q by finite order characters. While the theory of Heegner points was generalised to elliptic curves E/F defined over totally real number fields, Kato’s result has not found its natural extension to twists of E/F yet. More recently, the theory of diagonal cycles, arising from the work and collective effort of Bertolini, Darmon, Rotger, Seveso, and Venerucci, has proven to be a fertile environment for proving new instances of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals. The aim of this talk is to discuss joint work in progress with Daniel Barrera, Santiago Molina, and Victor Rotger on the generalisation of the theory of diagonal cycles to quaternionic Shimura curves over totally real number fields F and its application to extending Kato’s result for twists of elliptic curves E/F by Hecke characters of F of finite order.

**June 1, 2022 - 14:30 - room 2BC30 - Alexandr Buryak (Higher School of Economics, Moscow) **

**Counting meromorphic differentials with zero residues on the Riemann sphere and the KP hierarchy**

Abstract: Let us fix n integers (multiplicities) and consider the set of configurations of n points on the sphere such that there exists a meromorphic differential with zero residues and with the divisor given by the linear combination of the n points with the prescribed multiplicities. Let us factorize this set by the automorphism group of the sphere. The resulting set is finite if and only if the number of poles is equal to n-2. The cardinalities of these sets can be considered as natural analogs of Hurwitz numbers. In our joint work with Paolo Rossi and Dimitri Zvonkine we proved that these numbers are exactly the coefficients in the equations of the dispersionless KP hierarchy.

# Working Seminars

## Dasgupta-Kakde proof of the Hilbert 12 problem

We plan to run a seminar on the recent proof of Dasgupta-Kakde of the 12th Hilbert problem for totally real extensions. For more information, write to Luca dall'Ava dallava@math.unipd.it or Matteo Longo matteo.longo@unipd.it

Talk I (M. Longo). Thursday March 24, 2022, 10:30-11:30, 2AB40.

Talk II (M. Longo). Thursday March 31, 2022, 10:30-11:30, 2AB40.

Talk III (M. Longo). Thursday April 7, 2022, 10:30-11:30, 2AB40.

Talk IV (M. Longo, L. dall'Ava). Thursday April 7, 2022, 10:30-11:30, 2AB40.

Talk V (L. dall'Ava). Thursday April 14, 2022, 10:30-11:30, 2AB40.

Talk VI (L. dall'Ava). Thursday April 21, 2022, 10:30-11:30, 2AB40.

Talk VII (M. Baracchini) Thursday May 5, 2022, 10:30-11:30, 2AB40

Talk VIII (R. Walchek) Thursday May 12, 2022, 10:30-11:30, 2AB40

Talk IX (F. Zerman) Thursday May 19, 2022, 10:30-11:30, 2AB40

Talk X (S. Wang) Thursday May 19, 2022, 10:30-11:30, 2AB40

Talk XI (M.-R. Pati) Thursday June 23, 2022, 10:30-11:30, 2AB40

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