Geometry Seminar

Seminars

2 Oct

Luca Ferrigno (Roma Tre)

Unlikely intersections in families of abelian varieties

[click for abstract]

Abstract: We talk about "unlikely intersections" whenever we have a non-empty intersection between algebraic varieties that, for dimensional reasons, we do not expect to intersect. This expectation lies behind several landmark results and conjectures in Diophantine geometry, including Faltings’ Theorem (formerly the Mordell Conjecture), the Manin-Mumford Conjecture, the André-Oort Conjecture (proved by Pila, Shankar and Tsimerman), and the still open Zilber-Pink Conjecture. In this talk, I will give an introduction to unlikely intersections, focusing first on algebraic tori and abelian varieties, and then on families of abelian varieties. I will survey some key results by Masser-Zannier and Barroero-Capuano in this setting, and finally present results from my PhD thesis establishing some partial progress on the Zilber-Pink Conjecture for curves in abelian schemes.

9 Oct

Lorenzo Foscolo (Sapienza)

Special Hermitian metrics on non-compact Calabi-Yau 3-folds

[click for abstract]

Abstract: We present the construction of distinguished non-Kähler metrics on non-compact Calabi-Yau 3-folds. These metrics solve a system of equations known as the IIB system which arises in theoretical physics and is related to recent attempts to define notions of "canonical" metrics on non-Kähler Calabi-Yau manifolds. The examples we construct include infinitely many complete metrics obtained by deforming an asymptotically conical Calabi-Yau 3-folds in the direction of a non-trivial Äppli class and families of solutions on the ordinary double point and its smoothing that enjoy a cohomogeneity one symmetry (i.e. there is a symmetry group that acts with 1-dimensional orbit space) and produce a "conifold transition" in non-Kähler Calabi-Yau geometry. The talk is based on joint work with Mario Garcia-Fernandez.

16 Oct

Davide Lombardo (Pisa)

Rational points on modular curves & generalised Fermat equations

[click for abstract]

Abstract: In 1972, Serre proved that the Galois representations arising from the p-power torsion points of non-CM elliptic curves over \(\mathbb{Q}\) have open image in \(\operatorname{GL}_2(\mathbb{Z}_p)\), and Mazur later initiated a vast programme to determine all such possible images explicitly -- for fixed p, it is known that there are only finitely many possibilities. Much progress has been made for small primes p, but a complete classification remains open beyond \(p \in \{2,3,13,17\}\). In this talk, I will describe recent progress on this problem for p = 7, based on a surprising correspondence between rational points on modular curves and primitive integer solutions to certain generalised Fermat equations of signature (2,3,7), such as \( a^2 + 28b^3 = 27c^7. \) We show that these Diophantine equations can be reduced to determining the rational points on a finite collection of genus-3 curves. As a consequence, we are able for example to determine the rational points on a modular curve of genus 69 and establish that the 7-adic Galois images of elliptic curves over \(\mathbb{Q}\) are determined by their reduction modulo \(7^2\).

16 Oct

Harry Schmidt (Warwick)

On the \(L^2\)-norm of exponential polynomials

[click for abstract]

Abstract: In join work with Manon Parent we have investigated how small the \(L^2\)-norm of an exponential polynomial can get if we fix the norm of the its coefficient vector. We prove lower and upper bounds of this minimum in terms of the degree of the polynomial and we apply our methods to a variant of a problem of Hilbert. We prove that on any given interval the infimum of the \(L^2\)-norm of exponential polynomials with integer coefficients on this interval is 0.

23 Oct

(-)

No Seminar: Lauree

30 Oct

Federico Pieroni (Roma Tre)

TBA

[click for abstract]

Abstract: TBA

6 Nov

Stefan Schreieder (Hannover)

TBA

[click for abstract]

Abstract: TBA

13 Nov

(-)

TBA

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Abstract: TBA

20 Nov

Francesco Russo (Catania)

TBA

[click for abstract]

Abstract: TBA

27 Nov

(-)

TBA

[click for abstract]

Abstract: TBA

4 Dec

Antonio Rapagnetta (Tor Vergata)

TBA

[click for abstract]

Abstract: TBA

11 Dec

(-)

TBA

[click for abstract]

Abstract: TBA

Christmas Break

Info

The seminar is usually held every Thursday at 14:15-15:45 in Aula M1. This year seminars will be held in person but you can email one of the organizers if you are interested in attending remotely (through the platform Microsoft Teams).

The seminar is organized by Luca Schaffler and Amos Turchet and maintained by the Geometry Group and the Number Theory Group of the Department of Mathematics and Physics at the Roma Tre University.

We acknowledge the support of the grants PRIN2022: Moduli spaces and birational geometry and PRIN2022: Semiabelian varieties, Galois representations and related Diophantine problems, the 'Programma per Giovani Ricercatori "Rita Levi Montalcini"' and the support of the Department of Mathematics and Physics at Roma Tre University.

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Next Seminar: October 16

Davide Lombardo (Pisa) 14:00

Rational points on modular curves & generalised Fermat equations

Harry Schmidt (Warwick) 15:00

On the \(L^2\)-norm of exponential polynomials

Aula M1 - Largo S. L. Murialdo 1

2 Oct

Luca Ferrigno (Roma Tre)

Unlikely intersections in families of abelian varieties

[click for abstract]

9 Oct

Lorenzo Foscolo (Sapienza)

Special Hermitian metrics on non-compact Calabi-Yau 3-folds

[click for abstract]

16 Oct

Davide Lombardo (Pisa)

Rational points on modular curves & generalised Fermat equations

[click for abstract]

16 Oct

Harry Schmidt (Warwick)

On the \(L^2\)-norm of exponential polynomials

[click for abstract]

23 Oct

(-)

No Seminar: Lauree

30 Oct

Federico Pieroni (Roma Tre)

TBA

[click for abstract]

6 Nov

Stefan Schreieder (Hannover)

TBA

[click for abstract]

13 Nov

(-)

TBA

[click for abstract]

20 Nov

Francesco Russo (Catania)

TBA

[click for abstract]

27 Nov

(-)

TBA

[click for abstract]

4 Dec

Antonio Rapagnetta (Tor Vergata)

TBA

[click for abstract]

11 Dec

(-)

TBA

[click for abstract]

Christmas Break