Colombian Encounter of Tropical and Non-archimedean Geometry

#### Description

Tropical geometry is a piece-wise linear geometry which merges ideas from algebraic, symplectic and non-archimedean geometry with tools from combinatorics and convex geometry. Via the process of tropicalization, classical varieties and geometric problems can be connected to the tropical world and, in some cases, solved there. In non-archimedean geometry, the valuation on the underlying field makes the idea of tropicalization particularly natural and powerful. In recent years, this connection has received much attention and exhibited links to diverse topics such as Hodge theory, mirror symmetry and the study of zeta functions. The main goal of this event is to introduce students and young mathematicians to these exciting topics and to foster and strengthen the connections between local researchers and the international mathematical community.

The school will introduce the participants to tropical and non-archimedean geometry via two minicourses given by Ilia Itenberg (Sorbonne Université) and Marco Maculan (Sorbonne Université). Additionally, in the research talks we will explore the latest developments in the field.

We hope that this school can serve as the starting point for a local network of researchers and students and therefore can be continued in the coming years by follow-up events of a similar type.

#### Main objectives

- Promote the cutting-edge research areas of tropical and non-archimedean geometry in Latin America.
- Introduce students and young researchers to the field and connect them with internationally renowned senior researchers.
- Foster collaborations in the region and establish a local network of collaborators and students in the field.
- Strengthen the ties between Colombia and the international mathematical community.
- Create accessible material for participants of this and future schools, derived from the minicourses and talks.

#### Organizers:

Pablo Cubides (Universidad de los Andes)

Johannes Rau (Universidad de los Andes)

Registration

Please register if you plan to attend regardless of whether you need financial support or not.

We are in the final stage of registrations and cannot provide travel support anymore.
We still have limited funding for accomodation.
Please indicate in the registration form if you need support.
We will come back to you as soon as possible with an answer.

Registration form |

Programme

Monday | Tuesday | Wednesday | Thursday | Friday | |

08:00-09:00 | Registration | ||||

09:00-10:30 | Minicourse 1 Itenberg |
Minicourse 1 Itenberg |
Minicourse 1 Itenberg |
Talk 1 Cotterill |
Talk 5 Garay |

10:30-11:00 | Break | Break | Break | Break | Break |

11:00-12:30 | Problems 1 | Problems 1 | Minicourse 2 Maculan |
Talk 2 Mehmeti |
Talk 6 Jaramillo |

12:30-14:00 | Lunch | Lunch | Lunch | Lunch | Lunch |

14:00-15:30 | Minicourse 2 Maculan |
Minicourse 2 Maculan |
Excursion Free afternoon |
Talk 3 Zharkov |
Open problem session |
---|---|---|---|---|---|

15:30-16:00 | Break | Break | Break | Closing Toast | |

16:00-17:30 | Problems 2 | Problems 2 | Talk 4 Lombardi |

Minicourses

The conference will feature two minicourses aiming to introduce students and young researchers to the cutting-edge research areas of Tropical and Non-archimedean Geometry and their interactions. The courses are delivered by two leading experts in the two fields which are also renowned for the excellent exposition skills. Each course will be complemented by exercise sessions.

#### Minicourse 1

* Tropical geometry
*

Ilia Itenberg (Sorbonne Université)

Ilia Itenberg (Sorbonne Université)

Tropical geometry can be seen as algebraic geometry over the tropical (max-plus) semifield. On tropical varieties, one can define different types of homology and intersection theories. These can be related to classical (complex and real) algebraic varieties in various ways.

##### Introductory texts

- Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin, Kristin Shaw, Brief introduction to tropical geometry.
- Johannes Rau, A first expedition to tropical geometry.

##### Exercises

#### Minicourse 2

*Non-archimedean geometry
*

Marco Maculan (Sorbonne Université)

Marco Maculan (Sorbonne Université)

Serre’s GAGA theorem says that on a complex projective variety holomorphic objects (functions, vector bundles, etc.) are all algebraic. Without compactness assumptions this is false, as we will see through many examples. We can still look for weaker questions: for instance, an algebraic variety which can be holomorphically embedded in C^n is it necessarily affine? A classical example of Serre shows that this is not always the case. In this course, we will go through the classical complex picture and then move to the non-Archimedean framework. In this case, the situation is much more rigid leading to a full panoply of open questions.
found.

Some references:

ALGEBRAIC GEOMETRY

- J.-P. Serre, Faisceaux algébriques cohérents (French)
- J.-P. Serre, Géométrie algébrique et géométrie analytique, (French)

RIGID ANALYSIS

There are Conrad’s notes covering several viewpoints:

The courses have been filmed:

- Several approaches to non-archimedean geometry 1
- Several approaches to non-archimedean geometry 2
- Several approaches to non-archimedean geometry 3
- Several approaches to non-archimedean geometry 4
- Several approaches to non-archimedean geometry 5

##### Exercises

Talks

Ethan Cotterill, Universidade Estadual de Campinas (Unicamp), Brasil

##### Matroids, semirings and curve singularities

Abstract: Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the value semigroup, or its compactification, the value semiring. One natural problem is to explicitly determine the value semirings of distinguished infinite classes of singularities, with a view to understanding their asymptotic properties. In this paper, we introduce a matroidal framework for systematically resolving this problem. More precisely, we show how to associate to any curve singularity a support semiring that maps homomorphically to the value semiring. This is a tropical semiring with a finitary matroid structure, whose basic features explain well-known features of value semirings of singularities, including a natural characterization of minimal generating sets. In the case of either line arrangements (i.e., multiple points) or cusps, we can be more quantitatively precise; and our results have important consequences for the topology of Severi varieties of singular rational curves in projective space.

Note: this will serve as a geometric introduction to Cristhian Garay's talk on Friday.

Vlerë Mehmeti, Sorbonne Université, France

##### Variation of the Hausdorff dimension and degenerating Schottky groups

Abstract: I will talk about the Hausdorff dimension of the limit set of Schottky groups defined over an arbitrary complete valued field. In 2021, Poineau and Turchetti constructed a moduli space for these groups using the theory of Berkovich spaces over Z. I will present a result on the continuity of the Hausdorff dimension of the limit sets over said moduli space. We will conclude with an application on families of degenerating complex Schottky groups that can be extended to a continuous family by a non-Archimedean counterpart. This is joint work with Nguyen-Bac Dang.

Ilia Zharkov, Kansas State University, USA

##### Phase tropical hypersurfaces

Abstract: One can complement the tropical (log) map to (C*)^n \to R^n by the argument map and consider the resulting images of hypersurfaces. They are PL objects which share the topology of their complex counterparts. I will mention an application of this to constructing (topological and symplectic) Strominger-Yau-Zaslow torus fibrations starting from an integral affine base. If time permits I will also describe a recent work of Mikhail Shkolnikov on generalization of this combined map to other situations.

Luigi Lombardi, University of Milan, Italy

##### The Grothendieck ring of varieties and derived equivalence

Abstract: A question of Ito-Miura-Okawa-Ueda asks whether the classes of two smooth projective derived equivalent complex varieties in a modified version of the completed Grothendieck ring of complex varieties coincide. In this talk, I will examine the case of varieties fibered in varieties of general type under the Albanese map. This talk is based on a joint work with F. Caucci and G. Pareschi.

Cristhian Garay, Centro de Investigación en Matemáticas AC (CIMAT), México

##### A matroidal perspective on curve singularities

Abstract: Curve singularities are classical objects of study in algebraic geometry. The key object in their combinatorial structure is the value semigroup, or its compactification, the value semiring. In this talk, we show how to associate to any curve singularity a support semiring from which we can recover its value semiring as a quotient. The advantage of working with the support semiring over the value semiring is that it is a tropical object, in the sense that it carries a natural structure of a finitary matroid. We show how basic features of the matroid structure explain well-known features of value semirings of singularities, including a natural characterization of minimal generating sets. This is joint work with Ethan Cotterill (Unicamp, Brazil).

Andrés Jaramillo Puentes, Università degli studi di Napoli Federico II, Italy

##### Tropical Methods in Motivic Enumerative Geometry

Abstract: Over the complex numbers the solutions to enumerative problems are invariant: the number of solutions of a polynomial equation or polynomial system, the number of lines or curves in a surface, etc. Over the real numbers such invariance fails. However, the signed count of solutions may lead to numerical invariants: Descartes' rule of signs, Poincaré-Hopf theorem, real curve-counting invariants. Since many of this problems have a geometric nature, one may ask the same problems for arbitrary fields. Motivic homotopy theory allows to do enumerative geometry over an arbitrary base, leading to additional arithmetic and geometric information. The goal of this talk is to illustrate a generalized notion of sign that allows us to state a movitic version of classical theorems like the Bezout theorem, the tropical correspondence theorem and a wall-crossing formula for curve counting invariants for points in quadratic field extensions.