# ICG 2023

View detailed Congress program here## The 8th Iberoamerican Congress on Geometry will be held at the Universidad de La Frontera, Campus Pucón, located at 78 Caupolicán Street, Pucón, Araucanía Region, Chile.

The Research Center Geometry at the Frontier is pleased to announce the 8th Iberoamerican Congress on Geometry, which will take place from December 10th to 15th, 2023, in Pucón, Chile. The program consists of eight plenary talks and four special sessions on Higher Dimensional Algebraic Geometry, Dynamical Systems, Algebraic Surfaces and Riemann Surfaces.

The Iberoamerican Geometry Congress (IGC) has a history of bringing together researchers and students from various mathematical communities in America, Spain, Portugal, and other regions of the world. The aim is to foster an exchange of ideas between mathematicians in different fields, connected by the geometry of Riemann surfaces, abelian varieties and related areas.

**Registration Fee**: 60 USD (50.000 CLP / 55 EUR)

**Payment Option:**

On-site Payment in Pucón: The only available payment option for the registration fee is to make the payment in cash in person. Please note that the payment should be made in cash on the first day of the event when you register in Pucón (Monday, December 11th, from 9:30 – 11:00 a.m.)

Observations:

1. The Registration Fee only includes the cost of coffee breaks and printed material.

2. Every participant must pay the Registration Fee. This fee is the same for everyone, regardless of their status (professor, student, postdoc, etc.)

# A Brief History

ICG have a history of more than 35 years. It was in 1987 when several geometers from Chile, México, Spain and the United States met at a congress on Riemann surfaces held in Trieste, and decided to have periodic work meetings; Rubí E. Rodríguez and Sevín Revilas started to think about organizing a geometry meeting of Iberoamerican scope. From this idea the first workshop was born. Its name was «Workshop on Abelian varieties and Theta Functions», the place of birth Morelia (México) and the date july 1996. Indeed, in that occasion José María Muñoz took part and soon afterwards Irwin Kra also joined the idea. These four friends launched the project of carrying out such a congress periodically and rotating in venue.

Their efforts paid off in 1998, when the first Iberoamerican Congress on Geometry was held in Olmué (Chile). It was later organized in 2001 (Guanajuato, México), 2004 (Salamanca, Spain), 2007 (Ouro Preto, Brazil), 2010 (Pucón, Chile), 2014 (New York, USA), 2018 (Valladolid, Spain).

The congress proceedings have been published by Contemporary Mathematics in several occasions. In each congress, the interest and participation of geometers from Iberoamerica and other regions of the world have grown substantially, and the congress has become very dynamic with a high academic standard, with more and more mathematical fields being represented with lectures on exciting recent developments.

# Speakers

**Alicia Dickenstein**

University of Buenos Aires, Argentina

**Andrés Navas**

University of Santiago, Chile

**Valentino Tosatti**

New York University, USA

**Javier Fernández de Bobadilla**

Basque Center for Applied Mathematics, Spain

**Mihnea Popa**

Harvard University, USA

**Ravi Vakil**

Stanford University, USA

**Luna Lomonaco**

Instituto de Matemática Pura e Aplicada (IMPA), Brazil

**Cecília Salgado**

University of Groningen, Netherlands

# Plenary talks (Abstracts)

**Alicia Dickenstein**

##### Sparse systems with high local multiplicity

Consider a sparse system of \(n\) Laurent polynomials in \(n\) variables with complex coefficients and support in a finite lattice set \(A\). The maximal number of isolated roots in the torus of the system is known to be the normalized volume of the convex hull of \(A\) (the BKK bound). We explore the following question: if the cardinality of \(A\) equals \(n+m+1\), which is the maximum local intersection multiplicity at one point in the torus in terms of \(n\) and \(m\)? This study was initiated by Gabrielov in the multivariate case. In joint work with Frédéric Bihan and Jens Forsgård, we give an upper bound that is always sharp for circuits and, under a generic technical hypothesis, it is considerably smaller for any codimension \(m\). We also present a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.

**Javier Fernández de Bobadilla**

##### Symplectic degenerations at radius 0 and their applications

Given a normal crossings degeneration \(f:(X,\Omega_X)\to\Delta\) of complex Kahler manifolds, in recent work together with T. Pelka, we have shown how to associate a smooth locally trivial fibration \(f_A:A\to \Delta_{log}\) over the real oriented blow up of the disc \(\Delta\). It is moreover endowed with a closed \(2\)-form \(\omega_A\) giving it the structure of a symplectic fibration. The restriction of \(\omega_A\) to every fibre of \(f_A\) «at positive radius» (that is over a point of \(\Delta\setminus\{0\}\) is the modification by a potential of the restriction of \(\omega_X\) to the same fibre. The construction is so that:

(1) We can produce symplectic representatives of the monodromy with very special dynamics, and based on this and on a spectral sequence due to McLean prpve the family version of Zariski’s multiplicity conjecture.

(2) If \(f\) is a maximal Calabi-Yau degeneration we can produce Lagrangian torus fibrations over a the complement of a codimension 2 set over the (expanded) essential skeleton of the degeneration, satisfying many of the properties conjectured by Kontsevich and Soibelman.

**Luna Lomonaco**

##### Rational maps, kleinian groups and correspondences

The analogies between the iteration of holomorphic maps and the action of Kleinian groups are numerous.

In this talk we will discuss how these two worlds can be combined together. More precisely, we will see how the dynamics of rational maps and kleinian groups can be united in a holomorphic correspondence.

**Andrés Navas**

##### On the geometry and topology of diffeomorphisms groups

In this talk, I will discuss some topological and geometric properties of diffeomorphisms groups that are hard to tackle via classical methods because of the lack of local compactness. In particular, I will elaborate on Gromov’s notion of distortion in this context. I will also draw the ideas of a result recently obtained in collaboration with Hélène Eynard-Bontemps (Ins. Fourier, Grenoble): The space of pairs of twice-differentiable commuting diffeomorphisms of \(1\)-manifolds is path-connected.

**Mihnea Popa**

##### When do varieties map to each other?

A basic question in algebraic geometry is whether there can be any non-constant maps between (smooth, projective) varieties of different types. I will explain some basic and some more sophisticated obstructions to the existence of such maps, the latter revolving around the notion of Kodaira dimension of an algebraic variety. I will also discuss some recent conjectures.

**Cecília Salgado**

##### Mordell-Weil rank jumps on families of elliptic curves

**Valentino Tosatti**

##### The volume function on projective varieties

The volume of a line bundle on a smooth projective variety is a rough measure for the asymptotic growth of the dimension of the space of sections of its high tensor powers, and line bundles with positive volume are called big. The volume extends naturally to a continuous function on the real Neron-Severi group, which vanishes outside the big cone and is \(C^1\) differentiable inside of it, by work of Boucksom-Favre-Jonsson and Lazarsfeld-Mustata. An interesting question is then to understand the behavior of the volume near the boundary of the big cone. More precisely, given \(D\) a pseudoeffective \(R\)-divisor with volume zero and \(A\) an ample divisor, what is the behavior of the function \(vol(D+tA)\) as \(t\) decreases to zero? I will discuss joint work with Simion Filip and John Lesieutre where we construct examples where this volume function is \(C^1\) but not better, and answer negatively questions of Lazarsfeld, Fujino and others.

**Ravi Vakil**

##### Bott periodicity, algebro-geometrically

I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.

# Special Sessions

##### Higher Dimensional Algebraic Geometry

Session Organizers

Robert Auffarth

Universidad de Chile, Chile

Juan Carlos Naranjo

University of Barcelona, Spain

##### Dynamical Systems

Session Organizers

Mónica Moreno Rocha

CIMAT, México

Felipe Riquelme

Pontificia Universidad Católica de Valparaíso, Chile

##### Algebraic Surfaces

Session Organizers

Michela Artebani

Universidad de Concepción, Chile

Patricio Gallardo

University of California, Riverside, USA

##### Riemann Surfaces

Session Organizers

Paul Apisa

University of Wisconsin – Madison, USA

Sebastián Reyes Carocca

Universidad de Chile, Chile

## Schedule

December 10th – 15th, 2023

##### Monday, December 11th

##### Tuesday, December 12th

##### Wednesday, December 13th

##### Thursday, December 14th

##### Friday, December 15th

#### 9:30 – 11:00

Registration

**Special session talks: **

##### S1 – S2 (*)

**Plenary Speaker:** Prof. Alicia Dickenstein

**Special session talks:**

##### S1 – S3 – S4 (*)

**Special session talks:**

##### S3 – S4 (*)

#### 11:00 – 11:30

Coffee break

Coffee break

Coffee break

Coffee break

Coffee break

#### 11:30 – 12:30

**Plenary Speaker:** Prof. Mihnea Popa

**Plenary Speaker:** Prof. Cecília Salgado

**Plenary Speaker:** Prof. Javier Fernández de Bobadilla

**Plenary Speaker:** Prof. Luna Lomonaco

**Special session talks:**

##### S1 – S2 (*)

#### 12:30 – 15:00

Lunch break

Lunch break

Lunch break

Lunch break

Lunch break

#### 15:00 – 16:00

**Plenary Speaker:** Prof. Ravi Vakil

**Plenary Speaker:** Prof. Andrés Navas

Free

**Plenary Speaker:** Prof. Valentino Tosatti

#### 16:00 – 16:30

Coffee break

Coffee break

Free

Coffee break

#### 16:30 – 18:00

**Special session talks:**

##### S3 – S4 (*)

**Special session talks:**

##### S3 – S4 (*)

Free

**Special session talks:**

##### S1 – S2 – S4 (*)

#### 18:15 – 20:00

**Opening Ceremony**

#### 20:00

**Dinner**

*** Special session talks**

**S1: Higher Dimensional Algebraic Geometry **

**S2: Dynamical Systems**

**S3: Algebraic Surfaces**

**S4: Riemann Surfaces**

## Schedule

December 10th – 15th, 2023

Join us at the **MATH-AmSud School on Geometry: Group Actions, Symmetries, Moduli and Beyond**. This Satellite School on Geometry will take place from **December 4th to 7th** in the city of **El Quisco** (Valparaíso region). To learn more, click **here**.

Scientific Committee

**Ángel Carocca**

Chile

**Samuel Grushevsky**

USA

**Laura DeMarco**

USA

**Francisco Plaza Martín**

Spain

**Alexis García Zamora**

Universidad Autónoma de Zacatecas,

Mexico

**Rubí E. Rodríguez**

Chile