A tropical surface


Soy profesor asistente en la Universidad de los Andes. Mi área de investigación es la geometría tropical y sus conexiones con la geometría enumerativa, la geometría algebráica real y la teoría de intersecciones. Mas generalmente, mi investigación está radicada en la geometría algebráica, la geometría simpléctica y la combinatoria. Si quieren saber más, miran abajo o escribenme un correo.

Email: j.rau (at) uniandes.edu.co

A real amoeba


LAGARTO Seminario

Estamos organizando quincenalmente el seminario Latino-Americano Geometría Algebráica Real y TrOpical. En caso de interesa a participar, esribennos por favor.

Patchworking En Línea

Construye tu curva algebráica personal con esta pequena aplicación de browser ;)


Estoy preperando un libro sobre la geometría tropical en colaboración con Grigory Mikhalkin. El siguiente enlace dirige al borrador (más o menos) actual. Me alegro mucho recibir sus correciones y sugerencias.


(hasta ahora, solo en inglés)

My research area is tropical geometry (the fancy adjective tropical was originally used in the context of the max-plus algebra to honour earlier work of the Brazilian (Hungarian-born) mathematician Imre Simon). Even though the origins of the field are much older, tropical geometry emerged as an new trend in algebraic and symplectic geometry around 2000. Here is a list of keywords describing my research interests.

  1. (Real) enumerative geometry
    • Real Hurwitz numbers
    • Descendant Gromov-Witten invariants, Psi classes
  2. Topology of real algebraic varieties
    • Hilbert's 16th problem for nodal curves
    • Real algebraic surfaces
  3. Intersection theory and Hodge theory
    • Tropical intersection products
    • Matroids
    • Tropical Hodge groups
    • Rational equivalence
  4. Book
    I write a book on tropical geometry together with Grigory Mikhalkin. You can find a draft version here. Comments and corrections are highly welcome.
    Tropical geometry (draft)

Dinosaurs and skeletons

Tropical geometry for non-mathematicians

Tropical mathematics can be compared to the world of dinosaurs. When paleontologists want to learn more about these animals, they can't just watch them them in the zoo or the jungle, because unfortunately the poor things became extinct a long time ago. Instead, they work more like A tropical surface archeologists. They go digging for their bones, try to reassemble their skeletons, and from that draw conclusions about how these animals looked like, what they ate, how they hunted etc. In tropical geometry, we do exactly the same!

In our setting, the dinosaurs are called algebraic varieties. These are complicated geometrical shapes given by polynomial equations. Such algebraic varieties show up all the time in mathematics, science and real life, and therefore their study forms one of the oldest and most sophisticated fields in mathematics (called algebraic geometry). Algebraic varieties are often so complicated that it is impossible to get our hands on them directly like the extinct dinosaurs. However, in some cases mathematicians found a way to get hands on the skeletons of these mathematical dinosaurs. Technically, you first have to turn the dinosaurs into amoebas and then starve them out until only the skeletons are left this is where the analogy gets a little violent ;).

The mathematical skeletons are called tropical varieties, and in tropical geometry we play paleontologist and try to find out more about the original geometrical objects by studying their tropical skeletons (you can find some pictures on this page). The nice thing is that tropical varieties are much simpler objects than the original ones and can be studied in much more down-to-earh terms. Of course, we cannot work wonders and find answer all questions (it is easy to estimate the size of the real-life dinosaur from its skeleton, but did it have furry or smooth skin?), but by now some remarkable facts about algebraic varieties were deduced from the study of their tropical skeletons, and that is why tropical geometry is nowadays such an exciting and rapidly growing field.

Tropical geometry for students/mathematicians from other fields

You are bachelor/master student in mathematics (or a researcher from a different field) and want to embark on a first expedition to the tropics? Then have a look at these lecture notes. A combinatorial patchworking of degree 30

These notes grew out of a lecture series I gave for bachelor students without any prior knowlwedge of the topic. They are therefore on a very elementary level and give priority to intution and illustrations as opposed to rigour and depth.

Topology of real algebraic varieties

Combinatorial Patchworking

One of the origins of tropical geometry is Viro's patchworking method. The link below leads to a small browser app which allows you to do some experiments with this method.

You can use it to create your own real algebraic curves and pictures like the one to the right.


Two planes intersect in a line



2020-02 Introducción a la Geometría Tropical
2020-02 Cálculo Integral y Ecuaciones Diferenciales
2020-01 Cálculo Integral y Ecuaciones Diferenciales
Invierno 19/20 Geometría Enumerativa Tropical 2
Verano 19 Análisis 1
Verano 19 Geometría Enumerativa Tropical 2
Invierno 18/19 Geometría Tropical
Verano 18 Geometría Algebráica I
Invierno 17/18 Algebra Lineal I
Invierno 17/18 Mate básica para Refugiados
Verano 17 Geometría Algebráica I
Verano 17 Mates básicas para Refugiados
Invierno 16/17 Teoría de Hodge Tropical
Verano 16 Curvas Algebráicas Reales
Invierno 15/16 Lineare Algebra I
Verano 15 Algebra
Verano 14 Geometría Algebráica II
Invierno 13/14 Geometría Algebráica I
Verano 13 Algebra
Invierno 12/13 Geometría Enumerativa

Seminarios de Estudiantes

Verano 16 De la Geometría Hiperbólica a la Teoría de nudos

Seminarios de Investigación

2020-01 Matroides y Geometría Tropical
Verano 17 Recursion Topológica
Invierno 16/17 Recursion Topológica
Verano 16 Integrable systems and tropical geometry
Invierno 15/16 Moduli spaces of log-stable maps
Verano 15 Cluster algebras
Verano 14 Derived Categories
Invierno 13/14 Random matrices and Hurwitz numbers
Verano 13 Moduli stacks of algebraic curves
Invierno 12/13 Berkovich spaces
A real nodal quartic


Gluing pairs of pants

Correo electrónico

j.rau (at) uniandes.edu.co

Correo ordinario

Departamento de Matemáticas
Universidad de los Andes
Carrera 1 # 18A - 12
Bogotá, Colombia
Postal Code: 111711


Fon: +57 1 3394949
Ext: 5229


Oficina H-001
Edificio H
Cómo llegar