A tropical surface


I work at the Departament of Mathematics at the Universidad de los Andes, Bogotá, Colombia. My research area is tropical geometry and its connections to enumerative geometry, real algebraic geometry and intersection theory. More general, my mathematical interests focus on algebraic geometry, symplectic geometry and combinatorics. For more details, please check out the other sections and contact me.

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

A real amoeba



We are currently organizing a biweekly Latin-American Tropical and Real Algebraic Geometry seminar. Please contact us if you are interested in participating. A lagarto (Latino-Americano Geometría Algebráica Real y TrOpical) is lizard.

Patchworking Online

Construct your own real algebraic curve using this simple browser app ;)


I write a book on tropical geometry together with Grigory Mikhalkin. You can find a draft version here. Comments and corrections are highly welcome.


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 Introduction to Tropical Geometry
2020-02 Integral Calculus and Differential Equations
2020-01 Integral Calculus and Differential Equations
Winter 19/20 Tropical Enumerative Geometry 2
Summer 19 Analysis 1
Sommer 19 Tropical Enumerative Geometry 1
Winter 18/19 Tropical Geometry
Summer 18 Algebraic Geometry I
Winter 17/18 Linear Algebra I
Winter 17/18 Math Basics for Refugees
Summer 17 Algebraic Geometry I
Summer 17 Math Basics for Refugees
Winter 16/17 Tropical Hodge theory
Summer 16 Real algebraic curves
Winter 15/16 Linear algebra I
Summer 15 Algebra
Summer 14 Algebraic Geometry II
Winter 13/14 Algebraic Geometry I
Summer 13 Algebra
Winter 12/13 Enumerative Geometry

Students Seminars

Summer 16 From the hyperbolic plane to knot theory

Working group seminars

2020-01 Matroids and tropical geometry
Summer 17 Topological Recursion
Winter 16/17 Topological Recursion
Summer 16 Integrable systems and tropical geometry
Winter 15/16 Moduli spaces of log-stable maps
Summer 15 Cluster algebras
Summer 14 Derived Categories
Winter 13/14 Random matrices and Hurwitz numbers
Summer 13 Moduli stacks of algebraic curves
Winter 12/13 Berkovich spaces
A real nodal quartic


Gluing pairs of pants


j.rau (at) uniandes.edu.co

Postal address

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


Phone: +57 1 3394949
Ext: 5229


Room H-001
Building H
How to get there