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

LAGARTOS birthday poster



We are currently organizing the biweekly Latino-Americano Geometría Algebráica Real y Tropical Online Seminario (LAGARTOS). Please contact us if you are interested in participating. Lagartos is Spanish for lizards.

Patchworking Online

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

You can find more information on this topic in these slides.


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



A real amoeba 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 go watch 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. 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 dig out the skeletons of these mathematical dinosaurs. Technically, you first have to turn the dinosaurs into amoebas and then starve them out until only their skeletons are left ;).

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 to 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 at the moment such an exciting and steadily growing field.

Further Resources

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.


A few slides with more info on some projects.

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




Johannes Rau, Arthur Renaudineau, and Kris Shaw. Real phase structures on tropical manifolds and patchworks in higher codimension. Preprint (2023). arXiv: 2310.08313.


Jan Draisma, Sarah Eggleston, Rudi Pendavingh, Johannes Rau, and Chi Ho Yuen. The amoeba dimension of a linear space (2023). arXiv: 2303.13143.



Erwan Brugallé, Lucía López de Medrano, and Johannes Rau. Combinatorial patchworking: back from tropical geometry. Trans. Amer. Math. Soc. (accepted) (2022). arXiv: 2209.14043.


Johannes Rau. Real semi-stable degenerations, real-oriented blow-ups and straightening corners. Int. Math. Res. Notices 2023.18 (2023), pp. 15896–15927. doi: 10.1093/imrn/rnad005. arXiv: 2203.17097.


Johannes Rau, Arthur Renaudineau, and Kris Shaw. Real phase structures on matroid fans and matroid orientations. J. Lond. Math. Soc. 106.4 (2022), pp. 3687–3710. doi: 10.1112/jlms.12671. arXiv: 2106.08728.


Johannes Rau. On the tropical Lefschetz-Hopf trace formula. J. Algebraic Combin. (2023). doi: 10.1007/s10801-023-01220-y. arXiv: 2010.07901.


Johannes Rau. The tropical Poincaré-Hopf theorem. J. Combin. Theory Ser. A 196 (2023), p. 105733. doi: 10 . 1016 / j . jcta . 2023 . 105733. arXiv: 2007.11642.


Grigory Mikhalkin and Johannes Rau. Spines for amoebas of rational curves. Enseign. Math. 65 (2 2019), pp. 377–396. doi: 10.4171/LEM/65-3/4-3. arXiv: 1906.04500.


Jan Draisma, Johannes Rau, and Chi Ho Yuen. The dimension of an amoeba. Bull. London Math. Soc. 52.1 (2020), pp. 16–23. doi: 10.1112/blms.12301. arXiv: 1812.08149.


Johannes Rau. Lower bounds and asymptotics of real double Hurwitz numbers. Math. Ann. 375.1-2 (2019), pp. 895–915. doi: 10.1007/s00208-019- 01863-y. arXiv: 1805.08997.


Boulos El Hilany and Johannes Rau. Signed counts of real simple rational functions. J. Algebraic Combin. 52.3 (2020), pp. 369–403. doi: 10.1007/s10801- 019-00906-6. arXiv: 1712.05639.


Philipp Jell, Johannes Rau, and Kristin Shaw. Lefschetz (1,1)-theorem in tropical geometry. Épijournal Géom. Algébrique 2.11 (2018). doi: 10. 46298/epiga.2018.volume2.4126. arXiv: 1711.07900.


Ilia Itenberg, Grigory Mikhalkin, and Johannes Rau. Rational quintics in the real plane. Trans. Amer. Math. Soc. 370 (2018), pp. 131–196. doi: 10.1090/ tran/6938. arXiv: 1509.05228.


Hannah Markwig and Johannes Rau. Tropical Real Hurwitz numbers. Math. Z. 281.1-2 (2015), pp. 501–522. doi: 10.1007/s00209- 015- 1498- 4. arXiv: 1412.4235.


Mathieu Guay-Paquet, Hannah Markwig, and Johannes Rau. The Combinatorics of Real Double Hurwitz Numbers with Real Positive Branch Points. Int. Math. Res. Not. 2016.1 (2016), pp. 258–293. doi: 10. 1093/imrn/rnv135. arXiv: 1409.8095.


Lars Allermann, Simon Hampe, and Johannes Rau. On rational equivalence in tropical geometry. Canad. J. Math. 68.2 (2016), pp. 241–257. doi: 10. 4153/CJM-2015-036-0. arXiv: 1408.1537.


Georges François and Johannes Rau. The diagonal of tropical matroid varieties and cycle intersections. Collect. Math. 64.2 (2013), pp. 185–210. doi: 10.1007/s13348-012-0072-1. arXiv: 1012.3260.


Johannes Rau. Intersections on tropical moduli spaces. Rocky Mt. J. Math. 46.2 (2016), pp. 581–662. doi: 10.1216/RMJ-2016-46-2-581. arXiv: 0812.3678.


Hannah Markwig and Johannes Rau. Tropical descendant Gromov-Witten invariants. Manuscr. Math. 129.3 (2009), pp. 293–335. doi: 10.1007/s00229- 009-0256-5. arXiv: 0809.1102.


Lars Allermann and Johannes Rau. First steps in tropical intersection theory. Math. Z. 264.3 (2010), pp. 633–670. doi: 10.1007/s00209-009-0483-1. arXiv: 0709.3705.



Johannes Rau. A First Expedition to Tropical Geometry. Lecture notes for a mini course given at the International School on Topological and Geometric Combinatorics, Tehran, Iran, 13–16/02/2017. 2018. url: https://math.uniandes.edu.co/~j.rau/downloads/FirstExpedition.pdf.


Grigory Mikhalkin and Johannes Rau. Tropical Geometry. textbook in preparation. 2019. url: https://math.uniandes.edu.co/~j.rau/downloads/main.pdf.


Boulos El Hilany, Johannes Rau, and Arthur Renaudineau. Combinatorial patchworking tool. Javascript applet. 2017. url: https://math.uniandes.edu.co/~j.rau/patchworking/patchworking.html.


Johannes Rau. Tropical intersection theory and gravitational descendants. PhD-Thesis. Technische Universität Kaiserslautern, 2009. url: http://kluedo.ub.uni-kl.de/volltexte/2009/2370/.


Selected Talks

  • 17/12/2019 MATRIX conference “Tropical geometry and mirror symmetry”, Creswick, Australia
  • 01/10/2019 Workshop “Regensburg days on non-archimedean and tropical geometry”, Regensburg, Germany
  • 13/09/2019 CMO Workshop “Tropical Methods in Real Algebraic Geometry”, Oaxaca, Mexico
  • 29/04/2019 MFO-Workshop “Tropical Geometry: new directions”, Oberwolfach, Germany
  • 22/01/2019 Seminar “Diskrete Mathematik/Geometrie”, TU Berlin, Germany
  • 16/01/2019 Seminar “Topologie Algèbre et Géométrie”, Université de Nantes, France
  • 03/12/2018 Colloquium talk, Universität Bern, Switzerland
  • 02–04/05/2018 Lecture Series “Tropical methods in real algebraic geometry”, ASGARD math meeting, U Oslo, Norway
  • 22/03/2018 Workshop on Moduli spaces of curves and mirror symmetry, IML Stockholm, Sweden
  • 28/11/2017 Workshop “Young Researchers in String Mathematics”, MPI Bonn, Germany
  • 13–16/02/2017 Lecture series “Tropical geometry”, School on Topological and Geometric Combinatorics, Tehran, Iran
  • 11/01/2016 German-Israeli Workshop in Algebraic and Tropical Geometry, Tel Aviv, Israel
  • 12/06/2015 AMS-EMS-SPM International Meeting, Session on Moduli Theory, Porto, Portugal
  • 01/05/2015 MFO-Workshop “Tropical Aspects in Geometry, Topology and Physics”, Oberwolfach, Germany
  • 18/09/2014 Meeting of the German and Polish mathematical societies, Real Algebraic Geometry workshop, Poznán, Poland
  • 03/04/2013 Seminar “Géométrie tropicale”, Institut de Mathématiques de Jussieu, Paris, France
  • 23/05/2011 Seminar “Géométrie symplectique”, Institut de Recherche Mathématique Avancée, Strasbourg, France
  • 04/04/2008 Combinatorics seminar at the University of Michigan, Ann Arbor, USA

Organization of conferences

  • 27–31/03/2017 Conference “Tropical curve counts, motivic integration and nonarchimedean geometry”, Universität Tübingen (joint with Hannah Markwig)
  • 14–18/09/2015 Conference “Tropical Geometry in the Alps”, Les Diablerets, Switzerland (joint with Kristin Shaw, Grigory Mikhalkin)
  • 24–28/11/2014 Closing conference of the TROPGEO project, Saas Fee, Switzerland (joint with Grigory Mikhalkin)
  • 20/02/2013 Meeting of the seminar “Tropical Geometry in Europe”, Universität des Saarlandes (joint with Hannah Markwig)
  • 17–20/09/2012 Minisymposium of the DMV annual meeting in Saarbrücken (joint with Hannah Markwig)
  • 18–21/12/2011 Conference “Perspectives in Tropical Geometry 2011”, Arolla, Switzerland (joint with Grigory Mikhalkin)



2022-02 Algebraic Geometry
2022-02 Linear Algebra
2022-01 Abstract Algebra 1
2022-01 Linear Algebra
2021-02 Hodge theory
2021-02 Integral Calculus and Differential Equations
2021-01 Complex Variable
2021-01 Integral Calculus and Differential Equations
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

2022-02 Hodge theory of matroids
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: 2722


Room H-001
Building H
How to get there