Johannes RauDepartament of MathematicsFaculty of Science 
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
We are currently organizing the biweekly LatinoAmericano Geometría Algebráica Real y Tropical Online Seminario (LAGARTOS). Please contact us if you are interested in participating. Lagartos is Spanish for lizards.
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.
My research area is tropical geometry (the fancy adjective tropical was originally used in the context of the maxplus algebra to honour earlier work of the Brazilian (Hungarianborn) 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.
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 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 downtoearh terms. Of course, we cannot work wonders and find answer to all questions (it is easy to estimate the size of the reallife 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.
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.
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.
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.
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 semistable degenerations, realoriented blowups 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 LefschetzHopf trace formula. J. Algebraic Combin. (2023). doi: 10.1007/s1080102301220y. arXiv: 2010.07901.
Johannes Rau. The tropical PoincaréHopf 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/653/43. 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. 375.12 (2019), pp. 895–915. doi: 10.1007/s00208019 01863y. 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 019009066. 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. 281.12 (2015), pp. 501–522. doi: 10.1007/s00209 015 1498 4. arXiv: 1412.4235.
Mathieu GuayPaquet, 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/CJM20150360. 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/s1334801200721. arXiv: 1012.3260.
Johannes Rau. Intersections on tropical moduli spaces. Rocky Mt. J. Math. 46.2 (2016), pp. 581–662. doi: 10.1216/RMJ2016462581. arXiv: 0812.3678.
Hannah Markwig and Johannes Rau. Tropical descendant GromovWitten invariants. Manuscr. Math. 129.3 (2009), pp. 293–335. doi: 10.1007/s00229 00902565. 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/s0020900904831. 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. PhDThesis. Technische Universität Kaiserslautern, 2009. url: http://kluedo.ub.unikl.de/volltexte/2009/2370/.
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j.rau (at) uniandes.edu.co Postal address
Departamento de Matemáticas 
Phone
Phone: +57 1 3394949 Office
Room H001 