Séminaire des Doctorants et Doctorantes

An Introduction to Tropical Geometry

by Jules Chenal

BigBlueButton Platform (Webinar)

BigBlueButton Platform



The construction of Tropical Geometry began with optimisation problems. Back then we would have used the word idempotent instead of tropical. The idea was to reformulate some optimisation problems in the vocabulary of linear algebra by weakening the hypotheses that make a field. It provided a translation of optimisation problems into geometric ones. Even if tropical linear algebra doesn't behave completely as in the classical case the resemblance was close enough to leave the linear realm, consider polynomials and define a "tropical algebraic geometry". Which is what people usually mean by tropical geometry. In my talk I will present through examples the basics of tropical geometry and try to highlight (in dimension 2) the interactions between complex, realistic and tropical with 3 fundamental theorems: Maslov's dequantisation theorem by Kapranov, Itenberg/Katzarkov/Mikhalkin/Zharkov's theorem on tropical homology and Viro's patchworking.