In words of Diane Maclagan and Bernd Sturmfels at the beginning of their book, Introduction to Tropical Geometry (AMS, 2015), “Tropical geometry is an exciting new field at the interface between algebraic geometry and combinatorics with connection to many other areas”. It is now a wide field with applications in many branches of pure and applied mathematics like enumerative and real algebraic geometry, mirror symmetry, coding theory, etc.

This school is organized by the IMUVA-Mathematics Research Institute of the University of Valladolid. The main objective is to offer to future and young researchers of distinct branches of mathematics a series of courses at doctorate level on a new field connected to many areas. There will be an introductory course with minimal prerequisites and two courses on recent applications in different areas. The three courses are aimed to be as self-contained as possible in order to make the school attractive to a wide and diverse audience..

The target audience is master’s students, PhD students and postdocs from all areas of mathematics. Senior researchers interested in learning on this new topic are also welcome. The school is based on 3 courses organized in an exercise-based format (1 hour of lecture + 40 minutes of exercises for each session). We will also give the opportunity to young researchers to present their own work either through a series of short communications or a poster session.

Title: Introduction to Tropical Geometry.
LecturerDiane Maclagan, University of Warwick.
Tutor: Sara Lamboglia, Johann Wolfgang Goethe-Universität and Marta Panizzut, TU Berlin
Abstract: Tropical geometry is a combinatorial shadow of algebraic geometry, in which algebraic varieties are replaced by piecewise linear objects that retain core information about the original varieties. This allows techniques from combinatorics, including polyhedral geometry, to be brought to bear on problems from algebraic geometry, and has also allowed new applications of these techniques outside algebraic geometry, and also outside mathematics. This minicourse will give a basic introduction, with minimal prerequisites, to this exciting area.
Title: Introduction to tropical ideals.
LecturerFelipe Rincón, Queen Mary University of London.
TutorSara Lamboglia, Johann Wolfgang Goethe-Universität and Marta Panizzut, TU Berlin
Abstract: Tropical ideals are algebro-combinatorial objects introduced with the aim of giving tropical geometry a solid algebraic foundation. These ideals over the semiring of tropical polynomials have a rich structure dictated by a sequence of ‘compatible’ matroids, and satisfy many desirable properties. This minicourse will give a basic introduction to tropical ideals, discuss some of their main properties, and point out some of the main open questions in this direction.
Title: Tropical approach to singular points and its connections to Nagata’s conjecture and linear codes.
LecturerNikita Kalinin, National Research University Higher School of Economics in St. Petersburg and University of St. Petersburg, Russia.
Tutor: Nikita Kalinin
Abstract: Tropical geometry provides a tool to capture the influence of a singular point of a curve on a subdivision of the Newton polygon of this curve. To visualise this influence we will study some necessary facts about matroids and polyhedral geometry, so the students are advised to read somewhere the definition of a matroid and look at the main examples in advance. Then we discuss applications of the above ideas for Nagata’s conjecture and linear codes.