This is an EMS-European Mathematical Society summer school.

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 with the support of the European Mathematical Society, the Doctoral School of the University of Valladolid and the Foundation Compositio Mathematica. 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 three courses on recent applications in different areas. The four 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 4 courses organized in an exercise-based format (50 minutes of lecture + 50 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: Alheydis Geiger, Eberhard Karls Universität Tübingen and Jorge Alberto Olarte, 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.
Tutor: Alheydis Geiger, Eberhard Karls Universität Tübingen and Jorge Alberto Olarte, 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.
COURSE C                   
Title: Introduction to tropical curve counting.
Lecturer: Renzo Cavalieri, Colorado State University.
Tutor: Andy Fry, Colorado State University.
Abstract: The main goal of this minicourse is to introduce students to enumerative applications of tropical geometry. Tropical geometry is a combinatorialization of ordinary geometry. By either doing algebraic geometry over the tropical semi-field or over fields with a valuation, one may associate piecewise linear objects to ordinary algebraic varieties. Remarkably such highly degenerate objects retain a lot of geometric information about the original algebraic varieties. For example, the notion of dimension is preserved: thus to an algebraic curve is associated a graph. Further, the genus of the curve generically is equal to the genus of the graph. In the last twenty-some years many groups of researchers have explored these connections, providing a wealth of correspondence theorems: statements on how to recover classical information from its tropical counterpart. In this minicourse we will focus in particular on a celebrated result of Mikhalkin, which shows that the count of rational tropical plane curves of degree d through 3d − 1 points in general position equals the classical count. Time permitting we will show some further applications of tropical techniques to Hurwitz theory, the count of ramified covers of Riemann Surfaces.
Title: Tropical approach to singular points and its connections to Nagata’s conjecture and linear codes.
Lecturer: Nikita Kalinin, Saint Petersburg University and National Research University Higher School of Economics (HSE).
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.