This is an introduction to tropical geometry in two talks, mainly in case of curves. The stress will be put on the applications in complex and real algebraic geometry. At the end a few other approaches will be discussed briefly.

Linear systems on complex algebraic curves have been an interesting and rich area of study for a long time. It is known that a real structure on the initial curve induces a real structure on the varieties parameterizing these linear systems. In this talk, I will first present the classical Brill-Noether theorems for these varieties, and then address some fundamental topological questions in the real setting including the case of real trigonal curves.

Friezes, originally introduced by Coxeter, are simple objects which however turned out to be connected with surprisingly many seemingly unrelated concepts from a variety of subjects. These include combinatorics (polygon dissections), number theory (continued fractions), geometry (the Farey graph), and algebra (cluster algebras -- a hot topic of their own). I will survey these connections using interactive visualisations, introduce the family of basic examples of cluster algebras, and present a new connection between not-necessarily-positive friezes and cluster algebras.

This talk will be a survey on Viro’s combinatorial patchworking: a powerful method to construct real algebraic varieties. We will present the construction, give an idea of the proof, make the link with tropical geometry and then prove recent bounds on the Betti numbers of the real part. We will also see how the tropical perspective permits to generalize the initial construction.

Lecture Outline 1. Introduction to Binary Quadratic Forms and Conway’s Topographs We will begin with the basics of binary quadratic forms and their classification, followed by an introduction to Conway’s topographs—a visual and geometric framework for understanding them. Lecture Outline 2. Class Number Formula and Summation over Topographs Building on the first lecture, we will explore the class number formula and how summation identities arise naturally from the structure of topographs. Lecture Outline 3. Evaluation of Lattice Sums via Telescoping over Topographs The final lecture will focus on telescoping techniques, demonstrating how they can be used to evaluate intricate lattice sums—such as the one above—with geometric meaning.

The Geometry Seminar of the ICMS presents a talk by Lionel Lang (University of Gävle). On 27.11.2024, 16:00, Sofia time he will describe various tropical compactifications of the moduli spaces of curves. He will discuss some motivations and applications coming from tropical geometry (joint with M. Melo, J. Rau and F. Viviani).

Classical amoebas serve as a tool to study algebraic varieties and as one of the entry points to tropical geometry. The original definition involves a logarithmic projection of a subvariety of a complex algebraic torus, which can be interpreted as forgetting the phase, i.e. the arguments of complex numbers. In group-theoretic terms, this projection map may be thought of as passing to the quotient by the maximal compact subgroup. Suppose one replaces the algebraic torus with a complex three-dimensional matrix group $PSL_2 \mathbb{C}$. In that case, the analogous projection naturally has a three-dimensional hyperbolic space as its target, and it still makes sense to consider images, i.e. hyperbolic amoebas, of complex algebraic varieties under this map. I will review some of the basic properties of hyperbolic amoebas, extending a fascinating interplay between complex and hyperbolic geometries.

In order to classify objects in singularity theory and algebraic geometry, we define invariants associated to varieties or germs of singularities and hope to have enough tools to compute them easily. From classical projective duality, for any variety X, we can define a dual variety X∗ and we call the class of X the degree of X∗. Plücker’s formula allows one to compute this class for plane curves with a certain set of nodes, cusps and tacnodes.

In the last few decades, Berkovich’s theory of k -analytic space has extended classic rigid geometry. Since k -analytic space has good topological properties, and the Berkovich analytification of algebraic varieties could be also dealt with via the geometry of the model and it has a strong connection with tropical geometry, this allows us to use the combinatorial techniques to study algebraic varieties. Our lectures will mainly discuss the contents of Berkovich space and its application in other aspects of mathematics, we plan to go over the basic properties of Berkovich space in both algebraic and topological ways for the first lecture.