With next lecture we are continuing the mini-course started in November. We shall discuss O. Viro’s method of patchworking. For that we need some staff from real algebraic curves, which will be discussed first, as much as the time permits.

The emergence of continuum spacetime from a quantum substrate remains a key challenge in theoretical physics. Continuum geometry may arise dynamically through renormalization group (RG) flows in theory space. This talk reviews conceptual aspects of continuum emergence within quantum gravity, emphasizing non-perturbative RG methods in analyzing the global structure of theory space and its phase transitions. It then proposes, in a collaborative framework, an extension of this analysis to discrete quantum gravity approaches, investigating whether topological invariants of RG trajectories—such as fixed-point indices, flow connectivity, or higher-order topological characteristics—can serve as structural probes of the theory space, particularly within tensorial group field theory (TGFT). Further, we propose developing a computational framework based on Monte Carlo RG and lattice field theory techniques to construct alternative formulations not always accessible by analytical methods. The broader aim is to position quantum gravity research at the intersection of complexity science, computational physics, and pure mathematics.

In this talk I will introduce valuation-theoretic tools, such as the graded algebra grν(R) of a general valuation ν on a commutative ring R. This will serve as a tool to algebraically encode the leading terms of our analytic solutions. The graded algebra construction presents certain advantages, such as functoriality, which is essential for proving the following lifting theorem.

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).