Based on several past works with G. Kerr and H. Ruddat and on some ongoing projects with C.-Y. Mak, D. Matessi, H. Ruddat, and A. Vicente.

In this talk, the emphasis will be on Brill–Noether theorems using the language of linear systems on algebraic curves developed in the previous talk. We will focus on existence and dimension results for linear series, illustrating the theory through concrete examples. Finally, we will briefly discuss recent developments and related research directions, if time permits.

The notion of visibility from the origin along straight lines has been generalised by considering lattice points viewed through nonlinear trajectories. Inspired by this, we define polynomial Farey sequences, reducing to the classical Farey sequences for linear polynomials. In this talk, we discuss the distributional properties of polynomial Farey fractions and their subsequences using discrepancy and pair correlation measures

This will be a two-talk mini-series. In the first talk I will give a very gentle introduction to Brill–Noether theory for complex algebraic curves. I will start from the basic question: what kinds of functions can exist on a curve, and how many independent ones can we hope for? After introducing some necessary concepts, I will explain the classical results of Brill–Noether theory, which give a simple criterion for when such functions should exist on a “typical” curve. The emphasis will be on ideas and examples rather than technical details.

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.