Alexandre Benoist - Elliptic curves over finite fields and point counting
Elliptic curves are a fundamental research area in number theory. They are curves described by a cubic equation and equipped with a group law, so they are both algebraic and geometric objects. They have many applications, the most famous one being the proof of Fermat's Last Theorem by Wiles.
In many applications of elliptic curves in public-key cryptography, it is necessary to determine efficiently the number of rational points of an elliptic curve defined over a finite field. For fields of cryptographic size, the best method up to date for large characteristic is the Schoof-Elkies-Atkin (SEA) algorithm, which is an improvement of Schoof's algorithm.
The aim of this talk is to introduce what elliptic curves are and to explain how the SEA algorithm works.
Lota Copic (Aarhus University) - Quantitative bounds in high-dimensional CLTs
The Central Limit Theorem (CLT) is a fundamental result in probability and statistics, forming the basis of many statistical tests and estimators by enabling inference about model parameters.
In this talk, we focus on quantifying how the dimensionality of the data affects the rate of convergence in high-dimensional versions of the CLT. We begin with a brief overview of the classical univariate CLT and gradually move toward the main results of our current work. Specifically, we show that the convergence rate exhibits sub-polynomial dependence on the dimension, in the hyper-rectangular metric.
This talk is based on joint work with Andreas Basse-O'Connor and David Kramer-Bang (Aarhus University).
Clifford Chan - Explicit Bounds for the Density in Artin's Conjecture over Quadratic Fields
Let $K$ be a number field and let $\alpha\in K^\times$. Let $h\geq 1$ be the largest integer such that $\alpha\in K^{\times h}$.
In the talk, we will first briefly explain the setting of Artin's Conjecture on Primitive Roots. Under GRH there is a density $\mathrm{dens}(\alpha)$ of primes $\mathfrak{p}$ such that $\alpha$ is a primitive root modulo $\mathfrak{p}$; this density is a rational multiple of an Artin constant $A(h)$ that depends on $h$. Expanding on the work of A. Perucca and I. Shparlinski, we extend a technique by Hooley to calculate the ratio $\mathrm{dens}(\alpha)/A(h)$ and show that, if $\mathrm{dens}(\alpha)\neq 0$ and $K$ is quadratic, then this ratio is bounded optimally and uniformly by 8/15and 8/3.
Bruno Dular - Exploration of three-dimensional hyperbolic geometry
Topological surfaces are classified by their genus, the number of “donuts” they are made of. Surfaces of genus at least 2 admit hyperbolic geometries, and actually admit many of them. Thus one could say that most geometries in two dimensions are hyperbolic. It follows from Thurston's geometrization conjecture, proved by Perelman, that the same statement holds in dimension 3 as well.
The goal of this talk is to give an introduction to hyperbolic geometry in two and three dimensions. We will introduce and explore the space of all possible hyperbolic metrics that a surface or 3-manifold admits.
Annika Huch - Simon's Congruence and Absent Scattered Factors
In 1975 Imre Simon introduced the nowadays called Simon's congruence on words in the context of piecewise testable languages. Two words are called Simon k-congruent if they contain the same sets of scattered factors of length upto k, where scattered factors are (non-contiguous) subsequences. Until now the number of congruence classes of this congruence is still unknown. In this talk, we will have a look on the approach of characterising congruence classes by absent scattered factors (those that do not occur). This investigation led to a new factorisation that helped giving the index of Simon's congruence for the binary alphabet.
Thomas Lamby - Hölder Analysis of Thomae-Type Functions and their Rational-Irrational Dichotomy
The Thomae function has long served as a striking example in real analysis, showcasing the interplay between continuity and discontinuity. Introduced by Thomae in 1875 as a refinement of the Dirichlet function, within the framework of Riemann's concept of integration, it is defined as follows. Unless explicitly stated otherwise, any rational number x expressed as x=p/q (p in Z, q in N) with p and q coprime. The Thomae function is then given by
T_θ (x) = 1 if x=0,
q^(-θ) if x is rational with x=p/q,
0 if x is irrational.
with θ=1. The limiting case θ=0 corresponds to the Dirichlet function. For θ>0, the function exhibits the remarkable property of being continuous on the irrational numbers while discontinuous at every rational point. This duality, combined with its self-similar structure, renders the Thomae function an essential object of study for understanding irregular functions in analysis. Here, we will focus on the case where θ in (0,2].
Beyond its classical role in real analysis, the Thomae function has found relevance in broader mathematical and applied contexts. Recent studies have highlighted analogies between its spiked structure and distributions observed in empirical datasets, particularly in biology and clinical research.
This talk focuses on the Hölder regularity of the Thomae function, a key aspect of its behavior. First, we review its fundamental properties, offering a detailed account of its defining characteristics and self-similar nature. Then, we analyze the function's Hölder regularity, uncovering insights into its fractal-like properties through contemporary mathematical tools. By bridging its classical foundations with these contemporary perspectives, we aim to highlight both the theoretical elegance and the deeper structural nuances of this remarkable function.
Simon Lemal - Interpolation and amalgamation: A dual perspective
Craig interpolation is a very interesting property for a logic to have. It makes it significantly easier to automate proof-search, and allows one to prove theorems by working backwards from the goal to obtain a full deduction.
In this talk, we will use the perspective provided by algebraic logic and duality theory to show that classical logic has the Craig interpolation property.
Luis Maia - A very short introduction to Neural Networks
Abstract: Neural networks provide a powerful framework for approximating real-valued, discrete-valued, and vector-valued target functions. They are particularly effective when dealing with complex or difficult-to-interpret input data, and have recently achieved remarkable success in applications such as handwritten character recognition, speech recognition and natural language processing tasks.
In this talk, we will introduce the fundamental concepts of neural networks. We begin with a brief historical overview, followed by a general definition of neural networks. We will then present both single-layer and multilayer perceptrons. Furthermore, we will discuss supervised training of networks using gradient descent for parameter optimization. And will conclude with an introduction to the theoretical challenge of characterizing the properties of neural networks at initialization, when parameters are randomly initialized.
Anaïs Meunier – Topological Statistics of the Cosmic Web
The large-scale structure of the universe, commonly referred to as the cosmic web, emerges from the gravitational evolution of tiny fluctuations in the early universe. Initially described as a nearly Gaussian random field, the matter distribution becomes increasingly non-linear and anisotropic, forming a complex network of voids, walls, filaments, and nodes.
This talk will introduce a topological and geometric approach to studying this structure, as developed notably by Sandrine Codis and collaborators, with a focus on Minkowski functionals and Euler characteristics as statistical tools. These descriptors are particularly powerful in the non-linear regime, offering robustness against observational noise and bias. We will explore how these tools reveal information beyond standard correlation functions and provide insight into non-Gaussian and anisotropic features.
Finally, I will present an example of recent theoretical predictions in a non-Gaussian anisotropic model and discuss how perturbation theory and Gram-Charlier expansions can be used to make analytical progress in this rich geometrical framework.
Leolin Nkuete - Drinfeld Modules
Drinfeld modules, named after the eminent mathematician Vladimir Drinfeld, represent a profound and elegant synthesis of algebraic geometry, number theory, and the theory of function fields. Since their introduction in the 1970s, Drinfeld modules have been pivotal in understanding and solving various mathematical problems, such as the Stark conjecture in function fields, Langlands conjectures for GL2 over global function fields, explicit class field theory for global function fields, and the construction of curves over finite fields with many rational points, among others. Introduced by Drinfeld as elliptic modules [Dri76] and later renamed Drinfeld modules, these are algebraic objects that generalize the concept of elliptic curves. It is well known that elliptic curves play a central role in number field arithmetic geometry, and Drinfeld modules play an analogous role for function fields. This talk aims to provide a detailed introduction to the algebraic theory of Drinfeld modules.
Francisco Pina - High-Dimensional Inference and Diffusion Processes
High-dimensional statistics plays a key role in modern data analysis, especially when dealing with complex dynamical systems. In this talk, I will introduce some core ideas and motivations behind high-dimensional inference, with a focus on diffusion processes. We will discuss typical challenges, modelling approaches, and the role of structural assumptions like sparsity in estimating the drift component.
Antoine Renard - Automatic proofs in combinatorial game theory
In this talk, we consider Wythoff’s game, which a variation of the game of Nim. Given two piles of tokens, two players are taking turns removing them with the following rules: either you take some tokens from only one of the two piles, or you withdraw the same amount from both. The first player unable to play loses the game.
More precisely, we are interested here in characterising the P-positions of this game, which are exactly the losing positions. In fact, this has already been done (Fraenkel, 1982)… but! Thanks to Walnut, a free software commonly used in combinatorics on words, we can now provide automatic proofs of several results from the literature. Moreover, we are able to state new results and conjectures about generalisations of Wythoff’s game.
This talk will act as a stroll around game theory, first-order logic and non-standard numeration systems. Are you ready to play a bit…?
This talk is based on joint work with Bastien Mignoty, Michel Rigo and Markus Whiteland.
Tim Seuré - Spectral Theory on Cayley Graphs
In this talk, I am going to share a neat connection between algebra and graph theory. I’ll walk you through some of the basics of spectral graph theory and talk about why it’s worth caring about. After that, I will introduce Cayley graphs, and establish a fundamental result concerning spectral theory on Cayley graphs. Specifically, we will express the eigenvectors and eigenvalues of Cayley graphs in terms of linear characters.
Pierre Stas - A Solution to the “Topic Fitting Problem”
The main challenge I faced while preparing this talk was finding a topic suitable for such a general audience. Each time I thought of an idea, I couldn't decide whether it would be appropriate or not. So I tried to devise an algorithm to determine whether a topic would “fit”. That led me down quite the rabbit hole.
In this talk, I’ll share the journey of trying to figure out what I could possibly present to you.
With any luck, the explanation will last just long enough that I won’t actually have to pick a topic.
Francesco Tognetti - Three flavours of sheaves
Any category of sheaves on a site serves a three-fold purpose: To generalise the concept of topological space, to generalise the of universe of sets and to complete a base category maintaining some useful properties.
This presentation is an attempt to present these three –apparently separate- views on the same objects and the interplay between them.
List of speakers:
Alexandre Benoist
Anaïs Meunier
Bruno Dular
Clifford Chan
Francesco Tognetti
Francisco Pina
Leolin Nkuete
Lota Copic
Luís Maia
Simon Lemal
Tim Seuré
Thomas Lamby
Antoine Renard
Pierre Stas
Annika Huch