Abstracts

Alexandre Benoist

Counting points on elliptic curves over finite fields

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.

Quirijn Boeren

Cusps in the AdS/CFT correspondence

The AdS/CFT correspondence is a powerful tool in theoretical physics, relating string theories on hyperbolic (Anti-de Sitter) manifolds to a conformal field theory on a boundary manifold. It provides some of the most promising models of quantum gravity. As often in theoretical physics the theory struggles with divergences. I will walk you through one such divergence, caused by a construction from hyperbolic geometry: a manifold with cusp—a puncture at infinite distance—can generate infinite summands to the relation, producing a divergence.

Javier Fernández

Grassmannians and representations of Lie groups

Grassmannians are objects endowed with rich geometrical structures that have been studied in algebraic geometry since the 19th century. A useful way to understand these spaces is through the seemingly unrelated theory of representations of Lie groups. The goal of this talk is to present a brief overview of the interplay between these fields and to motivate how computers are useful in answering many related questions.

Anne Fisch

Modular forms in characteristic p: same objects, new behaviours

What happens to modular forms when we reduce them mod p? In this talk, I will give an informal introduction to modular forms in characteristic p and explain why this perspective is useful. I will describe how classical structures, such as Hecke operators and newforms, behave in this setting. A key phenomenon is that cusp forms mod p can exhibit Eisenstein-like behaviour, something that does not occur in characteristic zero.

Cara Hobohm

Title to be announced

Abstract to be announced.

Carl-Fredrik Lidgren

Ambidexterity in higher localization sequences

Localization sequences of Abelian categories, triangulated categories, or even stable (∞,1)-categories are important tools in homological algebra and the category-theoretic approach to geometry. Given a suitable notion of "stable (∞,n)-category", one can categorify the notion of a localization sequence to the setting of higher category theory. Meanwhile, higher category theory exhibits a funny property called ambidexterity, where, surprisingly often, left adjoints agree with right adjoints. As it turns out, this has a curious consequence for higher localization sequences.

Sahand Mahmoudian

From the Einstein universe to contact geometry in dimension three

This talk explores the interplay between three-dimensional conformal Lorentzian geometry and contact geometry. The starting point is the Einstein universe, the conformal compactification of Minkowski space, which has the conformal group O(2,3). To any conformal Lorentzian manifold, one associates its photon space — the space of unparametrized null geodesics.

For the Einstein universe in dimension three, the photon space is itself a contact three-manifold. Via the exceptional isomorphism so(2,3) ≅ sp(4, ℝ), it is identified with ℝP³ carrying its standard contact structure, on which PSp(4, ℝ) acts by contact symmetries. This places us within a natural family of (PSp(4, ℝ), ℝP³) geometries and motivates the question that will guide the rest of the talk: "What do three-dimensional contact geometries actually look like?"

Francesco Tognetti

Modal definability for polyhedra

Modal logic is an extension of propositional (quantifier-free) logic that admits "local" descriptions, allowing for a rich and interesting model theory. Polyhedra form a sound and complete semantics for a very common modal logic, and many "geometric" classes share the same logical properties. My joint work with Doctor David Gabelaia (Razmadze mathematical institute) unveils what constructions lead to classes of compact polyhedra that are definable in our semantics.

Pieter Belmans

Geometry, representations and algebra: classification and experiments

Replacing one word in the GRACE acronym, we end up with a list of my favorite things. I will give a very biased survey of some of the things I find exciting about the experimental nature of fundamental mathematics and what kind of beautiful classifications we can obtain.

Vincent Despré

Title to be announced

Abstract to be announced.

Tim Gehrunger

Benchmarking Agentic AI on Research-Level Mathematics

Can commercially available LLMs solve research-level mathematics problems? Can mathematicians use these models to support their own research? The current answer to both questions seems to be a cautious “yes,” which grows less cautious over time. In this talk, I will give an overview of recent benchmarking efforts for mathematical reasoning, with a special focus on agentic systems. I will also highlight best practices, limitations, and open questions surrounding the use of AI in mathematical research.

Aurel Page

Class field theory in Pari/GP

Class field theory, which gives a description of all abelian extensions of a number field, is a cornerstone of algebraic number theory, with applications in arithmetic geometry. In this talk, I will state the main theorems of class field theory in an elementary way, describe algorithmic methods to make it effective, and present their use in Pari/GP.

Florent Schaffhauser (Heidelberg University)

Interactive theorem proving in Lean — tutorial (Lean 1–8)

The goal of this tutorial is to provide an introduction to interactive theorem proving, using the Lean proof assistant. We start with a presentation of the basics of dependent type theory, before moving on to projects involving formal proofs at the level of a Bachelor program in mathematics (sequences of real numbers, divisibility theory, etc.). We assume no familiarity with dependent types at the outset: our goal is to introduce them from scratch and show how they can be used to formalize existential and universal statements in mathematics. This gives us an opportunity to discuss intuitionistic logic and briefly outline the differences with classical logic. We also emphasize the importance of treating propositions as types and proofs as programs, and we illustrate this notion with examples such as Euclidean division or the notion of prime element in a commutative ring without zero divisors. No prior programming skills are required to follow this tutorial. Practice files will be provided and no installation of Lean is necessary.

Marco Usula

Minimal surfaces, knots, and neural networks

In this talk, I will discuss a new computational approach to finding minimal surfaces in hyperbolic space using Physics-Informed Neural Networks (PINNs). By designing an ansatz that strictly enforces asymptotic boundary conditions, our algorithm finds explicit numerical minimal discs in Hⁿ. I will then explore applications of this method to existing conjectures involving knots bounding minimal surfaces in H⁴.

Sara Veneziale

AI for mathematics: opportunities and limitations

In recent years, there has been an explosion of applications of AI tools to advance research in pure mathematics, to conjecture new theorems, accelerate computations, and disprove conjectures. In this talk we give some examples of successful applications of AI to mathematics, trying to highlight the major points of friction: data representations, data availability, and algorithm choices.