18–19 November 2026 | BMS/MATH+ Anniversary Conference

In this talk I will tell the story of how together with J. Dolbeault and Michael Loss, and lately also with R. frank, we have been trying to develop methods to study the symmetry properties of optimizers of functional inequalities. I will present a series of examples to make the question clear and show the difficulties encountered during the journey.

Why do materials form stripes, branches, and other intricate patterns? The answer often lies in competing energetic effects: while some favor fine-scale structures, others penalize interfaces or bending.
In this talk, we explore how scaling laws for infimal energies help uncover the mechanisms behind pattern formation. Starting with basic variational models, we develop intuition for how competing energies determine characteristic length scales. We then turn to a two-phase biomembrane model and discuss a scaling-law result related to lipid raft formation in cell membranes, alongside interpolation inequalities which are used to show energy bounds. Finally, we explore the connection between energy minimization and gradient-flow equations, illustrated by numerical experiments for a two-phase Cahn—Hilliard model.

I will present and explain a new strategy for proving geometric inequalities based on potential theory and the divergence theorem. We will explore the method in the example of the classical Willmore inequality from differential geometry. We will also see how it can be generalized to several other geometric inequalities.
The results I will present are based on joint works with Florian Babisch, Albachiara Cogo, Benedito Leandro, Ariadna León Quirós, Benjamin Meco, Anabel Miehe, and João Paulo dos Santos.

This talk considers combinatorial properties of physical networks like water, gas, heat, and electrical networks. One common model is given by potential-based flows, for which, in general, the corresponding flows depend in a nonlinear way on potentials (e.g., pressures) and are unique. The talk will cover two different topics for such models. The first is how the fact that flows in physical networks are acyclic can be used to strengthen mixed-integer optimization formulations and to derive combinatorial models. The second deals with topology optimization, where a potential-based flow network should be designed in a cost-efficient way such that a given demand can be satisfied. The presented mathematical results concern valid inequalities and approximation algorithms.

We study completion of partial matrices with nonnegative entries to matrices of nonnegative rank at most r, where r is some natural number. For nonnegative matrices of rank one
and two, nonnegative rank is equal to rank. This is our main tool for studying nonnegative rank-1 and -2 completions. We show that a partial matrix with nonnegative entries has a nonnegative rank-1 completion if and only if it has a rank-1 completion. This is not true in general when r ≥ 2. For 3×3 matrices, we characterize all the patterns of observed entries when having a rank-2 completion is equivalent to having a nonnegativerank-2 completion. For nonnegative matrices of rank at least three, the nonnegative rank can differ from the usual rank, and therefore, studying the completion problem becomes more challenging. We introduce a geometric characterization for nonnegative rank-r completion employing families of nested polytopes which generalizes the geometric characterization for nonnegative rank and then use this characterization to obtain first results about nonnegative rank-3 completion.
This talk is based on joint work with Lilja Metsälampi.

Neural networks power modern AI systems, and AI is radically transforming our society. What kind of mathematics can we use to improve our theoretical understanding of neural networks? In this talk, I will make the case that one useful angle comes from polyhedral geometry, showing how polyhedral techniques can help us understand the capabilities and limitations of commonly used neural network architectures. Along the way, I’ll also tell tales of how AI has recently solved all my favorite problems, a fate I share with many other mathematicians these days.

Motions of graphs and frameworks under distance constraints have been classically studied in engineering and robotics. We will discuss the notion of rigidity for graphs and describe the Laman number of a minimally rigid graph and how to compute it. Subsequently, we also introduce configuration curves associated to calligraphs and describe an algorithm to compute the genus of these curves. An important theme throughout would be to witness the use of techniques from tropical geometry in proving results about configuration varieties of graphs.

Why can’t we simply walk through walls? One answer lies in Pauli’s exclusion principle, which governs how electrons can occupy quantum states. Formulated by Wolfgang Pauli, this principle earned him the Nobel Prize in Physics in 1945. Yet it also has a beautiful geometric interpretation – one that remains surprisingly underappreciated.
The key is to look at the eigenvalues of matrices describing quantum systems. For certain systems of fermions, these eigenvalues belong to a remarkably simple geometric object: the hypersimplex, obtained by slicing a hypercube with a hyperplane. This geometric viewpoint opens the door to a deeper question: what happens when the system is no longer at zero temperature?
Enter lineup polytopes: a generalization of hypersimplices that brings together ideas from combinatorics, discrete geometry, and quantum physics. Far more than a mathematical reformulation, these polytopes lead to new exclusion principles that refine Pauli’s principle and yield predictions that experimentalists can put to the test.
In this talk, I will trace the history of these ideas, introduce the geometry of lineup polytopes, and highlight some of their mathematical properties and physical implications. No prior knowledge of quantum physics will be assumed.
This talk is based on joint work with physicists Julia Liebert and Christian Schilling, and mathematicians Eva Philippe, Federico Castillo, and Arnau Padrol.

I will explain how questions about the irrationality and transcendence of certain numbers have fascinated mathematicians for over 2,000 years. I will describe some of the key ideas behind irrationality proofs, including Apéry’s celebrated proof of the irrationality of ζ(3) in 1978, and discuss recent developments in AI and their implications for research in this area.

How can we classify geometric objects? And what happens when these objects vary in families or begin to degenerate?
Moduli spaces provide a geometric framework for answering such questions: their points represent geometric objects, such as elliptic curves or higher-dimensional abelian varieties. Surprisingly, these spaces are closely connected to modular forms, highly symmetric functions that also play a central role in number theory.
In this talk, I will introduce these ideas through simple examples and explain how they come together in the moduli space of principally polarized abelian varieties. We will then explore what happens at the boundary of these spaces, where smooth geometric objects degenerate. This will lead us to compactifications and to a glimpse of how tropical geometry, with its combinatorial and piecewise-linear structures, provides new tools for understanding these degenerations.
The talk will be introductory and will not assume prior knowledge of algebraic geometry or modular forms.

A rose by any other name would smell as sweet, but the same cannot be said of mathematical objects: how we represent a number shapes what we can compute and what we can prove. After an illustration with the integers, I will venture into numbers defined by integrals of geometric origin, such as π. The goal is to explain a recent breakthrough showing that core properties of one-dimensional integrals, such as their transcendence, are entirely governed by the geometry of algebraic curves. Representing these integrals through curves, therefore, unlocks new computational abilities.

Unsupervised learning problems are prevalent in modern statistical, machine learning, and computer science literature. This includes a variety of practical situations ranging from clustering nodes on a large graph, or feature vectors in a large dataset, ranking experts/workers in crowd-sourced data, ranking players in a tournament or equivalently sorting objects based on pairwise comparisons. Because of the combinatorial nature of these problems, they akin to solving large non-convex optimization problems under uncertainty. So that a main challenge in this field is to construct algorithms for the task at hand, based on incomplete and noisy data.
In this talk, we focus on understanding a subset of these problems both from an informational – namely, characterizing the fundamental statistical thresholds for optimal estimation – and a computational – namely, also characterizing the fundamental limits of computationally efficient estimation – perspective. A core question for these problems is on whether statistical optimality is compatible with computational efficiency.
To do that, we first consider the simpler sub-problem of sub-matrix detection and estimation, which is useful to apprehend the more complex problems – and we will particularly focus on computational lower bounds through low-degree polynomials.

Stochastic control problems seek to minimize expected costs by influencing the evolution of a random system. When the dynamics or costs depend on the past, the current state alone is generally insufficient for determining an optimal control. Using the full history as the state leads to an infinite-dimensional control problem, and numerical methods therefore require a suitable finite-dimensional approximation. A natural approach is to represent the observed history through an additional state that evolves along the trajectory.
In this talk, we introduce path signatures, the collection of iterated integrals of a path, as a systematic starting point for constructing such representations. The signature itself solves a linear differential equation driven by the observed path; its truncations provide finite-dimensional representations of the history. Through simple examples, we discuss how the choice and dimension of the representation determine what information is retained and how this affects the approximation of the optimal value.