Plenary Talks

Transformer architectures have demonstrated remarkable empirical success, yet the mathematical principles underlying their ability to extract statistical structure from data remain only partially understood. In this talk, I will present a measure-based framework for the analysis of softmax attention, in which attention layers are viewed as operators acting on probability distributions rather than finite collections of tokens.

This perspective, combined with concentration arguments, allows one to pass from finite prompts to a population description and to derive a precise characterization of the training dynamics of softmax attention. In particular, the resulting infinite-prompt limit provides a tractable framework in which the evolution of the parameters can be analyzed rigorously.

I will illustrate the versatility of this approach on both unsupervised and supervised learning tasks. Overall, these results suggest that attention can be understood as a statistical operator that progressively adapts its parameters to the structure of the underlying data distribution, offering a unified mathematical framework for analyzing representation learning and attention-based inference.

The coupling of the Virtual Element Method (VEM) with the Boundary Element Method (BEM) has recently emerged as a novel and promising framework for the numerical solution of wave propagation problems in unbounded domains. By combining the geometric flexibility of VEM for the discretization of complex heterogeneous interiors with the natural treatment of radiation conditions provided by BEM, this approach offers an effective and versatile alternative to classical FEM-BEM formulations.

The talk will provide an overview of the recent developments in VEM-BEM coupling for time-dependent and time-harmonic wave propagation problems. The presentation will discuss formulations in both two and three spatial dimensions, highlighting the extension to three-dimensional problems as one of the most recent advances in the field. Particular attention will be devoted to the mathematical formulation of the coupled methods, their numerical implementation, and the challenges arising from the interaction between virtual element discretizations and boundary integral operators.

The lecture will also compare different boundary element discretization strategies, focusing on both Galerkin and collocation BEM formulations. Their respective advantages, computational properties, and performance within the coupled VEM-BEM framework will be illustrated through representative numerical examples. The presentation aims to provide a unified perspective on this research area, emphasizing recent theoretical and computational results while outlining promising directions for future developments in the numerical simulation of wave propagation in unbounded media.

Learning-based methods can reveal subtle structures in ciphers, spanning differential distinguishers, key recovery, and explicit classical attacks. Reversing the perspective, cryptanalytic techniques can reconstruct complex models from their observable behaviour. Through these two case studies, the talk explores how targeted perturbations can reveal hidden mathematical structure from input-output behavior.

We present a framework for solving inverse problems in solar physics using adaptive kernel-based methods. This approach combines Variably Scaled Kernels (VSKs) with greedy sampling strategies, enabling efficient reconstruction of solar phenomena from sparse, noisy data. VSKs adapt the shape of basis functions using spatially varying scale functions, enhancing resolution in areas with rapid variations while simplifying smoother regions. The greedy algorithm features an automatic stopping criterion based on Stein's Unbiased Risk Estimate (SURE), balancing approximation accuracy with model complexity without needing the true solution. Theoretical analysis confirms convergence rates and error bounds for the greedy VSK approximation. Numerical tests on STIX flare data show that this method achieves comparable or superior accuracy to leading approaches while reducing computational costs.

Many conservation and balance laws admit families of moving steady states that are crucial to preserve at the numerical level. While several techniques have been developed to achieve this in one dimension, few are available for multidimensional problems, and those often apply only to linear cases. The so-called Global Flux (GF) approach [Chertock, et al. 2018] integrates multiple terms into a single physical flux, yielding a unified differential operator acting on a more complex flux function. We extend this to multiple dimensions [Barsukow, et al. 2025, Barsukow, et al. 2026] both in finite element and finite volume formulations, in order to discretely preserve truly multi-dimensional equilibria as divergence free solutions. While natively defined on Cartesian grids, we also present a smooth extension to curvilinear domains using a ghost-point boundary extrapolation method.

We propose an approach based on the Hodge Laplacian for three different problems and settings. 

First, we present a cross-order Laplacian renormalization group (X-LRG) scheme for arbitrary higher-order networks. Our approach uses a diffusion process to group nodes or simplices, allowing information to flow between nodes and between simplices (i.e., higher-order interactions). 

Secondly, we discuss simplicial Kuramoto models, which have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than on nodes. We present a unified framework for describing variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. 

Lastly, we consider associated games in cooperative game theory, which enable meaningful characterizations of solution concepts. We view associated games through the lens of game maps and graph Laplacian, thus defining the novel Hodge Generalized Value (HGV). We characterize HGV via an axiomatic approach as a generalized value.