Termine

Solving physics-based models governed by differential equations is relevant across engineering and the natural sciences, often requiring large computational resources. Model order reduction offers a principled approach to derive low-dimensional, computationally efficient reduced-order models, and is particularly valuable when a full-order model must be solved repeatedly for varying parameter values or when fast evaluations are required. Classical projection-based model order reduction techniques rely on linear trial spaces, typically constructed via methods such as proper orthogonal decomposition. While effective for diffusion-dominated problems, linear subspace approaches suffer from a fundamental expressivity limitation for transport-dominated problems, where the slow decay of the Kolmogorov $n$-width renders linear approximations computationally infeasible. To overcome this limitation, recent research has shifted toward nonlinear MOR, where the reduced approximation is sought on a nonlinear trial manifold. A prominent realization of this idea is the use of convolutional autoencoders to learn nonlinear reduced manifolds directly from snapshot data.

In such autoencoders, convolutional layers are by construction equivariant w.r.t. translations: convolving a shifted input yields the same result as shifting the convolved output. Interpreting translations as the action of the group $G = (\mathbb{R}^2, +)$, a convolutional layer $\Phi$ satisfies $\Phi(\mathscr{L}_y f) = \mathscr{L}_y \Phi(f)$ for all $y \in G$. Group convolutional neural networks extend this principle to broader symmetry groups, such as rotations or reflections, yielding layers that are equivariant w.r.t. these additional transformations. In this work, we propose a group convolutional autoencoder framework for nonlinear model order reduction. By constructing networks where layers are equivariant with respect to a chosen group action, the learned reduced manifold generalizes to transformed, e.g. rotated, variants of the system without retraining or extensive data augmentation. We demonstrate this approach on a 2D linear wave equation parametrized by the wave speed, a problem known to exhibit a slowly decaying Kolmogorov $n$-width. Since the wave equation admits a Hamiltonian formulation, we further incorporate structure-preserving model order reduction by enforcing weak symplecticity of the decoder via a symplectic loss term during training, inducing a symplectic structure on the trial manifold and yielding reduced order models that conserve energy.

I will overview some directions and recent progress in the study of groups definable in NIP structures (a common generalization of stable and o-minimal structures). This includes questions around external definability, connections to tame topological dynamics (e.g.~the revised Newelski's conjecture), and the study of the convolution semigroup of measures.

Over the past decades, infinite-dimensional Riemannian geometry has developed into a vibrant area of research. Interest in the field has been driven by its emergence in a wide range of applications, notably in geometric data science, mathematical shape analysis, and geometric hydrodynamics. Although the fundamental definitions of Riemannian geometry extend almost effortlessly to infinite-dimensional spaces, many classical results from the finite-dimensional theory are known to fail in the infinite setting. In this talk, I will survey several phenomena unique to infinite dimensions and discuss conditions under which certain finite-dimensional properties can be partially recovered, including the non-degeneracy of the geodesic distance and Hopf-Rinow-type results. While the results will be illustrated using simple examples modeled on spaces of sequences, I will also discuss applications to the aforementioned areas of mathematical shape analysis and geometric hydrodynamics.

The study of p-adic deformations of automorphic forms was initiated by Hida in the 1980s, following his discovery of systematic congruences between the Fourier coefficients of modular forms. The eigencurve, introduced by Coleman and Mazur, offers a geometric framework for understanding these congruences, and has since become a central tool in tackling deep number-theoretic conjectures, such as the Birch and Swinnerton-Dyer conjecture. At non-critical classical points of integer weight k>1, the eigencurve is known to be smooth, thanks to the classicality theorems of Hida and Coleman. In contrast, the structure of the eigencurve at weight k=1 is significantly more subtle and intricate. In this talk, after reviewing the seminal work of Bellaïche and Dimitrov in the so-called p-regular case (i.e. when crystalline Frobenius does not act by a scalar), I will present joint work with Adel Betina and Alice Pozzi giving a complete description of the local geometry in the more delicate p-irregular case. Time permitting, I will also discuss ongoing work comparing the completed local rings of the eigencurve at p-irregular weight one points with suitable Galois deformation rings.

Recent years have seen substantial progress in understanding algebraic K-theory of singular varieties. Still, computations of (higher) K-theory of singular varieties are quite rare, especially in dimension >1. In this talk, I will explain some techniques that allow one to perform computations in higher dimensions, with a particular focus on cubic surfaces and threefolds, where complete computations can be obtained in many cases.

Equipments, a special kind of double categories, have shown to be a powerful environment to express formal category theory. We build a model structure on the category of double categories and double functors whose fibrant objects are the equipments, and combine this together with Makkai?s early approach to equivalence invariant statements in higher category theory via FOLDS (First Order Logic with Dependent Sorts) and Henry?s recent connection between model structures and formal languages, to show a result on the equivalence invariance of formal category theory.