Transformers excel when dealing with sequential data.
Generalizing transformer models to geometric domains, such as manifolds, we encounter the problem of not having a well-defined global order.
We propose a solution with attention heads following a space-filling curve.
As a first experimental example, we present the Spiroformer, a transformer that follows a polar spiral on the $2$-sphere.

We introduce a new approach to the reconstruction of hidden structures from incomplete data, unifying techniques from geometric integration and topological analysis within the pioneering frameworks of Vaisman and Neifeld. Our method transcends traditional iterative schemes by employing a refined geometric decomposition of configuration spaces into invariant foliations and moment maps, thereby resolving the intrinsic ambiguities of underdetermined inverse problems. By synergistically combining Vaisman’s deep insights into symmetry and Neifeld’s analytic methodologies, we establish a robust, noise-resistant paradigm that not only ensures computational tractability but also fundamentally redefines the landscape of reconstruction in imaging and structural analysis. This framework paves the way for transformative applications across diverse scientific domains, heralding a new era in the synthesis of geometry and topology for inverse problem solving.

Unlike conventional telemetry systems, this solution delivers unparalleled flexibility and scalability, as the number of nodes in the network can be expanded as needed. The system’s physical layer capabilities enable long-range, high-data-rate communication, while the unique network layer algorithm ensures reliable relaying capabilities.

This paper details the practical implementation of the telemetry system within the D328 Deutsche Aircraft flight test campaign, highlighting its advantages over traditional solutions. Key use cases include data acquisition and relay between multiple airborne systems and ground stations, demonstrating the system’s potential to significantly enhance range and reliability in demanding environments. By reducing dependency on direct line-of-sight, this approach paves the way for more robust and efficient telemetry operations in future aerospace applications.

The Distance Geometry Problem (DGP) asks whether a simple weighted undirected graph G can be realized in the Euclidean space so that the distances between embedded vertices correspond to the edge weights. The DGP is a rich and active research field, with many important applications. Several approaches to the DGP are based on the idea of directly placing the vertices of G in space. Our model uses a completely new approach: we focus our attention on the edges, and not on the vertices, and we attempt placing in space the “sticks” that can be associated to each edge of the graph. Sticks have fixed length (hence they always satisfy all distance constraints), and they admit three total degrees of freedom (position of one vertex, plus the stick orientation) in 2D. The automatic satisfaction of all DGP constraints comes at the cost of possibly having several distinct positions associated to the same vertex, potentially a different one for every stick where each vertex is involved. Therefore, we formulate a problem consisting in finding stick configurations where all vertices involved in multiple sticks can find a unique position in space, implying in turn the definition of a valid realization for the original DGP. We initially focus the attention on DGPs where the information on the stick orientations is a priori given, so that to formulate a convex quadratic optimization problem with linear constraints. For the general case, we propose a heuristic which solves, at each iteration, an instance of the quadratic problem.

This paper focuses on the challenge of interactively modeling street networks. In order to provide usable datasets for artificial intelligence applications, it is often necessary to generate random cities with adjustable parameters, such as the spatial extent of the city and its traffic distribution. Several models have been developed for this purpose, including the hyperfractal model introduced by Philippe Jacquet. This model offers a significant advantage as it accounts not only for the fractal geometry of urban structures but also for the statistical distribution of traffic within a city.

In this work, we extend the simple fractal model, which is particularly useful for describing small cities or individual districts, by constructing random cities based on a tiling structure over which hyperfractals are distributed. This approach enables the connection of multiple hyperfractal districts, providing a more comprehensive urban representation.

Furthermore, we demonstrate how this decomposition can be used to segment a city into distinct districts through fractal analysis. Finally, we present tools for the numerical generation of random cities following this model.

Reconstruction problems lie at the very heart of both mathematics and science, posing the enigmatic challenge: How does one resurrect a hidden structure from the shards of incomplete, fragmented, or distorted data? In this paper, we introduce a new approach that harnesses the profound insights of the Vaisman Atiyah–Molino framework. In stark contrast to conventional methods that depend on persistent homology, our approach exploits the concept of the Vaisman centroid—an intrinsic invariant that encapsulates the averaged geometry of a data set—to resolve the inherent ambiguities of inverse problems. In the present paper, we focus on the theory and applications of the Vaisman centroid, offering an innovative perspective for Topological Data Analysis that eschews persistent homology in favor of a unified geometric paradigm. The subsequent paper will extend these ideas to a full reconstruction scheme via the Atiyah–Molino framework. Our method not only provides a robust and computationally tractable framework for the recovery of hidden structures but also opens new avenues for the analysis of high-dimensional and noisy data across the mathematical sciences.