Analyse, Géométrie et Modélisation

In this paper, we study a question of Colliot-Thélène and Iyer concerning the existence of rational sections in families of homogeneous spaces over an abelian variety, after base change by a suitable étale isogeny of the abelian variety. Assuming characteristic zero and that the homogeneous spaces arise from connected reductive groups, the problem is reformulated in terms of torsors under reductive groups over an abelian variety $A$. Building on work of Moonen and Polishchuk, we construct a filtration on the motive of a Jacobian variety to analyze the action of isogenies on unramified cohomology and Witt groups. This approach allows for a positive response to the question for reductive groups whose root data do not contain a factor of type~$E_8$ when $\dim A > 2$ and $\mathrm{cd}(k) \leqslant 1$, and for all reductive groups when $\dim A = 2$ and $k$ is algebraically closed.

In [arXiv:2408.13806], two families of classical and quantum integrable hierarchies associated to arbitrary Cohomological Field Theories (CohFTs) were introduced: the meromorphic differential and twisted double ramification hierarchies. For trivial CohFT, the authors established a connection with the untwisted Double Ramification (DR) hierarchy. In this paper, we extend this study to the Hodge CohFT and prove an analogous correspondence with the untwisted DR hierarchy. This yields non-trivial identities between Hodge integrals over the DR cycle, the twisted DR cycle and the cycle of meromorphic differentials.

A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maximal image whose binding group is an abelian variety. The properties of such abelian reductions are investigated in the Galois-theoretic framework provided by stability theory.Several geometric consequences for the birational geometry of algebraic vector fields of characteristic zero are then deduced. In particular,(1)it is shown that if some cartesian power of an algebraic vector field admits a nontrivial rational first integral then already the second power does,(2)two-dimensional isotrivial algebraic vector fields are classified up to birational equivalence and(3)algebraic vector fields whose finite covers admit no nontrivial factors are studied in arbitrary dimension.Analogues of these results in bimeromorphic geometry are also obtained.

This paper explores the application of highly sensitive passive Electromagnetic Field (EMF) measurement systems to address critical challenges in maritime coastal areas, including environmental impact assessment, infrastructure integrity, and underwater surveillance and detection. We present the development and deployment of three innovative systems: PASSEM (vessel-towed), STATEM (static seafloor), and an ROV-mounted system, capable of measuring EMF variations at very low levels. These systems enable the accurate characterization of EMF signatures from anthropogenic and natural sources, crucial for understanding their effects and origins. Results from the SPECIES project demonstrate the systems' efficacy in assessing the EMFs emitted by subsea power cables, supporting sustainable marine energy development. Corrosion detection experiments highlight the systems' sensitivity to subtle EMF gradient changes, enabling the monitoring of cathodic protection evolution. Furthermore, the detection of a cargo vessel using STATEM data highlights the systems' potential for underwater surveillance and broader detection capabilities relevant to defense applications. This research establishes the potential of passive EMF measurement as a powerful tool for comprehensive marine infrastructure management, contributing to the safety, longevity, and environmental sustainability of maritime coastal areas.

Electromagnetic properties of Earth materials are important geophysical parameters and serve as good proxies for fluid-related processes, as their variations are directly linked to changes in porosity, connectivity, fluid flow, or magnetization of the porous network, as well as to the composition of fluids and the properties of mineral surfaces. In the oceanic crust, electromagnetic imaging provides key insights into the geometry of hydrothermal conduits and the distribution of fracturing and alteration. In sedimentary environments, it can also reveal the geometry of sedimentary bodies, map fluid migration conduits, and detect the presence and distribution of natural gas and gas hydrates within the upper tens to hundreds of meters of the seafloor. Within the Deepsea'Nnovation program (DSN) [1] this project proposes an innovative approach: adapting an ROV-based electromagnetic imaging methodology developed in our laboratory, originally inspired by large-scale controlled-source electrical transmitter/receiver systems. This method has already demonstrated its effectiveness in shallow water environments (less than 100 m depth). For high-resolution electromagnetic imaging in the deep sea, using an ROV is essential, as it allows deployment of an electrical source with sufficient power. The project aims to develop an ROV-mounted electromagnetic source paired with electric potential receivers. The system will be designed to accommodate different receiver configurations—varying in number, type (fixed or mobile), and deployment strategy. It will enable profile acquisition at exploration scale, with a penetration depth of up to a few meters below the seafloor.

Following recent work of T. Alazard and C. Shao on applications of para-differential calculus to smooth conjugacy and stability problems for Hamiltonian systems, we prove finite codimension stability of invariant surfaces (in finite differentiability classes) of flat geodesic flows on translation surfaces. The result is also based on work of the author on the cohomological equation for translation flows.