Internships

Numerous models have been proposed to describe the evolution of opinions in a population. These models are built from 2 parts: one describes the evolution of the opinions of two agents when they interact (with the underlying hypothesis that interactions tend to make opinions closer); the second describes the structure of interactions in the population, generally described by a network.

While many studies have focused on the impact of the network structure on how the opinions evolve, and on the possibility to reach a global consensus, few examine dynamical differences between models, for a fixed interaction structure. Studying such differences is however important to compare models to experiments, in order to discriminate between models and validate models.

The goal of the internship will thus be to define measures to describe the patterns of opinion evolution in a population, in a similar spirit as [3] in the case of spreading, and to investigate several models to show the discriminative power of these measures.

Numerous models have been proposed to describe the evolution of opinions in a population. These models are built from 2 parts: one describes the evolution of the opinions of two agents when they interact (with the underlying hypothesis that interactions tend to make opinions closer); the second describes the structure of interactions in the population, generally described by a network. While most studies consider such structures as fixed, they actually vary in time and need to be described through the framework of temporal networks.

Few studies have considered the impact of the temporality of interactions on the dynamics of opinions. The project will consist in the adaptation and simulation of models of opinion dynamics on temporal networks, given either by synthetic models or by empirical data sets (e.g. from SocioPatterns), as well as on randomized structures to examine the roles of the various temporal properties of the networks. The models will be further refined to take into account the diversity of social influence weights, and also to include adaptivity (feedback of the opinion differences on the network structure).

In the high-frequency limit, underwater acoustic waves admit an elegant Hamiltonian formulation. The resulting Hamiltonian is analogous to that of classical mechanics. The acoustic rays (or rays, for short) correspond to the trajectories governed by the canonical equations. This connection allows to investigate properties of rays using the geometrical and analytical tools of Hamiltonian theory. When the ocean sound-speed profile is allowed to vary with range as well as depth, the system becomes generally nonintegrable. In this regime, rays may exhibit chaotic behavior characterized by sensitive dependence on initial conditions – a phenomenon known as ray chaos, recognised by the community in the early 1990s. See [1-2] for an introduction to the topic.

The objectives of this internship are twofold:

1. Implementation and numerical treatment of the model. The first objective is to become familiar with the Hamiltonian formulation of underwater acoustic ray propagation. The student will develop its own symbolic and numerical tools to study the corresponding dynamics. This will involve to implement and validate numerical schemes to integrate numerically ray trajectories. This first stage will provide the foundations for the subsequent investigation of the system’s dynamical properties.

2. Billiard maps and phase-space exploration. The second and main objective is to investigate the dynamics of rays through a collision-based Poincaré map, analogous to those used in classical billiard systems. The construction allows to reduce the continuous dynamical system to a discrete dynamical system of lower dimension. Instead of sampling trajectories at fixed range intervals (Poincaré range map, see [1,2]), the collision map is based on successive encounters with the boundaries of the waveguide, providing a complementary description of the dynamics. The student will construct and analyse the corresponding map and explore the structure of its phase space for different values of the model parameters (such as the strength and spatial period of the sound-speed perturbation). The goal is to establish a zoology of dynamical regimes, identifying and characterising the main structures that emerge, such as regular regions, chaotic seas, and the possible presence of mixed phase space and sticky trajectories. If time permits, the study will be extended to waveguides with non-flat boundaries(non-flat sea surface or sea bottom), in order to explore how more complex geometries modify the phase-space structure and the resulting dynamics.

Throughout this project, the student will learn techniques of nonlinear dynamics and more particularly Hamiltonian chaos (phase space visualization and characterization of chaos) in a concrete physical setting. It will follow a set of seminal papers already identified. The internship contains a substantial part of numerical simulations.

References:
[1] I. P. Smirnov, A. L. Virovlyansky, and G. M. Zaslavsky, Ray chaos, travel time modulation, and sensitivity to the initial conditions, The Journal of the Acoustical Society of America 117, 1595 (2005)
[2] I. P. Smirnov, A. L. Virovlyansky, and G. M. Zaslavsky, Theory and applications of ray chaos to underwater acoustics, Physical Review E, Volume 64, 036221

Chaotic dynamics, characterized by sensitivity to initial conditions, are ubiquitous in nature and arise across a wide range of scales and physical systems, from the Solar System to fluid dynamics, plasma physics, or the motion of artificial satellites orbiting the Earth. The existence of chaos has strong consequences for the predictability of future states of the dynamical system. Despite being deterministic, small uncertainties in the initial conditions can grow rapidly, limiting our ability to make reliable predictions in the long-term. This fundamental property has motivated the development of several computational methods for detecting and characterizing chaotic dynamics over the past several decades.

The LIGO–Virgo–KAGRA collaboration is continuously expanding our knowledge of the population of merging black holes and neutron stars through gravitational-wave observations. The increasing number of detections offers new opportunities to investigate the astrophysical origin of compact binaries and their use as probes of cosmology.

This internship will focus on the analysis of the latest gravitational-wave transient catalog, including the data release expected in December 2026, using advanced population inference methods recently developed within our research group.

The project will address two main scientific questions: Is there evidence that the binary black hole mass distribution evolves with redshift? Can we identify characteristic mass scales in the binary mass distribution that could be used to constrain cosmological parameters, such as the Hubble constant?

The student will become familiar with gravitational-wave population inference and Bayesian statistical methods, and will apply and extend existing numerical tools developed by the group. The analysis will build on recent studies of redshift-dependent mass distributions and flexible models of the joint binary mass spectrum.

The internship will involve numerical data analysis, scientific programming in Python, and interpretation of astrophysical and cosmological results. Depending on progress, the student may contribute to ongoing research and scientific publications within the group.

Gravitational-wave observations provide a new way to study the expansion of the Universe. In particular, characteristic features in the mass distribution of merging compact binaries can act as statistical cosmological rulers, allowing us to extract information about the redshift of gravitational-wave sources and constrain cosmological parameters.

Recent population studies suggest the presence of previously unresolved structures in the joint mass distribution of binary black holes, including a possible excess involving black holes with masses around 20 solar masses. Understanding the origin and statistical significance of such features, and their potential use for cosmology, is an important open problem.

The goal of this internship is to develop and test an analytical population model capable of describing these emerging mass features, and to compare its performance with more flexible, data-driven approaches.

The student will investigate the statistical significance of the proposed structures using gravitational-wave observations and assess their impact on measurements of the Hubble constant. The project will combine analytical modelling, Bayesian inference, and numerical data analysis, building on an extension of the software developed within the research group.

The European gravitational-wave community is preparing for the next generation of ground-based observatories, with the Einstein Telescope (ET) as a flagship project. With a sensitivity significantly exceeding that of current detectors, ET is expected to observe a much larger population of compact binary mergers, opening new opportunities for studying black hole astrophysics and precision cosmology.

Accurately forecasting the scientific capabilities of these future observatories requires realistic simulations of gravitational-wave detections, including observational uncertainties and detector selection effects. Developing fast and reliable simulation tools is therefore an important challenge for the design and scientific preparation of next-generation experiments.

The goal of this internship is to develop and calibrate a computationally efficient gravitational-wave catalog simulator for the Einstein Telescope, capable of reproducing realistic measurement uncertainties and selection effects for different astrophysical populations.

The student will build on existing approaches for generating mock gravitational-wave catalogs and calibrate the simulator using more detailed ET-specific simulations. The resulting framework will be validated against reference simulations and used to explore the potential of Einstein Telescope observations to constrain compact binary populations and cosmological parameters.