Cosmological Inflation is one of the leading paradigms for explaining the
physical conditions that prevailed in the early Universe. It consists in a
phase of very high energy accelerated expansion that solves the hot
big bang model problems. When combined with Quantum Mechanics,
it also provides a causal mechanism for generating cosmological fluctuations
on large scales.

In this talk I will explain how the most up-to-date cosmological observations,
incorporated in the Bayesian model comparison approach, can be used to
extract the favoured models among hundreds of proposed scenarios.
I will insist on the role played by reheating, which connects inflation to
the subsequent radiation era, and during which ordinary matter as we see it
today is produced. I will also discuss some extensions to the
minimal single-field setups, where extra scalar fields are added and play
a role both during inflation and reheating.

Approaching a critical point, the divergence of the correlation length of fluctuations can be cut-off by a confinement of the system. This truncation fosters finite size effects with universal features, such as the critical Casimir force which has particularly drawn attention in the past decades. I will describe how the critical Casimir force arises in confined critical systems and how its universality has led to a clear separation between numerical/theoretical approaches and experimental approaches. On the one hand, theoretical and numerical works on the subject have almost systematically been performed in magnetic model systems, such as the Ising or XY models. On the other hand, experimental approaches were all realized in fluid systems, such as binary mixtures or helium IV close to the superfluid transition. I will then present two very different numerical approaches, both intended to bridge this gap. Firstly, I will present a numerical test of an experimental protocol for measuring the critical Casimir force in a magnetic layer and, secondly, a numerical approach in a Lennard-Jones binary mixture. These approaches could open the way to a variety of experimental approaches in magnetic thin films and to studying fluctuations or out-of-equilibrium aspects of the Casimir force in simulations of fluids.

A large class of materials, ranging from glasses to soft matter and biological systems, undergo structural arrest and physical aging. Following different experimental routes, like decreasing the temperature in a glass former liquid or increasing the packing fraction in a colloidal suspension, these materials can be driven in an out-of-equilibrium state, where their properties depend on the previous history of the system and spontaneously evolve with time. This process leads to an extremely complex dynamical scenario, characterized by a dramatic increase of the structural relaxation time and the emerging of multiple relaxation processes. The microscopic slow dynamics of out-of-equilibrium materials can be followed thanks to the unique properties of X-ray Photon Correlation Spectroscopy which probes the dynamics at the nanometric and atomic scale. Here I will present a detailed investigation of the relaxation processes occurring in metallic glasses during physical aging. Although the apparent differences, metallic glasses display an anomalous dynamics at the microscopic level which remembers that of complex soft materials, such as gels and concentrated colloidal suspensions. By combining dynamical and structural studies we have been able to quantitatively link for the first time this anomalous atomic motion to microscopic structural mechanisms usually observed in diffraction studies, providing a broader unique view of the glassy state.

I will present a one-parameter generalization of the symmetric simple exclusion process on a one dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rate to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. Then, we deduce the spectral gap in the thermodynamical limit. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multi-points correlation functions as well as the mean value of the lattice current and of the creation-annihilation current. Finally the variance of the lattice current is exactly computed for a finite size system. In the thermodynamical limit, it matches perfectly the value obtained from the associated macroscopic fluctuation theory. It provides a confirmation of the macroscopic fluctuation theory for dissipative system from a microscopic point of view.

Studying the underlying theory of composite Higgs models can reveal many features of their phenomenology.
In the context of fermionic gauge theories, which are naturally free of the hierarchy problem, I will discuss
how the Higgs may be accompanied by a Dark Matter candidate, and how di-boson signatures can be a
smoking gun at the LHC. The ATLAS/CMS di-photon resonance, if confirmed, would be a strong discriminator,
leading to neat predictions with the aid of Lattice calculations.

Acoustic or blast wave absorption by liquid foams is especially efficient and bubble size or liquid fraction optimization is an important challenge in this context. A resonant behaviour of foams has recently been observed in linear acoustics, but the main local dissipative process is still unknown. We address this issue by measuring the pressure attenuation of a weak shock wave in a liquid foam, in the linear and weakly non-linear regimes. I will show that the pressure attenuation depends on the nature of the gas composing the foam and that a maximum of attenuation is obtained for a given bubble size. Those results are interpreted with a thermal dissipation model, featuring an optimal attenuating bubble size close to the thermal boundary layer. If I have time, I will also present some results on the foam rupture after the impact of the blast wave.

Since Witten and Kontsevich’s work, regarding the description of two-dimensional topological gravity in terms of the $\tau$-function of a solution to the KdV hierarchy, the connection between Integrable Systems and 2D Topological Field Theory is known to be a deep one.
We will introduce the fundamentals of Symplectic Field Theory (SFT), a branch of symplectic topology that studies holomorphic curves with boundaries in symplectic manifolds. The potential counting these curves is interpreted as an Hamiltonian corresponding to a Quantum Integrable System. In particular, we will consider the commuting Hamiltonians obtained from the quantisation of the dispersionless KdV hierarchy, that arise naturally in the context of SFT. A complete set of common eigenvectors of these operators is found in terms of Schur polynomials and used to compute the SFT-potential of a disk.
We will finish by providing some remarks on some more recent developments in this area, namely the case of the Quantised Toda Lattice and Double Ramification Hierarchy.