Differential forms on Carnot groups

The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights, which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.

The molecular gas content of star-forming galaxies in the ALPINE fields.

In this thesis, I aim to study the evolution with redshift of the gas mass fraction of a sample of 53 sources (from z ∼ 0.5 to z > 5) serendipitously detected in ALMA band 7 as part of the ALMA Large Program to INvestigate C II at Early Times (ALPINE). First, I used SED-fitting software CIGALE, which is able to implement energy balancing between the optical and the far infrared part, to produce a best-fit template of my sources and to have an estimate of some physical properties, such as the star formation rate (SFR), the total infrared luminosity and the total stellar mass. Then, using the tight correlation found by Scoville et al. (2014) between the ISM molecular gas mass and the rest-frame 850 μm luminosity, I used the latter, extrapolating it from the best-fit template using a code that I wrote in Python, as a tracer for the molecular gas. For my sample, I then derived the most important physical properties, such as molecular gas mass, gas mass fractions, specific star formation rate and depletion timescales, which allowed me to better categorize them and find them a place within the evolutionary history of the Universe. I also fitted our sources, via another code I wrote again in Python, with a general modified blackbody (MBB) model taken from the literature (Gilli et al. (2014), D’Amato et al. (2020)) to have a direct method of comparison with similar galaxies. What is evident at the end of the paper is that the methods used to derive the physical quantities of the sources are consistent with each other, and these in turn are in good agreement with what is found in the literature.

Jeans modeling of massive early-type galaxies with observed stellar kinematic profiles up to 2-3 effective radii

A recent integral-field spectroscopic (IFS) survey, the MASSIVE survey (Ma et al. 2014), observed the 116 most massive (MK < −25.3 mag, stellar mass M∗ > 10^11.6 M⊙) early-type galaxies (ETGs) within 108 Mpc, out to radii as large as 40 kpc, that correspond to ∼ 2 − 3 effective radii (Re). One of the major findings of the MASSIVE survey is that the galaxy sample is split nearly equally among three groups showing three different velocity dispersion profiles σ(R) outer of a radius ∼ 5 kpc (falling, flat and rising with radius). The purpose of this thesis is to model the kinematic profiles of six ETGs included in the MASSIVE survey and representative of the three observed σ(R) shapes, with the aim of investigating their dynamical structure. Models for the chosen galaxies are built using the numerical code JASMINE (Posacki, Pellegrini, and Ciotti 2013). The code produces models of axisymmetric galaxies, based on the solution of the Jeans equations for a multicomponent gravitational potential (supermassive black hole, stars and dark matter halo). With the aim of having a good agreement between the kinematics obtained from the Jeans equations, and the observed σ and rotation velocity V of MASSIVE (Veale et al. 2016, 2018), I derived constraints on the dark matter distribution and orbital anisotropy. This work suggests a trend of the dark matter amount and distribution with the shape of the velocity dispersion profiles in the outer regions: the models of galaxies with flat or rising velocity dispersion profiles show higher dark matter fractions fDM both within 1 Re and 5 Re. Orbital anisotropy alone cannot account for the different observed trends of σ(R) and has a minor effect compared to variations of the mass profile. Galaxies with similar stellar mass M∗ that show different velocity dispersion profiles (from falling to rising) are successfully modelled with a variation of the halo mass Mh.