Observing planet formation

Planets are thought to form in the circumstellar disks orbiting young stars in formation. According to the core-accretion model, a candidate scenario for Earth-like planets, the interstellar sub-$\mu$m-sized dust particles grow thanks to collisions to mm/cm size and then form km-sized planetesimals via dynamical encounters. Eventually, the rocky planetary cores accrete gas and, depending on the total gas mass attained, a terrestrial planet or a gas giant forms. Modern sub-mm/mm/radio interferometers such as ALMA and VLA detect the thermal emission of dust grains and provide us with an unprecedented sharp view of protoplanetary disks at the spatial scales where planet formation occurs. In recent years, evidence of grain growth in disks has been obtained by extensive sub-mm/mm photometric studies, but so far they only provided disk-averaged estimates of the dust properties. Moreover, the derivation of dust properties from the observed spectral index was done under reasonable - but simplifying - assumptions rather than with a proper modeling of the disk emission. The thesis presents an analysis method that enables - for the first time - the disk structure and the dust properties to be constrained simultaneously by fitting multi-wavelength observations with a self-consistent physical model. The thesis presents also an accelerated version of the computer code that uses modern graphics cards and provides the computational breakthrough needed to exploit the new wealth of information now available. Applying the multi-wavelength analysis to observations of three disks in the Taurus and Ophiuchus star-forming regions, a key result is a radial gradient in the grain-size distribution, with large grains of up to $1\,\mathrm{cm}$ size confined to the inner disk and smaller grains of size $\ll 1\,\mathrm{mm}$ populating the whole disk. Similar results hold for another disk, HD~163296, where in addition the grain size radial profile supports the scenario of enhanced grain growth at the snowline location of the second most abundant volatile in disks, CO. The tool developed in the thesis is also designed to accelerate the analysis of high-resolution observations for demographic studies. By applying the analysis tool to an ALMA disk survey in the Lupus star-forming region, the physical structure of more than 20 disks is obtained, in particular the disks's size and dust mass among other physical parameters. To date, this is the largest sample of disks of the same star-forming region fitted homogeneously with a self-consistent model. Remarkably, the sample is complete in the mass range of 0.7$M_\odot$ to one $M_\odot$. The results are compatible with previous studies based on simpler analyses but also highlight a consistent difference in the disks's luminosity-size correlation between the older ($\sim3\,\mathrm{Myr}$) Lupus and the younger ($\sim1-2\,\mathrm{Myr}$ old) Taurus-Auriga region. The application of the analysis developed in this thesis to multi-wavelength observations of large samples of disks with ALMA will allow us to spatially resolve the early growth of solids in numerous protoplanetary disks, and therefore to provide measurements that will be crucial to inform, test, and refine theoretical models of planet formation.