Stars are formed in cold clouds of molecular gas and dust. The primary question in star formation is: why do such clouds fragment into such smaller pieces? It appears that the answer to this question is turbulence; density and velocity fluctuations develop which then can collapse away much faster than the entire cloud due to their local strong self-gravity.
Extinction map of the Orion region showing that the molecular clouds are produced by gas swept-up in bubbles, mostly driven by supernovae. From Rowles & Froebrich (2009).
Left: Schematic illustration of the generation of turbulence via perturbation of a shock interface as gas is being swept up (see text). Right: two-dimensional simulation of colliding (atomic) flows with an initially perturbed interface, showing density fluctuations. Instabilities plus cooling develop small-scale structure which can then become gravitationally unstable. See also Figure 14). Courtesy Fabian Heitsch
Where does this turbulence come from? In a series of papers, Fabian Heitsch (now at the University of North Carolina), along with other collaborators and myself, showed that the process of forming clouds, which generally involves supersonic flows, creates shocks which in turn produce density fluctuations, which can then collapse further under gravity to form stars.
An implication of this picture is that all star formation is “triggered” by flows in the interstellar medium. These flows are mostly supernova-driven, but massive star winds on small scales and spiral density waves on large scales can also contribute.
Left: The Orion Nebula cluster as seen at 4.5 (blue), 5.8 (green), and 24 microns (red) with the Spitzer Space Telescope. The dust emission highlights small bubbles driven by massive star winds. Right: Green dots show the positions of young stars and the blue dots the positions of protostars, superposed on the dust emission. From Megeath+ 2012.
As shown in the above figure, a typical result of gravitational collapse is the formation of dense filaments which fragment into pre-stellar cores and then collapse into stars. What explains this evolution? In a series of papers, graduate student Aleksandra Kuznetsova and collaborators investigated simulations in which a sub-virial cloud of gas, seeded with turbulent fluctuations as expected, collapses purely under its self-gravity. Starting with a cloud that is not spherical but elliptical imposes a geometry on the cloud which naturally results in the formation of filamentary structure, as shown in the figure below.
Time sequence of the collapse of a molecular cloud seeded with density fluctuations. The over-dense regions collapse first and form stars (sink particles in the simulation, shown as green circles). Clustering naturally occurs due to gravitational concentration. From Kuznetsova+ 2015.
Clustering naturally occurs due to the tendency of gravity to cause local runaway collapse. In the above simulation the densest regions form sink particles which are created to prevent the simulation from becoming overly slow – these are essentially unresolved stars (or binary/multiple star systems).
The basic theory of gravitational accretion – Bondi-Hoyle-Littleton – predicts that the accretion rate onto a central mass is
(1) 
where ρ is the ambient density and v is a relative velocity. Zinnecker (1980) showed that, starting with a small dispersion of initial masses, if ρ v-3 is constant, an asymptotic distribution of masses with dN/dlogM ~ M-1 is the result. This is intriguing because the cluster mass function is M-1, and the upper stellar mass function is M-1.35.
Left: evolution of group (cluster) mass function in a flattened system of particles. The dashed line has a slope of -1. Right: the "mass accretion rates" (adding particles) as a function of time for the corresponding times in the left panel. The dashed line has a slope of 2. From Kuznetsova+17.
The problem with applying the BHL formula is that, in realistic simulations neither the density nor the velocity are constant. Nevertheless, our previous simulations of sink formation in a turbulent initial medium also showed development of an M^-1 mass function (Ballesteros-Paredes+15). To explore this problem further, we considered a pure N-body calculation of cluster accretion to eliminate issues of gas dynamics (essentially, this is equivalent to a dark matter simulation without expansion of space). We (Kuznetsova+17) were able to reproduce the sink formation results; even though the mass addition rates do not follow the M2 law precisely (right panel of above figure), the resulting mass function asymptotes to the M-1 slope. Our results suggest that the star cluster mass function is a natural consequence of gravitational accretion- another indication that gravity dominates cluster formation – and suggests that the stellar upper mass function has a similar origin.