This paper explores 1/2 BPS geometries in type IIB string theory and M-theory, focusing on their dual descriptions in field theories as free fermions. These geometries correspond to smooth, horizonless configurations in ten or eleven dimensions, with no singularities. The key idea is that the phase space of free fermions maps to a two-dimensional plane in the geometry, and the topology of the geometry depends on the topology of the droplets (regions occupied by fermions) on this plane. The solutions also describe geometric transitions between branes and fluxes, and provide explicit realizations of these transitions. For plane wave geometries, the problem reduces to solving a Laplace or Toda equation with specific boundary conditions, leading to a large class of explicit solutions. The paper also discusses compactifications of M-theory preserving N=2 superconformal symmetry and smooth geometries corresponding to vacua of the mass-deformed M2 brane theory. Additionally, it presents a smooth 1/2 BPS solution of seven-dimensional gauged supergravity corresponding to a condensate of charged scalars. The solutions are analyzed in various cases, including AdS5×S5, AdS7×S4, AdS4×S7, and M-theory pp-waves. The paper also discusses the topology and charges of the solutions, showing that they can be related to fluxes and that the geometry can transition between branes and fluxes. The solutions are shown to have non-singular geometries, with the area of droplets quantized and related to the number of branes. The paper concludes with a discussion of the relation between these solutions and the DLCQ compactification of the pp-wave, as well as the correspondence between the solutions and Young diagrams in the dual field theory.This paper explores 1/2 BPS geometries in type IIB string theory and M-theory, focusing on their dual descriptions in field theories as free fermions. These geometries correspond to smooth, horizonless configurations in ten or eleven dimensions, with no singularities. The key idea is that the phase space of free fermions maps to a two-dimensional plane in the geometry, and the topology of the geometry depends on the topology of the droplets (regions occupied by fermions) on this plane. The solutions also describe geometric transitions between branes and fluxes, and provide explicit realizations of these transitions. For plane wave geometries, the problem reduces to solving a Laplace or Toda equation with specific boundary conditions, leading to a large class of explicit solutions. The paper also discusses compactifications of M-theory preserving N=2 superconformal symmetry and smooth geometries corresponding to vacua of the mass-deformed M2 brane theory. Additionally, it presents a smooth 1/2 BPS solution of seven-dimensional gauged supergravity corresponding to a condensate of charged scalars. The solutions are analyzed in various cases, including AdS5×S5, AdS7×S4, AdS4×S7, and M-theory pp-waves. The paper also discusses the topology and charges of the solutions, showing that they can be related to fluxes and that the geometry can transition between branes and fluxes. The solutions are shown to have non-singular geometries, with the area of droplets quantized and related to the number of branes. The paper concludes with a discussion of the relation between these solutions and the DLCQ compactification of the pp-wave, as well as the correspondence between the solutions and Young diagrams in the dual field theory.