"A fundamental problem of the theory of turbulence is to find a satisfactory mathematical framework linking the Navier-Stokes equations to the statistical theory of Kolmogorov. A central difficulty in this task is the inherent non-uniqueness and pathological behaviour of weak solutionsof the Euler equations, the inviscid limit of the Navier-Stokes equations. This non-uniqueness, rather than an isolated phenomenon, turns out to be directly linked to the celebrated construction of Nash and Kuiper of rough isometric embeddings and, more generally, to Gromov's h-principle in geometry. The central aim of this project is deepen the understanding of this link, with the following goals:I. Scaling Laws. Attack specific conjectures concerning weak solutions of the Euler equations that are motivated by the Kolmogorov theory of homogeneousisotropic turbulence. Most prominently the conjecture of Onsager, which relates the critical regularity requiring energy conservation to the scaling of the energy spectrum in the inertial range.II. Selection Criteria. Study the initial value problem for weak solutions, with the aim of characterizing the set of initial data for which an entropy condition implies uniqueness, and obtaining information on the maximal possible rate of energy decay and identifying selection criteria that single out a physically relevant solution when uniqueness fails.III. General Theory. Identify universal features of the construction, in order to be applicable to a large class of problems. This involves an analysis of the geometry induced by the equations in an appropriate state space, a better understanding of how an iteration scheme using only a finite number of ""cell-problems"" can be developed, and developing versions of convex integration that use higher-dimensional constructions."