The key mathematical model describing fluid flows is the Navier-Stokes system of nonlinear partial differential equations (PDEs) representing the conservation of mass and momentum in a viscous incompressible fluid. Its complexity for fully three dimensional flows has for a long time defied mathematicians, computational scientists and physicists. However, the last decade or so has witnessed a number of research efforts which began to shed new light on some unresolved open problem of theoretical fluid mechanics. Some examples are:

• the discovery of "edge states'' demarcating laminar and turbulent states in certain shear flows, such as pipe and channel flow, followed by their experimental observation,
• the construction of "minimal seeds'', i.e. finite-sized perturbation that lead to subcritical transitions in flows with edge states,
• the computation of an increasing variety of invariant solutions to the Navier-Stokes equations such as periodic orbits, invariant tori and connecting orbits, which together can form the "skeleton'' of turbulence
• the identification of localized vortex states in extremely violent events as possible precursors for singularity formation.

A common feature in all of these investigations is that, in addition to an accurate solution of the underlying PDEs, they rely on an innovative use of computational techniques allowing one to identify the initial data with some very special properties. For the different problems mentioned above, these correspond to initial data lying on the manifolds separating laminar states from turbulent ones (the so-called ``edge states''), states leading to unstable spatio-temporally complex periodic motions, or the initial data generating the most singular flow evolution (as quantified by the deterioration of certain measures of regularity). The computational approaches required to robustly identify such special initial data typically involve solution of constrained variational optimization problems or fixed-points problems for the underlying discretized PDEs. The structure of these problems is more often than not quite complicated and may suffer from stiffness, nonsmoothness as well as ill-posedness.

While certain progress has already been achieved along these lines, much remains to be done. The scientific community working on these problems, which is rather decentralized, is at a point where methodological advances are needed to warrant new breakthroughs. In the proposed thematic program we will therefore focus on the following open and emerging problems:

• reliable solution of large-scale, possibly nonsmooth, variational optimization and fixed-point problems for PDE systems arising in fluid mechanics applications,
• development of novel modeling and simulation techniques to describe the organization of fluid motion at high Reynolds numbers and on large domains
• discretization of PDE problems in the presence of potential singularities,
• optimal ways to computationally search for extreme behaviour (variational approaches, "instanton formulations'', Monte Carlo, approaches based on complexified equations, etc.),
• approaches to studying the existence and regularity of solutions to flow problems using computer-assisted proofs and rigorous computations.

One of the main objectives of the proposed thematic program will be to make some headway with the open problems identified above. There are natural connections between some of these issues and the open problems relevant in the context of the second and third main theme of the program offering valuable opportunities for exchange of ideas.