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Vici grant for Jan Bouwe van den Berg

VU mathematics professor Jan Bouwe van den Berg has been awarded an NWO Vici grant of 1.5 million Euro over the next 5 years for his research project "Connecting Orbits in Nonlinear Systems".

Patterns are all around us: our fingerprints, the red spots typical for measles, the stripes on zebras, and the convection rolls in the atmosphere and oceans which shape our climate. Much is known about how such patterns start to develop from a homogeneous state, but we are still unable to predict or analyze the features of fully developed patterns. These nonlinear problems require the integration of computations with topology.

Numerical simulation produces clear pictures of the dynamics, but the information it provides is local and non-rigorous, since in scientific computing one selects an approximate version of the problem, and computes on this prototype. The novelty of the proposed research lies in developing topological tools that unequivocally link the outcome of such calculations with the true problem. The goal is to create topologically validated computational machinery for finding the pivotal objects of interest in large amplitude pattern formation: the paths along which dynamical systems change from one state into another.

These connections play an organizing role. First, connecting orbits describe localized patterns such as pulses and boundary layers. Second, they act as building blocks from which more complicated, sometimes chaotic, patterns can be constructed. Third, the transitions between different patterns, and their emergence from a homogeneous background, are connections between structured and trivial equilibrium states.

A combination of computational methods and topological techniques can provide results that are both sufficiently detailed and robust, validating predictions beyond the scope of simulations. The outcomes are then fully justified, which is vital for our aim of making rock-solid predictions about concrete models.