Recent advances in experimental imaging, computational methods, and dynamical systems theory reveal that the unstable recurrent coherent structures observed in turbulent flows result from close passes to unstable invariant solutions of Navier-Stokes equations.

In presence of continuous translational and rotational symmetries these solutions form 2- and 3-dimensional families of equivalent traveling waves and relative periodic orbits, interconnected by a web of still higher-dimensional stable/unstable manifolds, all embedded in the PDE infinite-dimensional state spaces. In order to chart out this state space, one has to quotient the symmetry, i.e. replace the dynamics by an equivalent, symmetry reduced flow, in which each family of symmetry-related states is replaced by a single representative.

Physical systems often come equipped with symmetries, such as the reflection and rotation symmetries of various potentials, and symmetries simplify the dynamics in rather beautiful ways. The problem is much more general than its fluid dynamics version. If you care about atomic, nuclear or celestial physics, of general relativity or quantum field theory ("gauge fixing") you might be interested and perhaps help me with a better approach.

Happy news: The problem has been solved often, first by Hilbert and Weyl (1921), then by Cartan (1924), then by [...], then by this week's arXiv [...] submission. Turns out, it's not as easy as it looks.

Still, every unhappy family is unhappy in its own way: The Hilbert's solution (invariant polynomial bases) is computationally unfeasible for dimensions larger than ten. In Cartan's "method of moving frames" the state space is sliced in such a way that each group orbit of
symmetry-equivalent points is represented by a single point, but slices are local, and several slices might be needed to capture the flow globally. Work, but one is rewarded by much deeper insights into turbulent dynamics.

个人简介：Professor Cvitanovic was born in Yugoslavia and received his PhD in Cornell University where he did the sixth order corrections to compute the anomaly of the electron magnetic moment. He is highly regarded for his work in nonlinear dynamics, particularly his contributions to periodic orbit theory. Perhaps his best-known work is his introduction of cycle expansions—that is, expansions based on using periodic orbit theory—to approximate chaotic dynamics in a controlled perturbative way. This technique has proven to be widely useful for diagnosing and quantifying chaotic dynamics in problems ranging from atomic physics to neurophysiology.