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We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This establishes an intrinsic connection between ideal Euler hydrodynamics (via Arnolds approach), shock formation in the multidimensional Burgers equation and the Wasserstein geometry of the space of densities.
In this note we obtain the characterization for asymptotic directions on various subgroups of the diffeomorphism group. We give a simple proof of non-existence of such directions for area-preserving diffeomorphisms of closed surfaces of non-zero curv
We consider the Cauchy problem for the Burgers hierarchy with general time dependent coefficients. The closed form for the Greens function of the corresponding linear equation of arbitrary order $N$ is shown to be a sum of generalised hypergeometric
By studying the Higgs bundle equations with the gauge group replaced by the group of symplectic diffeomorphisms of the 2-sphere we encounter the notion of a folded hyperkaehler 4-manifold and conjecture the existence of a family of such metrics param
Let $X$ be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry $text{Nil}$. We show that the diffeomorphism group $text{Diff}(X)$ deformation retracts to the isometry group $text{Isom}(X)$. Combining this with earlier work by m
We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabais theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to sphe