These lectures intend to give a self-contained exposure of some techniques for
computing the evolution of plane curves. The motions of interest are the so-called
motions by curvature. They mean that, at any instant, each point of the curve moves
with a normal velocity equal to a function of the curvature at this point. This kind of
evolution is of some interest in differential geometry, for instance in the problem of
Inspired by Lorenz’ remarkable chaotic ﬂow, we describe in this paper the structure of all C 1 robust transitive sets with singularities for ﬂows on closed 3-manifolds: they are partially hyperbolic with volume-expanding central direction, and are either attractors or repellers. In particular, any C 1 robust attractor with singularities for ﬂows on closed 3-manifolds always has an invariant foliation whose leaves are forward contracted by the ﬂow, and has positive Lyapunov exponent at every orbit, showing that any C 1 robust attractor resembles a geometric Lorenz attractor. ...
An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function, e.g. a constant. In two dimensions, the uniformization theorem assures the existence of a conformal metric with constant Gauss curvature. Moreover, J. Moser  proved that for every positive function f on S 2 satisfying f (x) = f (−x) for all x ∈ S 2 there exists a conformal metric on S 2 whose Gauss curvature is equal to f . A natural conformal invariant in dimension four is 1 Q = − (∆R −...
Many classical integrable systems (like the Euler, Lagrange and
Kowalewski tops or the Neumann system) as well as finite dimensional reductions
of many integrable PDEs share the property of being algebraically
completely integrable systems4. This means that they are completely integrable
Hamiltonian systems in the usual sense and, moreover, their complexified
invariant tori are open subsets of complex Abelian tori on which the
complexified flow is linear. To such systems the powerful algebro-geometrical
techniques may be applied...