Sala Seminari (Dip. Matematica)
In this talk based on joint work with Kienzler and Vazquez I will
explain how Caffarelli's strategy of proving regularity for free
boundary problems can be implemented for the porous medium equation.
Elements of the proof is a global existence result for Lipschitz
perturbations of flat fronts of C. Kienzler following my strategy with
Tataru for the Navier-Stokes equations, a careful construction of
comparison solutions, and Gaussian estimates in a subelliptic setting
A classical theorem of Gromov states that the Betti numbers, i.e. the size of the free part of the homology groups, of negatively curved manifolds are bounded by the volume. We extend this theorem to the torsion part of the homology in all dimensions d > 3. From Gromov’s work it is known that in dimension 3 the size of torsion homology cannot be bounded in terms of the volume. In dimension 4 we give a somewhat precise estimate for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension d > 4 up to homeomorphism.
Consider a group G action on a metric space X by isometries, and subgroups M_1,..., M_l. One interesting and important problem is to study conditions under which non-emptyness of each M_i-fixed points ensures that of global (G-)fixed points. One extreme case is where G is a simple Lie group, X is an origin-excluded Hilbert space, and the action is given by a (strongly continuous) unitary representation. Then, the Howe--Moore property, based on the Mautner phenomenon, implies that for each(!) non-compact closed subgroup M, the existence of M-fixed points suffices that of G-fixed points.
Abstract. Spectral sequences are often applied to compute local cohomology functors.
In this talk I’m going to review their use in order to calculate local
cohomology from the primary decomposition of an ideal I in a commutative Noetherian ring R. By one hand, we shall deal with the
computation of several generalized local cohomology functors supported on I. On the other hand, we will be mainly concerned with
the computation of the local cohomology of R/I.