Data: 24/03/2017

Ora: 14:00:00

Descrizione:

]]>For the N-body problem with equal masses, there exists a special type of periodic solutions called simple choreographies, where all the masses chase each other on a single loop. Well known examples including the rotating N-gon, the Figure-Eight solution of three body and the Super-Eight solution of four body. Simple choreographies with different shapes have been found numerically. However for many of them, rigorous proofs are still unavailable. In this talk, we give a proof of a special family of simple choreographies called "linear chain", which looks like a sequence of consecutive bubbles along a given line.

Data: 27/01/2017

Ora: 11:30:00

Descrizione:

Some definitions of spectral flow require a subdivision of the interval of definition into subintervals, where it is possible to define continuously paths of spectral projections. This method has been used by J. Phillips (1996), with the intention of providing a definition of spectral flow for continuous paths without using smooth paths as intermediate step. In this presentation we show that this definition can be simplified further if just one refrains from requiring the projections to be spectral. This simplification allows us to characterize the kernel and the image of the spectral flow in Banach spaces, and exhibit examples of spaces, where the spectral flow is not a group isomorphism, unlike infinite-dimensional Hilbert spaces. This difference amounts to the following behavior:

(1) the existence of infinite-dimensional spaces which are not isomorphic to closed subspaces of finite co-dimension

(2) the existence of pairs of projectors which are conjugated to each other, but there is no continuous path of projectors connecting one to the other.

We will focus on examples of projectors as in (2) arising in Banach spaces which can be obtained as a finite direct sums of sequences spaces \(\mathcal{l}_p \).

]]>Data: 21/12/2016

Ora: 14:30:00

Descrizione:

We present some recent results in the study of two, closely related, nonlocal problems: the fractional Allen-Cahn equation and the nonlocal minimal surfaces. More precisely, we show sharp energy estimates for minimizers and stable objects for both problems and, as a consequence, we deduce some flatness results in low dimensions. These results are contained in several works in collaborations with X. CabrĂ©, J. Serra, and E. Valdinoci.

]]>Data: 16/12/2016

Ora: 13:30:00

Descrizione:

A celebrated result of to H. PoincarĂ© asserts that a closed minimising geodesic on a orientable surface is unstable when considered as an orbit of the geodesic flow. In this talk starting from this classical result we discuss the recently proved results on the instability and hyperbolicity of closed geodesics on higher (maybe nonoriented) Riemannian manifold. We also discuss the relevant role played by the Morse indices of the geodesic and its second iteration on its stability.

]]>Data: 16/12/2016

Ora: 14:30:00

Descrizione:

Starting from the result on the stability of closed Riemannian geodesics, we discuss its generalisation to geodesics of any causal character on Lorentzian (and, more generally, on semi-Riemannian) manifolds. Dropping the non-positivity assumption of the metric tensor is a quite challenging task, since the Morse index is truly infinite. If time allows we give an idea of the proof of a Birkhoff conjecture for closed Riemannian geodesics, recently proved.

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