|28/06/17||Colloquium||15:00||16:00||1201 Dal Passo||Tristan Rivière||ETH Zurich||How much does it cost...to turn the sphere inside out ?
How much does it cost...to knot a closed simple curve ? To cover the sphere twice ? to realize such or such homotopy class ? ...etc.
All these questions consisting of assigning a "canonical" number and possibly an optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together notions and techniques from topology, geometry and analysis.
In this talk we will concentrate on the operation consisting of everting the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the existence of such an operation the search for effective realizations of such eversions has triggered a lot of fascination and works in the math community. The absence in nature of matter that can interpenetrate and the quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for the difficulty to develop an intuitive approach on the problem.
We will present the optimization of Sophie Germain conformally invariant elastic energy for the eversion. Our efforts will finally bring us to consider more closely an integer number together with a mysterious minimal surface.
|20/06/17||Seminario||14:30||15:30||1201 Dal Passo||Livia Corsi||Georgia Tech (Atlanta, USA)||Billiards and rigid rotations|
Probably one of the most famous open problems concerning billiard systems is the Birkhoff conjecture: "If a billiard map is integrabile than the boundary of the billiard table is an ellipse". Recently Treschev conjectured that there might exist analytic billiards, different from ellipses, for which the dynamics in the neighborhood of the period-2 orbit is conjugated to a rigid rotation, suggesting a very interesting example of local integrability for billiard tables different from ellipses. However the result of Treschev is only formal in the sense that he finds only a formal power series. Our aim is to prove the convergence of such series.
This is a joint work (in progress) with M. Procesi.
|20/06/17||Seminario||11:00||13:00||1201 Dal Passo||Rick Miranda||Colorado State University||Matrix reduction approaches to interpolation problems: a review with remarks|
About ten years ago M Dumnicki developed some techniques for interpolation problems related to a careful study of the rank of the matrices involved. Using these techniques he was able to extend the state of the art at the time, bringing certain problems into the range of computer analyses; this enabled him to prove that linear systems in the plane satisfied the SGHH conjecture for homogeneous multiplicities up to 42, the record then. We'll review the method, and consider some refinements, and applications to systems with ten points. The talk should be accessible to non-experts.