Il Prof. Marcel BERGER (IHES), visitatore presso l’Universita’ di Roma « La Sapienza » nel periodo 2-16 ottobre 2004, terra’ un ciclo di 4 conferenze

ROMA 4 LECTURES ABSTRACTS

First lecture : “Jean-Victor PONCELET (Metz 1788-Paris 1867) : the metamorphoses of his life and of his fascinating theorem”

Abstract  : Prisoner of the Russians during the Napoleon retreat, during these two years, having only at his disposal the memory of Monge lectures at Ecole polytechnique where his was a pupil, Poncelet managed to become the father of projective geometry (real as well as complex), including also for this geometry three new concepts : that of continuity, that of duality, that of projective transformation. With these notions he found many new theorems, in particular for conic sections. But the one of them, which concerns polygons which are at the same time inscribed in a conic and circumscribed to another, exercised a real fascination on all mathematicians who became aware of it, and still exercises that fascination. Besides its beauty, another reason is that there is no simple proof of it. Jacobi, to deeply inderstand it, discovered that it is the elliptic functions which are at the root of it. We will present in the lecture few different proofs of the theorem, quote the answers to difficult questions of finding explicit formulas, questions which stem out directly from the theorem. There is no year today where does not appear one or more papers connected with Poncelet big theorem.

Second lecture : “Variations on the theme  « slicing a convex body » and the big conjecture for convex bodies”

Abstract : convex bodies look very simple and natural mathematical objects. However, even for some simple problems concerning (say in dimension 3) the relations beetween the area of various plane sections as compared to the total volume of the body, some of the answers have been : surprising, hard to prove and obtained only quite recently, and when one is interested in large dimensions, the asymptotic behavior of the relations between the total volume and the size (small or large volume) of the hyperplane sections is still partly conjectured , this under at least five quite different equivalent formulations

Third lecture : “ Introduction to Gromov’s theory and results for mm-spaces (« metric-measure space »)”

Abstract : for Riemanian manifolds the metric determines canonically the measure, but in many contexts the measure and the metric can be almost completely non related (an intermediate case is the Gaussian measure on R^N). Obtaining results for such spaces is basic in various contexts, e.g. biology, medical imagery, etc. where the spaces under consideration are of large dimensions; typically one looks for asymptotic behavior. Gromov results concern three topics : the “observable diameter” for observable (functions coming from a physical objects are called observable, since they are the only measurements one can do to study such an object, this study is also called “concentration phenomena”). The second topic is the “spectrum” which can be defined any mm-space, by the trick of using the Dirichlet principle and the Lipschitz constantss of functions, and Gromov gives an inequality between the first eigenvalue and the observable diameter. The third topic is (in the same way Gromov revolutionized Riemannian geometry by introducing the so-called “Gromov-Hausdorff metric”) the construction of a metric on the set of all mm-spaces.

Fourth lecture : “Nabutovsky-Weinberger results on the shape of the set of all Riemannian structures on a given compact manifold”

A natural question is to ask “how looks the set of all Riemannina structures on a given comppact manifold ?” For surfaces, the answer is known since the conformal representation theorem and Teichmüller : the desired set is homotopic to the set of all metrics of constant Gaussian curvature ( a set of finite dimension which is well understood), moreover one can get this by following a natural flow. For dimension three, the Thurston geometrization (even a weaker form) insures also a nice picture for the set under consideration. Nabutovsky-Weinberger show that starting dimension 5 (and probably also in dimension 4) the panorama of our set of metrics is AWFUL, indescriptible. This for any manifold, and it is enough to be prove it for the sphere itself. Precisely, for any known functional, as soon as one is far enough from the standard metric, one gets (in the Gromov-Hausdorff metric) a not connected set, but moreover with a exponential number of components. And the worse is that points in these disjoints components cannot be joined by a curve of “computable length” (not computable functions are an algorithmic notion, for example any iterated exponential is still computable). So when one follows any reasobale flow, one finally get in some of these components, and not to a nice metric. The proof of this awfulness is quite hard, it is a mixture of algorithmic (Turing machines and computable functions) and of of geometric results (of interest in themselves), which finally contradicts theorems about Turing. The philosophy consequence is one still has to find new functionals, new directions, to look at.