Beyond perimeters, between discrete and continuous structures
speaker: Umberto Mosco (Worcester Polytechnic Institute)abstract: This talk is dedicated to the memory of the young Ennio De Giorgi, active in Roma at the turning of the nineteen-fifties and -sixties. His profound ideas of those times about variational problems directly or indirectly involving discontinuities and sets of finite perimeter, more than half a century later still represent the benchmark to assess the scope of some contemporary research in the field. In our talk we focus on three related topics that receive indeed proper light if seen in the perspective open by that early De Giorgis work. We present a family of ODEs in the plane driving their orbits, beyond finite perimeters, toward attractors of all dimensions between 1 and 2. We show that such non-rectifiable attractors possess an intrinsic dynamics described by measure-valued local energies more general than those expressed by the discon tinuous gradients of the classic differential operators. We present a new model for the self-organized-criticality (SOC) behavior of sand piles, that bridges the gap between the discrete automata models of physics and the PDEs and extend the theory to a fractional range of dimensions adding universality to the SOC paradigm. What emerges from these examples is that there exists a variety of discrete dynamical structures, with well-defined geometric and dynamic intrinsic features at all scales, that cannot be described, neither directly nor asymptotically, in terms of differential operators. In exploring this new landscape, new ideas must be developed, new tools must be designed, merging analysis and computation. In this challenging task, the creative work of De Giorgi will provide insight and inspiration to young researchers for years to come
timetable:
Wed 21 Sep, 9:00 - 9:55, Palazzo dei Congressi
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