The Weizmann Institute of Science Faculty of Mathematics and Computer Science Mathematical Analysis and Applications Seminar Seminar Room, Room 261, Ziskind Building on Tuesday, January 3, 2012 11:00 - noon Note the unusual location V. Oliker Emory University will speak on Designing lenses and mirrors with help from geometry and optimal mass transport Abstract: Systems of lenses or mirrors converting an incident plane wave of a given cross section and intensity distribution into an output plane wave irradiating at a given target set with prescribed intensity are required in many applications. Most of the known designs of devices with such capabilities are restricted to rotationally symmetric mirrors/lenses. In many cases this is a significant limitation. In this talk I will discuss designs with free-form mirrors/lenses, that is, without a priori assumption of rotational symmetry. Assuming the geometrical optics approximation, it can be shown that the functions describing such free-form mirrors/lenses satisfy fully nonlinear Monge-Amp\`{e}re type partial differential equations (PDE's) derived from the basic laws of geometrical optics. However, because of strong nonlinearities investigation of such PDE's is difficult and many basic questions are open. Fortunately, by extending classical ideas from convexity and of the Legendre transform some of these problems can also be formulated (in weak form) as problems in calculus of variations in which instead of solving the nonlinear PDE's one needs to find extrema of some Fermat-like functionals. Moreover, these functionals arise in a physically natural way as cost functions in optimal (linear and nonlinear) transportation problems. This approach was successfully applied in several design problems and allowed to prove existence, uniqueness and numerically calculate solutions by methods based on mathematical programming. I will describe these ideas in the case of the "two-lens" design problem. --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Yaeli Malka <yaeli.malka@weizmann.ac.il>