(see the video of the first session in Porto)

Professor Luis Caffarelli is Sid W. Richardson Foundation Regents Chair at the University of Texas at Austin, and has been awarded in 2009 the AMS Leroy P. Steele Prize for Lifetime Achievement. He is also the Director of Mathematics at the University of Texas at Austin of the CoLab - UTAustin|Portugal partnership program.

Professor Luis Caffarelli is Sid W. Richardson Foundation Regents Chair at the University of Texas at Austin, and has been awarded in 2009 the AMS Leroy P. Steele Prize for Lifetime Achievement. He is also the Director of Mathematics at the University of Texas at Austin of the CoLab - UTAustin|Portugal partnership program.

- July 8, 2009 - Centro de Matemática da Universidade do Porto, sala 0.07, 18:00.
(
*With the support of CMUP*)

**Title:***Phase transition and minimal surfaces for non local operators*.

**Abstract:**Movement by mean curvature , i.e, when a surface evolves with normal speed proportional to its mean curvature, appears in the modelling of phase transition phenomena , for instance as a limit of phase field models In the case of slow decay of long range interactions, the corresponding limiting transition surface moves proportionally to an "integral version" of mean curvature. We will describe this phenomena, and the geometric properties of the corresponding "integral minimal surfaces".

- July 15, 2009 - Departamento de Matemática da Universidade de Coimbra, sala 17 de Abril, 14:30.
(
*With the support of CMUC*)

**Title:***Fully non linear equations for nonlocal diffusions*.

**Abstract:**Fully non linear equations arise in optimal control and game theory, when one is able to optimize a ( continuous) diffusion process ( or has incomplete information about it). A complete theory of existence and regularity for these equations was developed in the eighties thanks to the remarkable contributions of Krylov and Safanov ( the Harnack inequality) and of Evans- Krylov ( the Evans_ Krylov regularity theorem) In a series of papers , in collaboration with Luis Silvestre, we developed the parallel theory in the case of discontinuous (Levy type) diffusion. I plan to present the main steps of the theory, and give an idea of our proof of the Evans Krylov theorem in this case

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