## Preparatory Workshop

**Date and place**: Saturday, June 23, WMiIUJ building.

The __preparatory workshop__ will outline the basic concepts related to the main lecture. The schedule is as follows:

10:00-10:45 | Grzegorz Karch (UWr) On a nonlinear nonlocal diffusion equationAbstract: In the talk, I will describe an abstract framework for non-local non-linear diffusion, by which we mean a phenomenon with properties strongly associated to diffusive processes such as the conservation of mass, the maximum principle, and the comparison principle. This framework encompasses some of the known examples of equations like the fractional porous medium equation or the equation with the fractional p-Laplacian, but it also opens up the space for new examples to be constructed and studied. |

11:00-11:45 | Piotr Biler (UWr) Non-local diffusion and filtrationAbstract: Some aspects of the nonlinear diffusion and filtration equation will be discussed |

11.45-12.30 | pizza |

12.30-13.15 | Valentino Tosatti (Northwestern University) Regularity of solutions of concave PDEsAbstract: I will give an introduction to the C^{2,alpha} regularity theory of second-order fully nonlinear concave elliptic equations as pioneered by Evans and Krylov, and later substantially generalized by Caffarelli. I will also discuss a very neat proof of the Evans-Krylov theorem obtained recently by Caffarelli and Silvestre. |

13.30-14.15 | Tomasz Cieślak (IM PAN) Monotonicity formulasAbstract: I will be speaking about some monotonicity formulas in elliptic problems, including the famous Alt-Caffarelli-Friedman formula. I will show how to use them to obtain regularity estimates, in particular in obstacle problems. If time permits I will try to discuss some further developments including Weiss's formulas and Almgren's frequency formula. |

We invite all students, PhD candidates, and faculty to attend the workshop. Pizza will be served for all participants!