Scuola Normale Superiore – förlag

In 2013, a school on Geometric Measure Theory and Real Analysis, organized by G. Alberti, C. De Lellis and myself, took place at the Centro De Giorgi in Pisa, with lectures by V. Bogachev, R. Monti, E. Spadaro and D. Vittone.

This thesis deals with specific featuresof the theory of holomorphic dynamics in dimension 2 and then sets out to studyanalogous questions in higher dimensions, e.g. dealing with normal forms forrigid germs, and examples of Kato 3-folds.

The local dynamics of holomorphic mapsaround critical points is still not completely understood, in dimension 2 orhigher, due to the richness of the geometry of the critical set for alliterates.

In dimension 2, the study of thedynamics induced on a suitable functional space (the valuative tree) allows aclassification of such maps up to birational conjugacy, reducing the problem tothe special class of rigid germs, where the geometry of the critical set issimple.

In some cases, from such dynamical dataone can construct special compact complex surfaces, called Kato surfaces,related to some conjectures in complex geometry.

This study provides innovative mathematical models for assessing the eruption probability and associated volcanic hazards, and applies them to the Campi Flegrei caldera in Italy. Throughout the book, significant attention is devoted to quantifying the sources of uncertainty affecting the forecast estimates.

The Campi Flegrei caldera is certainly one of the world’s highest-risk volcanoes, with more than 70 eruptions over the last 15,000 years, prevalently explosive ones of varying magnitude, intensity and vent location. In the second half of the twentieth century the volcano apparently once again entered a phase of unrest that continues to the present. Hundreds of thousands of people live inside the caldera and over a million more in the nearby city of Naples, making a future eruption of Campi Flegrei an event with potentially catastrophic consequences at the national and European levels.

This book presents a series of lectures on three of the best known examples of free discontinuity problems: the Mumford-Shah model for image segmentation, a variational model for the epitaxial growth of thin films, and the sharp interface limit of the Ohta-Kawasaki model for pattern formation in dyblock copolymers.

These lecture notes are devoted to the analysis of a nonlocal equation in the whole of Euclidean space. In studying this equation, all the necessary material is introduced in the most self-contained way possible, giving precise references to the literature when necessary.

The results presented are original, but no particular prerequisite or knowledge of the previous literature is needed to read this text. The work is accessible to a wide audience and can also serve as introductory research material on the topic of nonlocal nonlinear equations.