In 2008, a school on the theory of Optimal Transportation and its applications took place in Pisa, with lectures by F.Barthe, W.Gangbo, F.Maggi and R.McCann. This book collects the notes of the first three lectures. These notes provide a deep insight into topics of concentration inequalities, evolution PDE's of Hamiltonian type, geometric and functional inequalities.

In 2008, a school on the theory of Optimal Transportation and its applications took place in Pisa, with lectures by F.Barthe, W.Gangbo, F.Maggi and R.McCann. This book collects the notes of the first three lectures. These notes provide a deep insight into topics of concentration inequalities, evolution PDE's of Hamiltonian type, geometric and functional inequalities.

This volume collects lectures given within the Matematica, Cultura e Società Public Lectures (year 2006). All contributions are in italian. «Matematica, Cultura e Società» è il titolo di una rassegna di conferenze pubbliche che dal 2003 il Centro De Giorgi dedica alla divulgazione della matematica e, più in generale, del sapere scientifico. L'idea che sta alla base della rassegna è quella di creare un canale di comunicazione tra società e mondo della ricerca scientifica. Muovendo da diverse prospettive, queste lezioni - che si differenziano, oltre che per argomenti (dalla storia, alla filosofia, al cinema), per forme e modalità di trattazione - si propongono di illustrare alcuni aspetti del ruolo culturale della matematica nella società del passato e del presente; di comunicare quanto la matematica sia diffusamente presente nella cultura, nella scienza e nella tecnologia, oggi ancor più di quanto non sia stato nel passato; di contribuire, infine, a fare in modo che diminuisca il numero di coloro che considerano un valore la divisione fra le due culture.

This volume collects some contributed papers by the participants to the workshop "Singularities in nonlinear evolution phenomena and applications", which has been held at the Centro di Ricerca Matematica E. De Giorgi, from May 26th to May 30th 2008. The topic of the workshop was the formation and the evolution of singular structures, like systems of points, curves and surfaces. Such questions naturally arise in many models from Physics, Biology, Image Processing and Applied Mathematics in general, and have attracted a lot of attention in recent years. Their analysis requires sophisticated tools and an interdisciplinary approach, and poses new challenging mathematical problems.

The main aim is to present at the level of beginners several modern tools of micro-local analysis which are useful for the mathematical study of nonlinear partial differential equations. The core of these notes is devoted to a presentation of the para-differential techniques, which combine a linearization procedure for nonlinear equations, and a symbolic calculus which mimics or extends the classical calculus of Fourier multipliers. These methods apply to many problems in nonlinear PDE's such as elliptic equations, propagation of singularities, boundary value problems, shocks or boundary layers. However, in these introductory notes, we have chosen to illustrate the theory on two selected and relatively simple examples, which allow becoming familiar with the techniques. They concern the well-posedness of the Cauchy problem for systems of nonlinear PDE's, firstly hyperbolic systems and secondly coupled systems of Schrödinger equations which arise in various models of wave propagation.

Little more than one hundred years have gone by since the birth of Ettore Majorana, a highly renowed theoretical physicist. His career was brief and irregular but very intense, and he disappeared in March 1938 in circumstances that still are not completely clear. This volume is a contribution to a better understanding of the scientific, academic and human personality of Ettore Majorana, beyond the layers of legendary aspects which have accumulated over the years. Based on primary sources alone-scientific literature of the period and numerous archival documents-the figure of Ettore Majorana emerges in a completely new light. The young scientist is intensely involved in scientific research, completely independent, always striving to offer innovative contributions of the highest level according to the most advanced international standards, and very determined to make his results known by following a shrewd publication strategy. He is profoundly interested in his academic career, very scrupulous with institutional relationships, and attentive to his students. Moreover, his documented scholarly activity is much wider than previously reported. This historical analysis shows also that Majorana had a very important role in orienting research in Rome, especially in the sector of the statistical model of the atom and in Nuclear Physics. It is also clear that there are aspects of Ettore Majorana's life which transmit a solid historical legacy from the cultural and human points of view, besides the scientific legacy which is universally recognized to him. A rich reproduction of original documents completes the volume.

The book contains research papers on Diophantine Geometry, written by participants to a related workshop held at the Centro De Giorgi of the Scuola Normale of Pisa during the period April-July 2005. The authors are eminent experts in the field; actually, several interacting subfields of the main topic are represented here, which is particularly useful to get a broad overview of recent research developments.

In the most recent years growing attention has been dedicated to many body quantum systems from the point of view of quantum information. Indeed, after the initial investigation of simple systems as single or two qubits, the need for understanding the characteristics of a realistic quantum information device necessarily leads to the study of many body quantum systems. These studies are also driven by the very fast development of experiments which in recent years have reached the goal of coherent control of a few qubits with a roadmap for further scaling and improvement of coherent control and manipulation techniques. This book gives a selection of the current research topics in the field of quantum information for many body quantum systems together with open problems.

This book contains papers presented by leading experts at the "Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT)" held at the Centro di Ricerca Matematica (CRM) Ennio De Giorgi at Pisa, Italy, from October 15 to 19, 2007. The book consists of research and review papers by leading experts and practitioners in biomedical imaging and intensity-modulated radiation therapy (IMRT). The topics include mathematical aspects and practical problems in current major and emerging technologies for the diagnostic and therapeutic medicine and biology research. The contributed work signifies the interdisciplinary cooperation between mathematicians and scientists from medical physics, engineering, clinical medicine and biology, that leads to mathematically-based better solutions of practical problems in biomedical imaging and IMRT. This book is also a survey of important research directions and opportunities in mathematics induced by problems in biomedical imaging and radiation planning applications.

This volume collects lectures given within the Matematica, Cultura e Società Public Lectures (year 2005). All contributions are in italian.

«Matematica, Cultura e Società» è il titolo di una rassegna di conferenze pubbliche che dal 2003 il Centro De Giorgi dedica alla divulgazione della matematica e, più in generale, del sapere scientifico. L'idea che sta alla base della rassegna è quella di creare un canale di comunicazione tra società e mondo della ricerca scientifica. Muovendo da diverse prospettive, queste lezioni - che si differenziano, oltre che per argomenti (dalla storia, alla filosofia, al cinema), per forme e modalità di trattazione - si propongono di illustrare alcuni aspetti del ruolo culturale della matematica nella società del passato e del presente; di comunicare quanto la matematica sia diffusamente presente nella cultura, nella scienza e nella tecnologia, oggi ancor più di quanto non sia stato nel passato; di contribuire, infine, a fare in modo che diminuisca il numero di coloro che considerano un valore la divisione fra le due culture.

This volume deals with the problem of characterizing the limit points of sequences of smooth maps from the unit ball of Rⁿ with values into a smooth boundaryless Riemannian manifold and with equibounded "integral energies". After surveying some known results about Cartesian currents and graphs with finite area and finite boundary area, we do characterize, as in the title, weak limits of sequences of smooth maps with equibounded W^1,2_ W^12_ or BV-energies.

This volume collects lectures given within the Matematica, Cultura e Società Public Lectures (year 2004). All contributions are in italian. «Matematica, Cultura e Società» è il titolo di una rassegna di conferenze pubbliche che dal 2003 il Centro De Giorgi dedica alla divulgazione della matematica e, più in generale, del sapere scientifico. L'idea che sta alla base della rassegna è quella di creare un canale di comunicazione tra società e mondo della ricerca scientifica. Muovendo da diverse prospettive, queste lezioni - che si differenziano, oltre che per argomenti (dalla storia, alla filosofia, al cinema), per forme e modalità di trattazione - si propongono di illustrare alcuni aspetti del ruolo culturale della matematica nella società del passato e del presente; di comunicare quanto la matematica sia diffusamente presente nella cultura, nella scienza e nella tecnologia, oggi ancor più di quanto non sia stato nel passato; di contribuire, infine, a fare in modo che diminuisca il numero di coloro che considerano un valore la divisione fra le due culture.

This book wants to honor Franco Conti's memory and his contribution to mathematics.

These two volumes collect the contributions from the participants to the Research Trimester on Phase Space Analysis of Partial Differential Equations, which took place from February through April 2004 at the Centro di Ricerca Matematica "Ennio De Giorgi".

These two volumes collect the contributions from the participants to the Research Trimester on Phase Space Analysis of Partial Differential Equations, which took place from February through April 2004 at the Centro di Ricerca Matematica "Ennio De Giorgi".

In May 2004 Nicolai Krylov (University of Minnesota) visited the Centro di Ricerca Matematica Ennio De Giorgi as UMI-Lecturer. This volume is collects the lectures given by Krilov at the center.

This volume collects lectures given within the Matematica, Cultura e Società Public Lectures (year 2003). All contributions are in italian. «Matematica, Cultura e Società» è il titolo di una rassegna di conferenze pubbliche che dal 2003 il Centro De Giorgi dedica alla divulgazione della matematica e, più in generale, del sapere scientifico. L'idea che sta alla base della rassegna è quella di creare un canale di comunicazione tra società e mondo della ricerca scientifica. Muovendo da diverse prospettive, queste lezioni - che si differenziano, oltre che per argomenti (dalla storia, alla filosofia, al cinema), per forme e modalità di trattazione - si propongono di illustrare alcuni aspetti del ruolo culturale della matematica nella società del passato e del presente; di comunicare quanto la matematica sia diffusamente presente nella cultura, nella scienza e nella tecnologia, oggi ancor più di quanto non sia stato nel passato; di contribuire, infine, a fare in modo che diminuisca il numero di coloro che considerano un valore la divisione fra le due culture.

These two volumes collect the contributions from the participants to the Research Trimester on Dynamical Systems held from February through April 2002 at the Centro di Ricerca Matematica "Ennio De Giorgi". They are the first of a new series of publications that the Scuola Normale has launched to give account of the research activity carried out at the Centro De Giorgi.

These two volumes collect the contributions from the participants to the Research Trimester on Dynamical Systems held from February through April 2002 at the Centro di Ricerca Matematica "Ennio De Giorgi". They are the first of a new series of publications that the Scuola Normale has launched to give account of the research activity carried out at the Centro De Giorgi.