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The birth of condensed matter physics in Italy is linked to a small number of very distinguished scientists. Mario Tosi, Professor of Physics of Matter at the Scuola Normale Superiore, is unquestionably among the leading figures, a true founder of the theoretical activity in the country and a true catalyst of novel research directions internationally. This volume collects the proceedings of a symposium held at Scuola Normale Superiore di Pisa, designed to show Mario Tosi's broad, deep influence in very diverse areas of the quantum theory of condensed matter. The topics covered in the volume represent the breadth of his interests and the highlights in the quantum theory of condensed matter: liquids, electronic states in complex structures, quantum degenerate gases, many-body physics.

Mathematical theory of discrete time decision processes, also known as stochastic control, is based on two major ideas: backward induction and conditioning. It has a large number of applications in almost all branches of the natural sciences. The aim of these notes is to give a self-contained introduction to this theory and its applications. Our intention was to give a global and mathematically precise picture of the subject and present well motivated examples. We cover systems with complete or partial information as well as with complete or partial observation. We have tried to present in a unified way several topics such as dynamic programming equations, stopping problems, stabilization, Kalman-Bucy filter, linear regulator, adaptive control and option pricing. The notes discuss a large variety of models rather than concentrate on general existence theorems.

We consider in Rn a differential operator P(D), P a polynomial, with constant coefficients. Let U be an open set in Rn and A(U) be the space of real analytic functions on U. We consider the equation P(D)u=f, for f in A(U) and look for a solution in A(U). Hormander proved a necessary and sufficient condition for the solution to exist in the case U is convex. From this theorem one derives the fact that if a cone W admits a Phragmen-Lindeloff principle then at each of its non-zero real points the real part of W is pure dimensional of dimension n-1. The Phragmen-Lindeloff principle is reduced to the classical one in C. In this paper we consider a general Hilbert complex of differential operators with constant coefficients in Rn and we give, for U convex, the necessary and sufficient conditions for the vanishing of the H1 groups in terms of the generalization of Phragmen-Lindeloff principle.

These notes contain lectures on the theory of group representations and its applications to the physics of atoms, molecules and crystals, given at Purdue University, Scuola Normale Superiore (Pisa, Italy) and Universidad Técnica Federico Santa Maria (Valparaiso, Chile) on and off over a period of over 25 years. The topics selected reflect my special interests and their scope is limited by the time available to the students. The style is somewhat concise and will require careful attention on the part of the reader.

This book contains expositions of talks of the Colloquio De Giorgi, given by eminent experts and addressed to a 'general' public (audience) of mathematicians. The topics are prepared in such a way that readers will get some idea of recent progress in several mathematical fields.

In the academic year 1994/95 I lectured on semigroups in the Scuola Normale Superiore and in the Politecnico di Torino. The purpose of the lectures was to present to an audience of graduate students a self-contained introduction to the theory of strongly continuous semigroups of linear operators acting on a Banach space. The main trust was concentrated on laying down the basic geometrical aspects of the theory as a background for applications to concrete problems in the analysis of differential operators.

Neuronal plasticity is the term generally used to describe a great variety of changes in neuronal structure and functions, in particular activity-dependent, prolonged functional changes, accompanied by corresponding biochemical and possibly morphological alterations. The first insights about neuronal plasticity have been obtained on simple forms of life. Today, many areas in the mammalian central nervous system are under investigation, in particular the cortex and the hippocampus. In these areas, neuronal plasticity is considered to be at the basis of learning and memory. The study of brain plasticity in sensory systems has been particularly fruitful, because these systems can be experimentally manipulated in a very precise manner by changes in the sensory imput.

Low energy neutrino astrophysics studies the evolutionary life of the stars via the neutrinos emitted during the quiescent phase (solar neutrinos) and during the explosive death of big mass stars (supernova neutrinos). The neutrino mean free path in matter is about twenty orders of magnitude greater than that of light; therefore neutrinos reaching us can be produced also in deep and high-density levels of stars. Since massless neutrinos are unaffected by their travel in the interstellar space, their energies and arrival directions carry information on the star history. The subject of this thesis is the search for neutrino bursts from galactic stellar gravitational collapses performed in the MACRO experiment, a large area modular detector, operating since autumn 1989.