From the mathematical point of view, these models have very different behaviors. Their analysis therefore requires various mathematical methods which this book aims to present in a systematic, painstaking, and exhaustive way.

The first part of this work is devoted to the systematic formal analysis of viscous hydrodynamic limits of the Vlasov–Maxwell–Boltzmann system, leading to a precise classification of physically relevant models for viscous incompressible plasmas, some of which have not previously been described in the literature.

In the second part, the convergence results are made precise and rigorous, assuming the existence of renormalized solutions for the Vlasov–Maxwell–Boltzmann system. The analysis is based essentially on the scaled entropy inequality. Important mathematical tools are introduced, with new developments used to prove these convergence results (Chapman–Enskog-type decomposition and regularity in the $v$ variable, hypoelliptic transfer of compactness, analysis of high frequency time oscillations, and more).

The third and fourth parts (which will be published in a second volume) show how to adapt the arguments presented in the conditional case to deal with a weaker notion of solutions to the Vlasov–Maxwell–Boltzmann system, the existence of which is known.]]>

All the essays are self-contained and most of them can be understood by the general educated mathematician. They should be useful to researchers and to students of non-Euclidean geometry, and they are intended to be references for the various topics they present.]]>

The book focuses on estimates up to the boundary of a domain. It contains a great variety of inequalities for differential and pseudodifferential operators with constant coefficients. Results of final character are obtained, without any restrictions on the type of differential operators. Algebraic necessary and sufficient conditions for the validity of the corresponding a priori estimates are presented. General criteria are systematically applied to particular types of operators found in classical equations and systems of mathematical physics (such as Lame’s system of static elasticity theory or the linearized Navier–Stokes system), Cauchy–Riemann’s operators, Schrödinger operators, among others. The well-known results of Aronszajn, Agmon–Douglis–Nirenberg and Schechter fall into the general scheme, and sometimes are strengthened.

The book will be interesting and useful to a wide audience, including graduate students and specialists in the theory of differential equations.]]>

These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov–Sobolev type. In particular, it will be of interest for researchers dealing with approximation theory, numerical integration and discrepancy.]]>

The free boundary is the shock hypersurface and the boundary conditions are jump conditions relative to a prior solution, conditions following from the integral form of the mass, momentum and energy conservation laws. The prior solution is provided by the author‘s previous work which studies the maximal classical development of smooth initial data. New geometric and analytic methods are introduced to solve the problem. Geometry enters as the acoustical structure, a Lorentzian metric structure defined on the spacetime manifold by the fluid. This acoustical structure interacts with the background spacetime structure. Reformulating the equations as two coupled first order systems, the characteristic system, which is fully nonlinear, and the wave system, which is quasilinear, a complete regularization of the problem is achieved.

Geometric methods also arise from the need to treat the free boundary. These methods involve the concepts of bi-variational stress and of variation fields. The main new analytic method arises from the need to handle the singular integrals appearing in the energy identities. Shocks being an ubiquitous phenomenon, occuring also in magnetohydrodynamics, nonlinear elasticity, and the electrodynamics of noninear media, the methods developed in this monograph are likely to be found relevant in these fields as well.]]>

The book will be interesting and useful for a wide audience. It is intended for specialists and graduate students working in the theory of partial differential equations.]]>

Ten plenary, thirty-three invited and four special lectures formed the core of the program. As at all the previous EMS congresses, ten outstanding young mathematicians received the EMS prizes in recognition of their research achievements. In addition, two more prizes were awarded: The Felix Klein prize for a remarkable solution of an industrial problem, and – for the second time – the Otto Neugebauer Prize for a highly original and influential piece of work in the history of mathematics. The program was complemented by forty-three minisymposia with about 160 talks as well as contributed talks, spread over all areas of mathematics. Several panel discussions and meetings were organized, covering a variety of issues ranging from the future of mathematical publishing to public awareness of mathematics.

These proceedings present extended versions of most of the plenary and invited lectures which were delivered during the congress, providing a permanent record of the best what mathematics offers today.]]>

- to offer an accessible, reader-friendly and self-contained introduction to Kac–Moody algebras and groups;

- to clean the foundations and to provide a unified treatment of the theory.

The book starts with an outline of the classical Lie theory, used to set the scene. Part II provides a self-contained introduction to Kac–Moody algebras. The heart of the book is Part III, which develops an intuitive approach to the construction and fundamental properties of Kac–Moody groups. It is complemented by two appendices, respectively offering introductions to affine group schemes and to the theory of buildings. Many exercises are included, accompanying the readers throughout their journey.

The book assumes only a minimal background in linear algebra and basic topology, and is addressed to anyone interested in learning about Kac–Moody algebras and/or groups, from graduate (master) students to specialists.]]>

This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions.

The target audience are students from the upper undergraduate level onwards. We assume only a familiarity with linear algebra and basic group theory. Graph theory, finite fields, and character theory for abelian groups receive a concise overview and render the text essentially self-contained.]]>

These notes focus on the geometry and topology of Coxeter groups and buildings, especially nonspherical cases. The emphasis is on geometric intuition, and there are many examples and illustrations. Part I describes Coxeter groups and their geometric realisations, particularly the Davis complex, and Part II gives a concise introduction to buildings.

This book will be suitable for mathematics graduate students and researchers in geometric group theory, as well as algebra and combinatorics. The assumed background is basic group theory, including group actions, and basic algebraic topology, together with some knowledge of Riemannian geometry.]]>