Distinct scientific communities are usually involved in the three fields of quasi-crystals, of liquid crystals, and of systems having modulated crystalline structures. However, in recent years, there has been a growing feeling that a number of common problems were encountered in the three fields. These comprise the need to recur to "exotic" spaces for describing the type of order of the atomic or molecular configurations of these systems (Euclidian "superspaces" of dimensions greater than 3, or 4-dimensional curved spaces); the recognition that one has to deal with geometrically frustrated systems, and also the occurence of specific excitations (static or dynamic) resulting from the continuous degeneracies of the stable structures considered. In the view of discussing these problems, aNA TO-Advance Research Workshop has assembled in Preveza (Greece), in september 1989,50 experts of the three considered fields (with an equal proportion of theorists and experimentalists). 35 hours of conferences and discussions have led to a more detailed evaluation of the similarities and of the differences in the approaches implemented in the studies of the three types of systems. The papers contained in this NATO-series book provide the substance of this workshop. The reader will find three types of papers. Some very short papers giving the main ideas stated on a subject. Papers comprising 8-10 pages which stick closely to the contents of the talks presented. Longer papers providing more extensively the background and results relative to a given topic. It is worth summarizing the principal outputs of the workshop.

This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.

This textbook aims to briefly outline the main directions in which the geometrization of thermodynamics has been developed in the last decades. The textbook is accessible to people trained in thermal sciences but not necessarily with solid formation in mathematics. For this, in the first chapters a summary of the main mathematical concepts is made. In some sense, this makes the textbook self-consistent. The rest of the textbook consists of a collection of results previously obtained in this young branch of thermodynamics. The manner of presentation used throughout the textbook is adapted for ease of access of readers with education in natural and technical sciences.

Because it is grounded in math, chemical thermodynamics is often perceived as a difficult subject and many students are never fully comfortable with it. The first authoritative textbook presentation of equilibrium chemical and phase thermodynamics in a reformulated geometrical framework, Chemical and Phase Thermodynamics shows how this famously difficult subject can be accurately expressed with only elementary high-school geometry concepts. Featuring numerous suggestions for research-level extensions, this simplified alternative to standard calculus-based thermodynamics expositions is perfect for undergraduate and beginning graduate students as well as researchers.

This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge–Ampère equations. Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.