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Reflection Positivity
Refection Positivity is a central theme at the crossroads of Lie group representations, euclidean and abstract harmonic analysis, constructive quantum field theory, and stochastic processes. This book provides the first presentation of the representation theoretic aspects of Refection Positivity and discusses its connections to those different fields on a level suitable for doctoral students and researchers in related fields. It starts with a general introduction to the ideas and methods involving refection positive Hilbert spaces and the Osterwalder--Schrader transform. It then turns to Reflection Positivity in Lie group representations. Already the case of one-dimensional groups is extreme...
Provability, Computability and Reflection
Provability, Computability and Reflection
Unitary Reflection Groups
A unitary reflection is a linear transformation of a complex vector space that fixes each point in a hyperplane. Intuitively, it resembles the transformation an image undergoes when it is viewed through a kaleidoscope, or an arrangement of mirrors. This book gives a complete classification of all finite groups which are generated by unitary reflections, using the method of line systems. Irreducible groups are studied in detail, and are identified with finite linear groups. The new invariant theoretic proof of Steinberg's fixed point theorem is treated fully. The same approach is used to develop the theory of eigenspaces of elements of reflection groups and their twisted analogues. This includes an extension of Springer's theory of regular elements to reflection cosets. An appendix outlines links to representation theory, topology and mathematical physics. Containing over 100 exercises, ranging in difficulty from elementary to research level, this book is ideal for honours and graduate students, or for researchers in algebra, topology and mathematical physics. Book jacket.
Reflection Groups and Coxeter Groups
This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.
The Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures
This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann's conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development. Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws—PDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, ...
A View to a Death in the Morning
What brought the ape out of the trees, and so the man out of the ape, was a taste for blood. This is how the story went, when a few fossils found in Africa in the 1920s seemed to point to hunting as the first human activity among our simian forebears—the force behind our upright posture, skill with tools, domestic arrangements, and warlike ways. Why, on such slim evidence, did the theory take hold? In this engrossing book Matt Cartmill searches out the origins, and the strange allure, of the myth of Man the Hunter. An exhilarating foray into cultural history, A View to a Death in the Morning shows us how hunting has figured in the western imagination from the myth of Artemis to the tale of...
Reflection Groups and Invariant Theory
Reflection groups and invariant theory is a branch of mathematics that lies at the intersection between geometry and algebra. The book contains a deep and elegant theory, evolved from various graduate courses given by the author over the past 10 years.
The Reflection of Life
A. H. Louie’s The Reflection of Life: Functional Entailment and Imminence in Relational Biology is a continuation of the exploratory journey in relational biology which began with his 2009 monograph More Than Life Itself: A Synthetic Continuation in Relational Biology. The theme of his first book was ‘What is life?’; the theme of this sequel is “How do two life forms interact?” Biology is a subject concerned with organization of relations. Relational biology is the approach that advocates ‘function dictates structure”, rather than ‘structure implies function’. It is mathematics decoded into biological realizations. The book demonstrates some of the powers of the ap...
