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The Story of Proof
How the concept of proof has enabled the creation of mathematical knowledge The Story of Proof investigates the evolution of the concept of proof—one of the most significant and defining features of mathematical thought—through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival g...
Complexes Associated to Two Vectors and a Rectangular Matrix
This book is intended for graduate student and research mathematicians interested in commutative rings and algebras.
Frames, Bases and Group Representations
This work develops an operator-theoretic approach to discrete frame theory on a separable Hilbert space. It is then applied to an investigation of the structural properties of systems of unitary operators on Hilbert space which are related to orthonormal wavelet theory. Also obtained are applications of frame theory to group representations, and of the theory of abstract unitary systems to frames generated by Gabor type systems.
Rational Homotopical Models and Uniqueness
The main goal of this paper is to prove the following conjecture of Baues and Lemaire: the differential graded Lie Tlgebra associated with the Sullivan model of a space is homotopy equivalent to its Quillen model. In addition we show the same for the cellular Lie algebra model which we build from the simplicial analog of the classical Adams-Hilton model. It turns out that this cellular Lie algebra model is one link in a chain of models connecting the models of Quillen and Sullivan.The key result which makes all this possible is Anick's correspondence between differential graded Lie algebras and Hopf algebras up to homotopy. In addition we show that the Quillen model is a rational homotopical equivalence, and we conclude the same for the other models using our main result. Theconstruction of the three models is given in detail. The background from homotopy theory, differential algebra, and algebra is presented in great generality.
Iwasawa Theory and Its Perspective, Volume 2
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to $p$-adic $L$-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory. The goal of this second part of the three-part publication is to explain various aspects of the cyclotomic Iwasawa theory of $p$-adic Galois representations.
Modular Forms: A Classical And Computational Introduction (2nd Edition)
Modular Forms is a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to various subjects, such as the theory of quadratic forms, the proof of Fermat's Last Theorem and the approximation of π. The text gives a balanced overview of both the theoretical and computational sides of its subject, allowing a variety of courses to be taught from it.This second edition has been revised and updated. New material on the future of modular forms as well as a chapter about longer-form projects for students has also been added.
The Defect Relation of Meromorphic Maps on Parabolic Manifolds
This book is intended for graduate students and research mathematicians working in several complex variables and analytic spaces.
Number Theory
Monumental proceedings (very handsomely produced) of a major international conference. The book contains 74 refereed articles which, apart from a few survey papers of peculiar interest, are mostly research papers (63 in English, 11 in French). The topics covered reflect the full diversity of the current trends and activities in modern number theory: elementary, algebraic and analytic number theory; constructive (computational) number theory; elliptic curves and modular forms; arithmetical geometry; transcendence; quadratic forms; coding theory. (NW) Annotation copyrighted by Book News, Inc., Portland, OR
Iterated Function Systems and Permutation Representations of the Cuntz Algebra
This book is intended for graduate students and research mathematicians working in functional analysis.
