This is a systematic and well-paced introduction to mathematical logic. Excellent as a course text, the book presupposes only elementary background and can be used also for self-study by more ambitious students.
Starting with the basics of set theory, induction and computability, it covers propositional and first-order logic ? their syntax, reasoning systems and semantics. Soundness and completeness results for Hilbert's and Gentzen's systems are presented, along with simple decidability arguments. The general applicability of various concepts and techniques is demonstrated by highlighting their consistent reuse in different contexts.
Unlike in most comparable texts, presentation of syntactic reasoning systems precedes the semantic explanations. The simplicity of syntactic constructions and rules ? of a high, though often neglected, pedagogical value ? aids students in approaching more complex semantic issues. This order of presentation also brings forth the relative independence of syntax from the semantics, helping to appreciate the importance of the purely symbolic systems, like those underlying computers.
An overview of the history of logic precedes the main text, while informal analogies precede introduction of most central concepts. These informal aspects are kept clearly apart from the technical ones. Together, they form a unique text which may be appreciated equally by lecturers and students occupied with mathematical precision, as well as those interested in the relations of logical formalisms to the problems of computability and the philosophy of logic.
A History of Logic:
Patterns of Reasoning
A Language and Its Meaning
A Symbolic Language
1850?1950 ? Mathematical Logic
Modern Symbolic Logic
Summary
Elements of Set Theory:
Sets, Functions, Relations
Induction
Turing Machines:
Computability and Decidability
Propositional Logic:
Syntax and Proof Systems
Semantics of PL
Soundness and Completeness
First-Order Logic:
Syntax and Proof Systems of FOL
Semantics of FOL
More Semantics
Soundness and Completeness
Why is First Order Logic gFirst Orderh?
Readership: Undergraduates learning logic, lecturers teaching logic, any professionals who are non-experts in the subject but wish to learn and understand more about logic.
280pp Pub. date: Dec 2011
ISBN: 978-981-4343-86-2
ISBN: 978-981-4343-87-9(pbk)
This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimura?Taniyama conjecture, is given. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction.
In this new second edition, a detailed description of Barsotti?Tate groups (including formal Lie groups) is added to Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated at the end of Chapter 2 (in order to make the proof of regularity of the moduli of elliptic curve more conceptual), and in Chapter 4, though limited to ordinary cases, newly incorporated are Ribet's theorem of full image of modular p-adic Galois representation and its generalization to ebigf ©-adic Galois representations under mild assumptions (a new result of the author). Though some of the striking developments described above is out of the scope of this introductory book, the author gives a taste of present day research in the area of Number Theory at the very end of the book (giving a good account of modularity theory of abelian -varieties and -curves).
An Algebro-Geometric Tool Box
Elliptic Curves
Geometric Modular Forms
Jacobians and Galois Representations
Modularity Problems
468pp (approx.) Pub. date: Dec 2011
ISBN: 978-981-4368-64-3
Louis Kauffman was born in 1945. He graduated as valedictorian of his class at Norwood Norfolk Central High School in 1962. He received his BS at MIT in 1966 and his PhD in mathematics from Princeton University in 1972. Kauffman has been a prominent leader in Knot Theory, one of the most active research areas in mathematics today. His discoveries include a state sum model for the Alexander?Conway Polynomial, the bracket state sum model for the Jones polynomial, the Kauffman polynomial and Virtual Knot Theory. He is the Editor-in-Chief of JKTR, Editor of the Series on Knots and Everything, full professor at UIC and author of numerous books related to the theory of knots ? including gKnots and Physicsh, gKnots and Applicationsh, gOn Knotsh, and gFormal Knot Theoryh.
This invaluable book is an introduction to knot and link invariants as generalized amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas.
The book is divided into two parts: Part I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II includes topics such as frictional properties of knots, relations with combinatorics, and knots in dynamical systems.
In this new edition, articles on other topics, including Khovanov Homology, have been included.
Physical Knots
States and the Bracket Polynomial
The Jones Polynomial and Its Generalizations
Braids and the Jones Polynomial
Formal Feynman Diagrams, Bracket as a Vacuum-Vacuum Expectation and the Quantum Group SL(2)q
Yang?Baxter Models for Specializations of the Homfly Polynomial
Knot-Crystals ? Classical Knot Theory in a Modern Guise
The Kauffman Polynomial
Three Manifold Invariants from the Jones Polynomial
Integral Heuristics and Witten's Invariants
The Chromatic Polynomial
The Potts Model and the Dichromatic Polynomial
The Penrose Theory of Spin Networks
Knots and Strings ? Knotted Strings
DNA and Quantum Field Theory
Knots in Dynamical Systems ? The Lorenz Attractor
and selected papers
Readership: Physicists and mathematicians.
800pp (approx.) Pub. date: Scheduled Fall 2012
ISBN: 978-981-4383-00-4
ISBN: 978-981-4383-01-1(pbk)