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The interface between Physics and Mathematics has been increasingly
spotlighted by the discovery of algebraic, geometric, and
topological properties in physical phenomena. A profound example is
the relation of noncommutative geometry, arising from algebras in
mathematics, to the so-called quantum groups in the physical
viewpoint. Two apparently unrelated puzzles - the solubility of
some lattice models in statistical mechanics and the integrability
of differential equations for special problems - are encoded in a
common algebraic condition, the Yang-Baxter equation. This backdrop
motivates the subject of this book, which reveals Knot Theory as a
highly intuitive formalism that is intimately connected to Quantum
Field Theory and serves as a basis to String Theory.This book
presents a didactic approach to knots, braids, links, and
polynomial invariants which are powerful and developing techniques
that rise up to the challenges in String Theory, Quantum Field
Theory, and Statistical Physics. It introduces readers to Knot
Theory and its applications through formal and practical
(computational) methods, with clarity, completeness, and minimal
demand of requisite knowledge on the subject. As a result, advanced
undergraduates in Physics, Mathematics, or Engineering, will find
this book an excellent and self-contained guide to the algebraic,
geometric, and topological tools for advanced studies in
theoretical physics and mathematics.
The interface between Physics and Mathematics has been increasingly
spotlighted by the discovery of algebraic, geometric, and
topological properties in physical phenomena. A profound example is
the relation of noncommutative geometry, arising from algebras in
mathematics, to the so-called quantum groups in the physical
viewpoint. Two apparently unrelated puzzles - the solubility of
some lattice models in statistical mechanics and the integrability
of differential equations for special problems - are encoded in a
common algebraic condition, the Yang-Baxter equation. This backdrop
motivates the subject of this book, which reveals Knot Theory as a
highly intuitive formalism that is intimately connected to Quantum
Field Theory and serves as a basis to String Theory.This book
presents a didactic approach to knots, braids, links, and
polynomial invariants which are powerful and developing techniques
that rise up to the challenges in String Theory, Quantum Field
Theory, and Statistical Physics. It introduces readers to Knot
Theory and its applications through formal and practical
(computational) methods, with clarity, completeness, and minimal
demand of requisite knowledge on the subject. As a result, advanced
undergraduates in Physics, Mathematics, or Engineering, will find
this book an excellent and self-contained guide to the algebraic,
geometric, and topological tools for advanced studies in
theoretical physics and mathematics.
An in depth exploration of how Clifford algebras and spinors have
been sparking collaboration and bridging the gap between Physics
and Mathematics. This collaboration has been the consequence of a
growing awareness of the importance of algebraic and geometric
properties in many physical phenomena, and of the discovery of
common ground through various touch points: relating Clifford
algebras and the arising geometry to so-called spinors, and to
their three definitions (both from the mathematical and physical
viewpoint). The main points of contact are the representations of
Clifford algebras and the periodicity theorems. Clifford algebras
also constitute a highly intuitive formalism, having an intimate
relationship to quantum field theory. The text strives to
seamlessly combine these various viewpoints and is devoted to a
wider audience of both physicists and mathematicians. Among the
existing approaches to Clifford algebras and spinors this book is
unique in that it provides a didactical presentation of the topic
and is accessible to both students and researchers. It emphasizes
the formal character and the deep algebraic and geometric
completeness, and merges them with the physical applications.
This text explores how Clifford algebras and spinors have been
sparking a collaboration and bridging a gap between Physics and
Mathematics. This collaboration has been the consequence of a
growing awareness of the importance of algebraic and geometric
properties in many physical phenomena, and of the discovery of
common ground through various touch points: relating Clifford
algebras and the arising geometry to so-called spinors, and to
their three definitions (both from the mathematical and physical
viewpoint). The main point of contact are the representations of
Clifford algebras and the periodicity theorems. Clifford algebras
also constitute a highly intuitive formalism, having an intimate
relationship to quantum field theory. The text strives to
seamlessly combine these various viewpoints and is devoted to a
wider audience of both physicists and mathematicians. Among the
existing approaches to Clifford algebras and spinors this book is
unique in that it provides a didactical presentation of the topic
and is accessible to both students and researchers. It emphasizes
the formal character and the deep algebraic and geometric
completeness, and merges them with the physical applications. The
style is clear and precise, but not pedantic. The sole
pre-requisites is a course in Linear Algebra which most students of
Physics, Mathematics or Engineering will have covered as part of
their undergraduate studies.
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