Topics in Physical Mathematics

As many readers know, the 20th century was a time when the fields of science and mathematics were seen as two separate entities. Physical Topics in Mathematics demonstrates how various physical theories have played a crucial role in the developments of mathematics and geometric topology.

As many readers will know, the 20th century was a time when the fields of mathematics and the sciences were seen as two separate entities. Caused by the rapid growth of the physical sciences and an increasing abstraction in mathematical research, each party, physicists and mathematicians alike, suffered a misconception; not only of the opposition's theoretical underpinning, but of how the two subjects could be intertwined and effectively utilized. One sub-discipline that played a part in the union of the two subjects is Theoretical Physics. Breaking it down further came the fundamental theories, Relativity and Quantum theory, and later on Yang-Mills theory. Other areas to emerge in this area are those derived from the works of Donaldson, Chern-Simons, Floer-Fukaya, and Seiberg-Witten. Aimed at a wide audience, Physical Topics in Mathematics demonstrates how various physical theories have played a crucial role in the developments of Mathematics and in particular, Geometric Topology. Issues are studied in great detail, and the book steadfastly covers the background of both Mathematics and Theoretical Physics in an effort to bring the reader to a deeper understanding of their interaction. Whilst the world of Theoretical Physics and Mathematics is boundless; it is not the intention of this book to cover its enormity. Instead, it seeks to lead the reader through the world of Physical Mathematics; leaving them with a choice of which realm they wish to visit next.

A self-contained book that includes the basic material in physics and mathematics, prior to a discussion of their interaction Demonstrates how various physical theories have played a crucial role in recent developments in mathematics, in particular, in geometric topology Theories that are not yet accepted as physical theories are used, when they have led to new ideas and results in mathematics Includes supplementary material: sn.pub/extras

Texte du rabat
The roots of 'physical mathematics' can be traced back to the very beginning of man's attempts to understand nature. Indeed, mathematics and physics were part of what was called natural philosophy. Rapid growth of the physical sciences, aided by technological progress and increasing abstraction in mathematical research, caused a separation of the sciences and mathematics in the 20th century. Physicists' methods were often rejected by mathematicians as imprecise, and mathematicians' approach to physical theories was not understood by the physicists. However, two fundamental physical theories, relativity and quantum theory, influenced new developments in geometry, functional analysis and group theory. The relation of Yang-Mills theory to the theory of connections in a fiber bundle discovered in the early 1980s has paid rich dividends to the geometric topology of low dimensional manifolds. Aimed at a wide audience, this self-contained book includes a detailed background from both mathematics and theoretical physics to enable a deeper understanding of the role that physical theories play in mathematics. Whilst the field continues to expand rapidly, it is not the intention of this book to cover its enormity. Instead, it seeks to lead the reader to their next point of exploration in this vast and exciting landscape.

Contenu
Algebra.- Topology.- Manifolds.- Bundles and Connections.- Characteristic Classes.- Theory of Fields, I: Classical.- Theory of Fields, II: Quantum and Topological.- YangMillsHiggs Fields.- 4-Manifold Invariants.- 3-Manifold Invariants.- Knot and Link Invariants.