Ebook: Topics in physical mathematics

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.

This title adopts the view that physics is the primary driving force behind a number of developments in mathematics. Previously, science and mathematics were part of natural philosophy and many mathematical theories arose as a result of trying to understand natural phenomena. This situation changed at the beginning of last century as science and mathematics diverged. These two fields are collaborating once again; 'Topics in Mathematical Physics' takes the reader through this journey.

The author discusses topics where the interaction of physical and mathematical theories has led to new points of view and new results in mathematics. The area where this is most evident is that of geometric topology of low dimensional manifolds. These include the theories of Donaldson, Chern-Simons, Floer-Fukaya, Seiberg-Witten, and Topological (Quantum) Field Theory.

The author also discusses the interaction of CFT, Supersymmetry, String Theory and Gravity with diverse areas of mathematics. Several of these ideas have led to new insights into old mathematical structures and some have led to surprising new results The term "Physical Mathematics'' has been coined to describe collectively these new and fast growing areas of research, and regards the work of Donaldson and Witten as belonging to this new area of physical mathematics. Study of this work forms an important part of this book.