In the last years there have been great advances in the
applications of topology and differential geometry to problems in
condensed matter physics. Concepts drawn from topology and geometry
have become essential to the understanding of several phenomena in
the area. Physicists have been creative in producing models for
actual physical phenomena which realize mathematically exotic
concepts and new phases have been discovered in condensed matter in
which topology plays a leading role. An important classification
paradigm is the concept of topological order, where the state
characterizing a system does not break any symmetry, but it defines
a topological phase in the sense that certain fundamental
properties change only when the system passes through a quantum
phase transition. The main purpose of this book is to provide a
brief, self-contained introduction to some mathematical ideas and
methods from differential geometry and topology, and to show a few
applications in condensed matter. It conveys to physicists the
basis for many mathematical concepts, avoiding the detailed
formality of most textbooks.
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