This little book is based on a series of lectures given by its author at an NSF/CBMS Regional Conference in the Mathematical Sciences, 2009.

Its goal is to give a concise introduction to $K$-theory and the use of $K$-theory in the context of modern physics, in particular in string theories and their dualities. A central role in the book is played by $T$-duality which is presented in detail. After some introductory remarks on string theory the following topics are discussed (a collection from the table of content): A quick review of topological $K$-theory. $K$-theory and D-brane charges. $K$-homology and $D$-brane charges. A few basics of ${C}^{*}$ algebras and crossed products. Continuous trace algebras and twisted $K$-theory. The theory of gerbes. Connes’ Thom isomorphism. The Pimsner-Voiculescu sequence. The topology of $T$-duality and the Bunke-Schick construction. $T$-duality via crossed products. Higher-dimensional $T$-duality via topological methods. Higher-dimensional $T$-duality via ${C}^{*}$-algebraic methods. As more advanced topics, mirror symmetry and Fourier-Mukai duality are discussed.

The book introduces the necessary concepts in a very lively manner concentrating on essential aspects of the theory. The reviewer considers the book as a highly welcome introduction to a field of ongoing mathematical research. It gives an excellent overview of the methods and results. For the reader who wants to know more, further references are given.

##### MSC:

81T30 | String and superstring theories |

81T75 | Noncommutative geometry methods (quantum field theory) |

19K99 | $K$-theory and operator algebras |

46L80 | $K$-theory and operator algebras |

58B34 | Noncommutative geometry (á la Connes) |

55R10 | Fiber bundles |

55P65 | Homotopy functors |

55R50 | Stable classes of vector space bundles, $K$-theory |

14J32 | Calabi-Yau manifolds |

53Z05 | Applications of differential geometry to physics |