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The book covers recent hot topics of condensed matter physics, such as topological insulators and graphene, through a pedagogical approach. Starting from the treatment of energy bands in solids through advanced mathematical instruments, the transport properties of surface states in three-dimensional topological insulators and graphene are analyzed. The application of Dirac equation on curved space time to condensed matter systems is also treated, with the study of topological defects and local curvatures as possible applications. A study of the possible use of condensed matter system to the relevation of gravitational waves concludes the book, together with a detailed treatment of all the mathematical prerequisites needed to follow the calculations in the book.