Topology is a mathematical discipline which was born in the 18th century and knew a rigorous and impressive development in the last century, eventually becoming one of the great unifying ideas of mathematics. With the discovery of the quantum Hall effect in 1980, topology began to be used also in  condensed matter physics. The study of topologically non-trivial systems is nowadays extremely important both at the level of fundamental research and in view of promising applications which exploit the intrinsic robustness of topology. In this regard, topological protection is believed to be extremely relevant for the future development of quantum computation.

In this seminar, I will discuss the main ideas behind both the topological classification of  condensed matter systems and the quantum computation. I will then bring these two topics  together by introducing the so-called "Majorana zero modes". These exotic and topological excitations, which feature anyonic  exchange statistic, represent indeed the building block of the so-called "topological quantum computation" and are extensively studied both theoretically and experimentally.