Superconductivity is a collective state of matter displaying remarkable properties. For example,
superconductors are characterized by a vanishing static electrical resistivity and a
perfect diamagnetism, leading to the expulsion of a magnetic eld from the interior of a sample.
These properties originate from the "super
uid" nature of a superconductor: despite
being fermions, in a superconducting material a large fraction of the conduction electrons
condense in a collective state which extends over the whole volume and is able to move as
a whole. Thus, superconductivity is a striking example of a quantum phenomenon occuring
on a macroscopic scale. At zero temperature the condensation is complete, even if only
electrons near the Fermi surface mostly contribute to the properties of the condensate.
It is the scope of this lecture to describe the phenomenon of superconductivity from a
microscopic point of view. In the fi rst part we shall focus on classical aspects of superconductivity,
like the microscopic theory of Bardeen, Cooper and Schrie er (BCS) and its
explanation of the general features of bulk superconductors. In a second part we shall move
to more exotic forms of superconductivity found typically in low dimensional systems, e.g.
topological and nodal superconductivity. Finally, recent applications, especially promising in the context
of quantum information technology with superconducting platforms, will be mentioned.
The course assumes knowledge of the standard material from electrodynamics, quantum
mechanics I, and quantum statistics. The second-quantization formalism, usually introduced
in quantum mechanics II, will be used. Basic knowledge of solid state physics is useful.