noindent Outline of the course: 1) Brief historical introduction to RMT: 1applications. vspace{-8pt} begin{itemize} addtolength{itemsep}{-8pt} item[-] Discussion of basic properties of matrices, different random matrix ensembles, rotationally invariant ensembles such as Gaussian ensembles etc. end{itemize} vspace{-8pt} 2) Gaussian ensembles: derivation of the joint probability distribution of eigenvalues, starting from the joint distribution of matrix entries. 3) Analysis of the spectral properties of eigenvalues: given the joint distribution of eigenvalues, how to calculate various observables such as: vspace{-8pt} begin{itemize} addtolength{itemsep}{-8pt} item[-] Average density of eigenvalues — Wigner semi-circle law item[-] Counting statistics, spacings between eigenvalues etc item[-] Distribution of the extreme (maximum or minimum eigenvalues) end{itemize} vspace{-8pt} 4) Two complementary approaches to study spectral statistics: vspace{-8pt} begin{itemize} addtolength{itemsep}{-8pt} item[-] Large N (for an NxN matrix) method by the Coulomb gas approach: saddle point method item[-] Finite N method: for Gaussian unitary ensemble: orthogonal polynomial method (essentially quantum mechanics of free fermions at zero temperature). end{itemize} vspace{-8pt} 5) Tracy-Widom distribution: prob. distribution of the top eigenvalue. Its appearance in a large number of problems, universality and an associated third order phase transition. 6) Perspectives and summary.

LPTMS