### 1. Introduction to the extreme value statistics (EVS): statistics of the maximum (or minimum) of a set of random variables

EVS for uncorrelated random variables: the three limiting
distributions (Gumbel, Frechet and Weibull). Simple examples of EVS
for correlated random variables: (i) Weakly Correlated:
Ornstein-Uhlenbeck process (particle moving in a harmonic potential),
limiting distribution of the maximum is the same as that of
uncorrelated variables; (ii) Strongly Correlated: Brownian motion -
the limiting distribution is half-Gaussian: different from that of the
uncorrelated case.

### 2. EVS for two strongly correlated systems with many applications

Variety of constrained Brownian motions. Random Matrices.

### 3. Distribution of maximum for constrained Brownian motions:

Path-integral (Feynman-Kac) computation of the maximum distribution.
Application to: (i) Brownian Bridge; (ii) Brownian Excursion; (iii)
Brownian Meander; (iv) Fluctuating (1+1)-dimensional Interfaces:
Airy distribution function (applications in computer science).

### 4. Large deviations and rare events in random matrices

Large deviations of the maximum eigenvalue: probability of rare
fluctuations. Wishart radom matrices: average density of states
(Marcenko-Pastur law), maximum eigenvalue (Tracy-Widom again),
minimum eigenvalue (when different from Tracy-Widom), application to
Quantum Entanglement problem.

### 5. Ubiquity of the Tracy-Widom law:

Longest increasing subsequence problem. Hamersley process -
interacting particle systems. Directed polymer. (2+1)-dimensional
directed percolation. Sequence matching problem. Maximum Agreement
SubTree problem in Phylogeny. Etc.