This course presents fermion techniques applied to the study of random plane partitions or 3D Young diagrams. The main objects are Schur processes, namely random processes that can be described as the discrete time evolution of Young diagrams. The diagrams sweep a 3D discrete random stepped surface, whose geometry can be probed by solving the initial probabilistic model.

Plan of the course:

1. Combinatorics of Young diagrams: skew Young diagrams, Schur functions and Semi-Standard Young Tableaux, Gelfand-Zeitlin rule.
2. 3D Young diagrams as Schur processes: height function, correlations, probabilistic model.
3. Fermion representations as semi-infinite wedge products: Hilbert space, creation/annihilation operators, fermion bilinears and vertex operators, commutation relations.
4. Representing Skew Schur functions with vertex operators: Schur process partition function, correlation functions, existence of limit shapes.