Sala Conferenze (Puteano, Centro De Giorgi).
It is well known that the classical potential theory is based on the class of subharmonic functions and on the Laplace operator. The pluripotential theory, constructed in the 80s of the last century, is based on plurisubharmonic functions and on the Monge-Ampère operator. During 1976-90, intensive research was carried out in building the pluripotential theory: basic objects of the theory, such as extremal Green function, P−measure, pluripolar sets, capacity values, etc. have been introduced and studied, and the foundation of pluripotential theory was practically built. All the basic fundamental theorems of the theory have been identified and the method of their application has been developed. Most importantly, these studies have been used successfully in solving various problems that have accumulated in multidimensional complex analysis and in the theory of plurisubharmonic functions. Nowadays, this theory is one of the main subfield of complex analysis, being one of the basic technique of investigating the space of analytic functions of several variables. In the 1990s there were many attempts to develop and expand pluripotential theory to broader classes such as, the class of m−subharmonic functions (1 ≤ m ≤ n). In this talk we will discuss some of the most important results of the theory of m−subharmonic functions as well as the difficulties and problems of constructing a potential theory in the class of m-subharmonic functions.