Noncommutative Potential Theory (F. Cipriani)

In the last two decades classical Potential Theory has been extended to encompass themes and problems pertaining Noncommutative Geometry, as created by A. Connes, and Free Probability, invented by D.V. Voiculescu. A canonical differential calculus, underlying Dirichlet energy forms, has been developed to study the main themes of the theory, namely, potentials, finite energy states and multipliers on von Neumann and C*-algebras.

Applications to other fields of mathematics include the Hodge-de Rham theory and K-theory on post critically finite fractals, spectral characterizations of amenability and Haagerup property of von Neumann algebras, factoriality and rigidity of von Neumann algebras, construction of Levy processes and Spectral Triples on compact Quantum Groups (à la Woronowicz) and Logarithmic Sobolev Inequalities in continuous Quantum Statistical Mechanics.

Spectral Theory of Vector Operators and Applications (F. Colombo)

The theory of holomorphic functions has several applications in various fields of science and technology. Using the Cauchy formula for holomorphic functions it is possible to define the so called Riesz-Dunford functional calculus. Such calculus allows to define functions of operators such as the fractional powers of a linear operator like the Laplace operator.

One of the reasons to replace the Laplace operator in the heat equation, with fractional powers of the Laplace operator is that this new equation produce finite speed propagation of the heat. This is equivalent in some cases to a modification of the Fourier law.

Such modification can be generalized also for more general vector operators using a new functional calculus, the so called S-functional calculus, based on the notion of S-spectrum, that has been developed in the last ten years.

We propose the study of new classes of fractional diffusion problems.

Politecnico di Milano - Dipartimento di Matematica