Differential forms on Carnot groups

The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights, which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.

Active Translation into English and Italian of Hosseini's adaptation of the Persian epic,"Shahname"

This thesis proposes a translation from Persian into Italian and English of an ancient Persian epic called Shahname, or literally “The Book of Kings,” by Ferdosi, first published in the 11th century CE. The translation proposed, however, is not based on the original book by Ferdosi, which is written all in verse, but rather, an edited, shorter, and simplified version written in prose, by Mohamad Hosseini, first published in 2013. Nonetheless, in his version, Hosseini included some of the verses from the original poems in order to show the value and the beauty of Ferdosi’s writing. Many translations of Ferdosi’s book have been made into English, but only one translation has been made into Italian, by one Italo Pizzi, in 8 volumes, all in verse, in 1886. This thesis analyses and discusses the choices made for the two translations presented into English and Italian. My project is not only to propose translations of Hosseini’s version, but to also introduce the reader to the Persian culture, and to the life of the most famous Iranian epic writer, Ferdosi, and his masterpiece, Shahname.