Aspetti Algebrici e Combinatori della Teoria degli Operad

In my thesis, I deal with operad theory, introducing it with the classical definition of operad in the symmetric monoidal category of k-modules, with k a commutative ring, and later giving what is called its partial definition, demonstrating their equivalence. I present some preliminary notions, such as non-unital operads and non-symmetric operads, with relating examples. I bring up the concepts of algebra over an operad, module over an algebra over an operad, module over an operad, explaining the connections occurring between these objects and referring to some specific cases, recovering, for example, the classical definitions of associative algebra and module over an associative algebra. I then pave the way for the close link between operads and the concept of tree in graph theory, showing the notion of free operad, whose construction leads to a new definition of operad, also known as combinatorial definition, equivalent to the others, which makes use of the concept of algebra over a triple. By generalizing this definition, I bring up other types of objects related to operads, such as cyclic operads, PROPs, properads, dioperads, providing pertaining examples. Subsequently, I present some operads intrinsically linked to trees, such as the dendriform operad, the pre-Lie operad, the duplicial operad, the non-associative permutative operad. I then broach the concept of P-group associated with a certain operad P, giving some examples of these groups' elements for the operads mentioned above. I conclude with an example of a combinatorial problem, the solution of which may be found using operads as tools to enumerate families of combinatorial objects. Finally, I deal with the concept of colored operad, specifying its principal notions and providing an interesting example of a colored operad that can be seen as a model for wiring diagrams and electric circuits.