Cohomology ring of arrangement complements

The aim of this talk is to provide a uniform and intuitive description of the cohomology ring of arrangement complements.
We introduce complex hyperplane arrangements and state the Orlik-Solomon theorem (1980). Then, we describe the real case and the Gelfand-Varchenko ring (1987). We define toric arrangements and present their cohomology ring (De Concini, Procesi 2005 and Callegaro, D'Adderio, Delucchi, Migliorini and P. 2020).
Finally, we show a new technique to prove the Orlik-Solomon and De Concini-Procesi relations from the Gelfand-Varchenko ring. The technique applied to abelian arrangements provides a presentation of their cohomology. This is a work in progress with Evienia Bazzocchi e Maddalena Pismataro.