Abstract: The geometry and topology of the Quot schemes of rank zero quotients on smooth projective curves has been studied quite extensively in the works of Strømme (1981), Bifet (1989), Chen (2001), Bifet-Ghione-Letizia (1994), Marian-Oprea (2007), Marian-Negut (2023) and several others. The Quot scheme in this case is a smooth projective variety. The work of Marian-Negut describes the additive structure on the cohomology, which provides a natural way of computing Poincaré polynomials (these had been known before by the work of Bifet and Chen). In this seminar I would like to explain the multiplicative structure on the cohomology. One can give a satisfactory account of this for the Quot scheme over P^1. Such a description for a general curve is a work in progress; I will highlight some difficulties one encounters in this case. Finally, if time permits, I would like to talk about the connection of the Quot scheme on P^1 to the stack of bundles and use it to give a geometric description of certain relations appearing in the presentation of its cohomology ring.