In this seminar, we will explore three significant concepts in modern mathematical physics and algebraic geometry: quantum cohomology, Frobenius manifolds, and derived categories.

Quantum cohomology extends classical cohomology theories by incorporating quantum corrections derived from enumerative geometry, providing deep insights into the intersection theory of moduli spaces. Quantum cohomology defines a rich symplectic invariant of a smooth projective variety, and conjecturally, it also governs its topology and complex geometry.

Frobenius manifolds, originally introduced by B. Dubrovin, provide a rich algebraic structure that encapsulates the solutions to certain integrable systems and links to the deformation theory of complex structures. We will discuss the definition of Frobenius manifolds, their properties, and their connections to both quantum cohomology and mirror symmetry.

Derived categories offer a framework for homological algebra that extends the classical categories of sheaves, enabling a deeper understanding of the geometry and topology of spaces. We will review the basics of derived categories, including their construction and fundamental properties.

Throughout the seminar, we will discuss open conjectural relationships between these structures and present some results of the speaker in this regard.