Moduli spaces are geometric spaces whose points parametrize other mathematical objects, and they provide a useful way of studying families of objects simultaneously. In this talk, I will discuss moduli spaces of semistable sheaves on bielliptic surfaces, a class of complex algebraic surfaces closely related to products of elliptic curves.

I will first introduce the basic geometric ideas behind these moduli spaces and explain how, for sheaves supported on curves, they can be viewed as families of compactified Jacobians associated with a linear system of curves on the surface. I will then describe some of the topological properties of these spaces obtained in my recent work, including computations of their first two Betti numbers and information about their fundamental groups. Finally, I will discuss the fibers of their Albanese maps and explain how, in certain cases, these fibers are closely related to Calabi–Yau varieties.

No prior knowledge of moduli spaces or algebraic geometry beyond basic familiarity with complex geometry will be assumed.