June 19–June 30

Research Group:

  • Research Leader: Jonathan Mboyo Esole
  • Researcher: Saber Ahmed
  • Researcher: Dwight Anderson Williams II

Research topic: Elliptic fibrations, flops, and collisions of singularities

Description: Elliptic fibrations are beautiful and elegant geometries at the interface of algebraic geometry,number theory, and string theory. While ubiquitous, they still have many secrets to teach us. In particular, their uses by physicists indicate deep connections of their birational geometry with representation theory and hyperplane arrangements of weight systems. The geometric singular fibers over generic points of the discriminant locus of a smooth elliptic fibration are famously classified by Kodaira and Neron. They have dual graphs corresponding to twisted affine Dynkin diagrams. Under mild conditions, an elliptic fibration is birationally equivalent to a (singular) Weierstrass model. Two different crepant resolutions are connected by a sequence of flops and have some matching topological invariants, such as their Euler characteristics. We are interested in the structure of crepant resolutions of Weierstrass models of elliptic fibrations corresponding to given types of Kodaira fibers. We will explore their flops and compute their topological invariants.