Topology of Symplectic 4-Manifolds

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This research project investigates the properties of symplectic manifolds. Symplectic structures arise naturally in classical mechanics, the branch of physics that describes the motion of macroscopic objects, and these mathematical structures underlie the fundamental equations of classical and quantum physics. The project aims to gain new understanding of the general shape of symplectic four-manifolds using a variety of techniques from various branches of mathematics, such as topology, algebraic geometry, and differential geometry. The research concerns several aspects of four-dimensional topology and geometry. Some of the projects on closed four-manifolds are: defining smooth and symplectic invariants and studying their properties; classifying symplectic Calabi-Yau four-manifolds and classifying configurations of Lagrangian and symplectic surfaces in rational and ruled manifolds; and investigating smooth and symplectic mapping class groups and finite symmetries. For four-manifolds with boundary, the project investigates special symplectic caps, including unruled caps and Calabi-Yau caps, with an eye towards understanding various types of symplectic fillings. A new aspect of the work is investigation of concave symplectic four-manifolds as structures of independent interest; in particular, the project aims to show that concave symplectic four-manifolds share many properties with closed symplectic four-manifolds.