Geometry of Non-abelian Hodge Structures

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

Tony Pantev uses techniques from both the abelian and non-abelian Hodge theory to approach concrete geometric questions as well as problems in cosmology, mathematical physics and string theory. The investigator and his collaborators study four problems. The first one describes a conjectural Hodge theoretic criterion for deciding whether a given homotopy type underlies a smooth projective variety. The second concerns the symplectic topology of four manifolds, fibered over the two sphere. The third project designs an explicit construction of sheaves on a gerby K3 surface and discusses its relations to the physical notion of non-commutative deformation of fields. The fourth project proposes an explicit construction of the special coordinates on the moduli space of Eucledian $D$-branes by means of secondary Abel-Jacobi maps. The understanding of these questions is essential for unifying various linearization procedures in algebraic geometry, symplectic topology and mathematical physics. It brings us closer to understanding the basic structure of algebraic varieties and enhances the geometric arsenal of techniques for building explicit models of string theory vacua. This is a research in the field of algebraic geometry. Algebraic geometry is not only one of the most intensively researched parts of modern mathematics, but also a crucial testing ground for the recent influx of revolutionary mathematical ideas coming from string theory physics. The investigator's research sets the stage for a rigorous treatment of the hard conjectures generated by our current understanding of string dualities and enlarges the arsenal of available tools for understanding the physics of strings. The investigator's methods are applicable to a wide range of concrete problems in astrophysics, quantum gravity and string theory.