Geometry of Non-abelian Hodge Structures

点击翻译按钮获取中文摘要

Tony Pantev uses techniques from both the abelian and non-abelian Hodge theory to approach concrete geometric questions as wellas problems in cosmology, mathematical physics and string theory. The investigator and his collaborators study four problems. The first onedescribes a conjectural Hodge theoretic criterion for deciding whethera given homotopy type underlies a smooth projective variety. Thesecond concerns the symplectic topology of four manifolds, fiberedover the two sphere. The third projectdesigns an explicit construction of sheaves on a gerby K3 surfaceand discusses its relations to the physical notion of non-commutativedeformation of fields. The fourth project proposes an explicitconstruction of the special coordinates on the moduli space ofEucledian $D$-branes by means of secondary Abel-Jacobi maps.The understanding of these questions is essential for unifying variouslinearization 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 explicitmodels of string theory vacua.This is a research in the field of algebraic geometry. Algebraicgeometry is not only one of the most intensively researched partsof modern mathematics, but also a crucial testing ground for therecent influx of revolutionary mathematical ideas coming from stringtheory physics. The investigator's research sets the stage for arigorous treatment of the hard conjectures generated by ourcurrent understanding of string dualities and enlarges the arsenal ofavailable tools for understanding the physics of strings. Theinvestigator's methods are applicable to a wide range of concreteproblems in astrophysics, quantum gravity and string theory.