Hodge Theory, Dualities and Non-Commutative Geometry

点击翻译按钮获取中文摘要

This is a research in the field of algebraic geometry - a classicalsubject studying the solutions to systems of polynomial equations. Thecurrent project addresses four problems providing exciting interfacesbetween complex geometry and string theory and quantum physics. Thefirst problem aims to build a toolbox of Hodge theoretic techniques forcomputing invariants of complex projective varieties related to higherhomotopy. The second explores the limits of the classical Shafarevichuniformization conjecture in complex geometry. The third problemproposes an explicit construction of sheaves on a gerby Calabi-Yaumanifold which is fibered by K3 surfaces, and studies its impact on thephysical notion of non-commutative deformation of fields. The fourthproject probes the relationship between the extremal transitions indegenerating families of Calabi-Yau manifolds, the limiting behavior ofintegrable systems, and non-commutative algebraic geometry.The understanding of these questions is essential for unifying variouslinearization procedures in algebraic geometry, symplectic topology,theoretical and mathematical physics. The project sets the stage forunderstanding the basic structure of algebraic varieties in a waysuitable for pragmatic use in a broad spectrum of applications. Thework proposed will be immediately relevant to deep questions incategory theory, combinatorial group theory, the theory of integrablesystems, string theory, quantum gravity and cosmology. The projectproposes concrete interdisciplinary applications to matrix quantummechanics, string dualities and particle physics model building.