This is a research in the field of algebraic geometry - a classical subject studying the solutions to systems of polynomial equations. The current project addresses four problems providing exciting interfaces between complex geometry and string theory and quantum physics. The first problem aims to build a toolbox of Hodge theoretic techniques for computing invariants of complex projective varieties related to higher homotopy. The second explores the limits of the classical Shafarevich uniformization conjecture in complex geometry. The third problem proposes an explicit construction of sheaves on a gerby Calabi-Yau manifold which is fibered by K3 surfaces, and studies its impact on the physical notion of non-commutative deformation of fields. The fourth project probes the relationship between the extremal transitions in degenerating families of Calabi-Yau manifolds, the limiting behavior of integrable systems, and non-commutative algebraic geometry.
The understanding of these questions is essential for unifying various linearization procedures in algebraic geometry, symplectic topology, theoretical and mathematical physics. The project sets the stage for understanding the basic structure of algebraic varieties in a way suitable for pragmatic use in a broad spectrum of applications. The work proposed will be immediately relevant to deep questions in category theory, combinatorial group theory, the theory of integrable systems, string theory, quantum gravity and cosmology. The project proposes concrete interdisciplinary applications to matrix quantum mechanics, string dualities and particle physics model building.
