Interactions between the topology and the geometry of algebraic varieties

This project is about solutions of systems of polynomial equations with complex numbers as coefficients. These solution sets are called complex algebraic varieties. We will study the interaction between the topology and the algebraic geometry of such objects. Topology views a complex algebraic variety as being equivalent to any continuous deformation of any of its triangulations. Algebraic geometry is more rigid and views an algebraic variety as being equivalent to any other one with a similar space of algebraic functions on it. We will study topological invariants such as the spaces of representations of the fundamental groups of algebraic varieties and their natural strata of special representations. This will shed light on how special the topology of a variety is. We will also study algebraic invariants such as the spaces of vector bundles on algebraic varieties and their natural strata of special vector bundles. The common thread will be the development of a widelyapplicable deformation theory. We will address the Monodromy Conjecture, one of the most interesting open problems at the intersection of topology, geometry, and arithmetic of algebraic varieties.