The use of differential geometry to understand topological properties precedes the formal establishment of Algebraic Topology as a mathematical discipline and has been a central theme in mathematics since the early 20th Century.
The goals of this project fall in this tradition, in that we propose to study a variety of geometric objects and their topological properties using tools from Differential Geometry and Analysis. The main areas of study are the following:
- Foliations of singular spaces. The main objective is to generalize the theory for regular spaces by using tools of non-commutative geometry.
- Holonomy and Lie algebroids. A central question is the extension of the notion of parallel transport to higher dimensions, beyond the case of curves.
- Topology and geometry of moduli spaces. Among the central objects of study are character varieties for surface groups. Through the holonomy representation and non-abelian Hodge theory these spaces are viewed as moduli spaces of gauge theoretic objects, known as Higgs bundles.