Grant: Discovery Grants Program - Individual

My research mainly concerns certain geometric objects, called complex homogeneous manifolds, that display a large degree of symmetry. In the nineteenth century, the physicist, H. v. Helmholtz, made a simplifying assumption and considered spaces for which any two points have neighbourhoods that are basically equivalent, i.e., spaces that are homogeneous. Examples of such spaces abound in mathematics, e.g., spheres (under rotations), planes in 3-space (under translation), sets of vector subspaces of a fixed dimension of a given vector space (one can transform one such object into another by multiplying by a suitable matrix), etc. The basic object of our studies are complex manifolds (spaces modelled locally on complex vector spaces) that possess enough global transformations to map any given point in the space to any other point. The set of all such symmetries forms a group and we mostly restrict to settings where this is a Lie group. There is an extensive Lie theory, started by S. Lie in the 1870's and advanced by Killing, E. Cartan, H. Weyl, and many others since then. Along with some basic tools from complex function theory in several variables and algebraic topology and algebraic geometry, we can elucidate the structure of complex homogeneous manifolds that satisfy certain rather natural assumptions. For example, we would like to understand what are the obstructions for the existence of complex objects on the spaces in question, e.g., certain sorts of functions called holomorphic.