The principal homogeneous space concept is a special case of that of principal bundle : it means a principal bundle with base a single point.
32.
In other words the local theory of principal bundles is that of a family of principal homogeneous spaces depending on some parameters in the base.
33.
The prize citation highlighted their work on stationary measures and closed orbits for non-abelian group actions on homogeneous spaces, solving a long-standing conjecture of Hillel Furstenberg.
34.
Harmonic analysis has also been extended from the analysis of functions on a group " G " to functions on homogeneous spaces for " G ".
35.
The geometric relationship of the sphere and group of operators is that the unit sphere is a homogeneous space for the unitary group " U ".
36.
Later, moving frames were developed extensively by �lie Cartan and others in the study of submanifolds of more general homogeneous spaces ( such as projective space ).
37.
This notion is often used with G being a Lie group and X a homogeneous space for G . Foundational examples are hyperbolic manifolds and affine manifolds.
38.
Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space.
39.
In collaboration with Fran�ois Labourie and Shahar Mozes, cocycle superrigidity ideas were applied to the basic problem of the existence of compact locally homogeneous spaces of certain types.
40.
He does research on harmonic analysis, group representation theory, analysis on LIe groups and homogeneous spaces, special functions, Fourier analysis, and mathematical applications to physics and multivariate statistics.
