The notion of symmetry is of fundamental importance in geometry and physics. Mathematically, symmetries are described by Lie groups, or, more generally, Lie groupoids. The latter notion generalizes the former, playing a special role in the study of geometric structures with singularities, in which case the local symmetries may vary from one point to another. The importance of symmetries in geometry is illustrated by their role in symplectic reduction, a key operation in symplectic geometry. In physical terms, this procedure amounts to the reduction of the number of degrees of freedom of a mechanical system. Another central concept associated with geometric structures and of extreme relevance in physics is that of a "deformation". Deformation theory of mathematical structures has its roots in the work of Kodaira in complex geometry, but its modern formulations provide powerful tools for the study of various problems in mathematics and mathematical physics, including stability and quantization of classical mechanical systems. In the context of deformation theory, a crucial role is played by "rigid" mathematical structures (i.e., those that cannot be deformed) as well as the interactions between deformations and symmetries. For example, the presence of rigid symmetries (e.g. compact semisimple Lie groups) often forces the rigidity of the system. The present research project is centered around several fundamental questions concerning deformations and symmetries, including their relationship. We plan to work mainly on problems of geometric nature, although methods of algebraic topology such as K-theory, operads and cyclic homology will be necessary. Special attention will be given to Poisson geometry and its relationship to mathematical physics since the most fundamental problems in this field involve both symmetries and deformations. |