Duality (mathematics)In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings.
Large deformation diffeomorphic metric mappingLarge deformation diffeomorphic metric mapping (LDDMM) is a specific suite of algorithms used for diffeomorphic mapping and manipulating dense imagery based on diffeomorphic metric mapping within the academic discipline of computational anatomy, to be distinguished from its precursor based on diffeomorphic mapping. The distinction between the two is that diffeomorphic metric maps satisfy the property that the length associated to their flow away from the identity induces a metric on the group of diffeomorphisms, which in turn induces a metric on the orbit of shapes and forms within the field of Computational Anatomy.
Poincaré dualityIn mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the ()th homology group of M, for all integers k Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.
