Geometric and algebraic reduction for singular momentum maps

Geometric and Algebraic Reduction for Singular Momentum Maps

This paper concerns the reduction of singular constraint sets of symplectic manifolds. It develops a “geometric” reduction procedure, as well as continues the work of Sniatycki and Patrick on reduction a la Dirac. The relationships among the Dirac, geometric, and Sniatycki-Weinstein algebraic reduction procedures are studied. Primary emphasis is placed on the case where the constraints are give...

Singular solutions, momentum maps and computational anatomy

This paper describes the variational formulation of template matching problems of computational anatomy (CA); introduces the EPDiff evolution equation in the context of an analogy between CA and fluid dynamics; discusses the singular solutions for the EPDiff equation and explains why these singular solutions exist (singular momentum map). Then it draws the consequences of EPDiff for outline mat...

Momentum Maps, Dual Pairs and Reduction in Deformation Quantization∗

This paper is a brief survey of momentum maps, dual pairs and reduction in deformation quantization. We recall the classical theory of momentum maps in Poisson geometry and present its quantum counterpart. We also discuss quantization of momentum maps and applications of quantum momentum maps to quantum versions of Marsden-Weinstein reduction. This paper is organized as follows. We recall the c...

Geometric representation of interval exchange maps over algebraic number fields

This paper is concerned with the restriction of interval exchange transformations (IETs) to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically, and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable IETs with zero and non-zero drift vector and carry out some investi...

Momentum Maps and