Geometric phases are finding an important role in applied mathematics and engineering via mathematical control theory. Two striking examples are in the description of the kinematics of a self deforming body such as a satellite with robotic appendages (Shapere and Wilczek 1989, Montgomery 1990), and the swimming motions of swimming microorganisms (Shapere and Wilczek 1989). The formulation is very natural, and the geometric quantities obtained are useful for both the quantitative, and qualitative description of the dynamics.; In 1952, J. Lighthill showed that a sphere can swim at low Reynolds number by passing axially symmetric waves along its surface (Lighthill 1952). In 1971, J. Blake revisited Lighthill's model for the sphere in the context of ciliary propulsion (Blake 1971a). He also showed that an infinitely long circular cylinder can propel itself by passing axially symmetric waves down its cross section (Blake 1971b). In 1989, A. Shapere and F. Wilczek proposed a gauge theoretic model for low Reynolds number swimming that allowed for the generalization of the results of Blake and Lighthill to include the rotations and translations due to arbitrary small amplitude boundary motions of spheres and circular cylinders (Shapere and Wilczek 1989). The main tool for predicting swimming velocities in Shapere and Wilczek's model is the curvature of the Stokes connection.; In this thesis, we calculate the Stokes curvature for two biologically relevant shapes, the elliptical cylinder, and prolate spheroid. We then apply the formalism of Shapere and Wilczek (1989) to an analogous problem of transferring fluid using a deformable membrane. We define a new Stokes connection for this problem. We obtain a model for the low Reynolds number peristaltic pump, and use it to calculate pumping rates for arbitrary small amplitude pumping motions of a cylindrical tube.
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