Dennis DeTurck

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To Manfredo do Carmo in friendship and admiration, on his 80 th birthday Abstract. Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one(More)
The writhing number of a curve in Euclidean 3-space, introduced by C˘ alug˘ areanu (1959-61) and named by Fuller (1971), is the standard measure of the extent to which the curve wraps and coils around itself; it has proved its importance for molecular biologists in the study of knotted duplex DNA and of the enzymes which affect it. The helicity of a vector(More)
The helicity of a smooth vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid mechanics, magnetohy-drodynamics and plasma physics. In this report we show how the relation between energy and helicity of a vector field is influenced by the(More)
This paper presents a mathematically complete derivation of the minimum-energy divergence-free vector fields of fixed helicity, defined on and tangent to the boundary of solid balls and spherical shells. These fields satisfy the equation ∇×V = λV , where λ is the eigenvalue of curl having smallest non-zero absolute value among such fields. It is shown that(More)
Acknowledgments I would like to thank James Haglund, who embodies all the characteristics of a great advisor and a great mathematician. He made the difficult task of writing a dissertation an enjoyable experience. Many other people from the Penn mathematics department, including helped to make this dissertation possible. Thanks also to the mathematics(More)
Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibers as(More)