mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Seifert conjecture states that every nonsingular, continuous vector field on the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
has a closed orbit. It is named after
Herbert Seifert
Herbert Karl Johannes Seifert (; 27 May 1897, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology.
Biography
Seifert was born in Bernstadt auf dem Eigen, but soon moved to Bautzen, where he attend ...
. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase non-existence as a conjecture. He also established the conjecture for perturbations of the
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
.
The conjecture was disproven in 1974 by
Paul Schweitzer
Paul Alexander Schweitzer SJ (born July 21, 1937) is an American mathematician specializing in differential topology, geometric topology, and algebraic topology.
Schweitzer has done research on foliations, knot theory, and 3-manifolds. In 1974 h ...
, who exhibited a counterexample. Schweitzer's construction was then modified by
Jenny Harrison
Jenny Harrison is a professor of mathematics at the University of California, Berkeley.
Education and career
Harrison grew up in Tuscaloosa, Alabama. On graduating from the University of Alabama, she won a Marshall Scholarship which she used to ...
in 1988 to make a
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
for some . The existence of smoother counterexamples remained an open question until 1993 when
Krystyna Kuperberg
Krystyna M. Kuperberg (born ''Krystyna M. Trybulec''; 17 July 1944) is a Polish-American mathematician who currently works as a professor of mathematics at Auburn University, where she was formerly an Alumni Professor of Mathematics.A -smooth counterexample to the Hamiltonian Seifert conjecture in ', Ann. of Math. (2) 158 (2003), no. 3, 953–976
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*P. A. Schweitzer, ''Counterexamples to the Seifert conjecture and opening closed leaves of foliations'',
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...
(2) 100 (1974), 386–400.
*H. Seifert, ''Closed integral curves in 3-space and isotopic two-dimensional deformations'', Proc. Amer. Math. Soc. 1, (1950). 287–302.