The P.I. intends to continue her previous work that resulted in a real analytic counterexample to the Seifert Conjecture. The flexibility of the constructions introduced by the P.I. and further developed in a joint work with Greg Kuperberg allows the expansion of the already large list of smooth aperiodic flows on the 3-dimensional sphere. The P.I. plans to investigate various properties of dynamical systems without compact leaves in dimension 3 and higher as well as higher-dimensional foliations, and the minimal sets of such foliations. Is is still not known whether or not there is a flow on the 3-dimensional sphere with every orbit dense, although there are partial results. The assertion that there is no such flow is known as the Gottschalk Conjecture. The P.I. expects to answer some of the questions concerning the existence of minimal flows on the 3-sphere. "According to the hairy ball theorem, it is impossible to smooth down all the hairs on a hairy ball. ... This theorem explains why, for example, at any instant somewhere on Earth the horizontal wind speed is zero. Although the hairy ball theorem was proved long ago, its higher dimensional cousins have resisted attack. The most notorious is the Seifert Conjecture, a question asked in 1950 by Herbert Seifert of the University of Heidelberg. ... The surprising answer, just announced by Krystyna Kuperberg of Auburn University ... destined to change the face of higher-dimensional dynamics." Hairy Balls in Higher Dimensions, by Ian Stewart, New Scientist, 23 November 1993, page 18. Seifert proved that under certain conditions a non-singular vector field on the 3-dimensional sphere has a periodic orbit. The statement that there are no aperiodic vector fields on the 3-sphere, the Seifert Conjecture, remained unsolved until a 1974 counterexample of P.A.Schweitzer (class C-1, improved to C-2 by J.Harrison). The vector field constructed by the P.I. to answer Seifert's question is much smoother than the previous examples- C-infinity, and even real analytic. As usual, new methods raise more questions and more possibilities for theoretical investigations, and in this case also computerized simulations.
