Krystyna Kuperberg

Generalized counterexamples to the Seifert conjecture

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. In particular, the 3-sphere S^3 has a real analytic dynamical system such that all limit sets are 2-dimensional. We also prove that a 1-dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise-linear fashion so that there are no closed leaves but all minimal sets are 1-dimensional. These theorems provide new counterexamples to the Seifert conjecture, which asserts that every dynamical system on S^3 with no singular points has a periodic trajectory.

Formal Mathematics for Mathematicians

The collection of works for this special issue was inspired by the presentations given at the 2011 AMS Special Session on Formal Mathematics for Mathematicians: Developing Large Repositories of Advanced Mathematics. The issue features a collection of articles by practitioners of formalizing proofs who share a deep interest in making computerized mathematics widely available.

Fixed points of orientation reversing homeomorphisms of the plane

Let h be an orientation reversing homeomorphism of the plane onto itself. If X is a plane continuum invariant under h, then h has a fixed point in X. Furthermore, if at least one of the bounded complementary domains of X is invariant under h, then h has at least two fixed points in X.

On the bihomogeneity problem of Knaster

The author constructs a locally connected, homogeneous, finitedimensional, compact, metric space which is not bihomogeneous, thus providing a compact counterexample to a problem posed by B. Knaster around 1921.

A short proof of nonhomogeneity of the pseudo-circle

The pseudo-circle is known to be nonhomogeneous. The original proofs of this fact were discovered independently by L. Fearnley and J. T. Rogers, Jr. The purpose of this paper is to provide an alternative, very short proof based on a result of D. Bellamy and W. Lewis.

BIHOMOGENEITY AND MENGER MANIFOLDS

Abstract It is shown that for every triple of integers (α,β,γ) such that α ⩾ 1, β ⩾ 1, and γ ⩾ 2, there is a homogeneous, non-bihomogeneous continuum whose every point has a neighborhood homeomorphic to the Cartesian product of Menger compacta μ α × μ β × μ γ . In particular, there is a homogeneous, non-bihomogeneous Peano continuum of covering dimension four.

A Lower Bound for the Number of Fixed Points of Orientation Reversing Homeomorphisms

Let h be an orientation reversing homeomorphism of the plane onto itself. Let X be a plane continuum, invariant under h. If X has at least 2 k invariant bounded complementary domains, then h has at least k + 2 fixed points in X.

Total Curvature and Spiralling Shortest Paths

Abstract This paper gives a partial confirmation of a conjecture of Agarwal, Har-Peled, Sharir, and Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in R3 cannot be arbitrarily large. It is shown here that the conjecture holds for a class of polytopes for which the ratio of the radii of the circumscribed and inscribed ball is bounded. On the other hand, an example is constructed to show that the total curvature of a shortest path on the boundary of a convex polyhedron in R3 can exceed 2π. Another example shows that the spiralling number of a shortest path on the boundary of a convex polyhedron can be arbitrarily large.

Almost-Tiling the Plane by Ellipses

Abstract. For any λ > 1 we construct a periodic and locally finite packing of the plane with ellipses whose λ -enlargement covers the whole plane. This answers a question of Imre Bárány. On the other hand, we show that if C is a packing in the plane with circular disks of radius at most 1, then its (1+16-5) -enlargement covers no square with side length 4.

Andrzej Trybulec – in Memoriam

Andrzej Trybulec was born on January 29, 1941, in Kraków, Poland. He was the first child of Jan and Barbara Trybulec, both pharmacists, who owned a pharmacy in a small town of Szczucin in southern Poland. Andrzej went to high school in Ruda Śląska, Silesia, staying with his aunt, and then on his own initiative, he transferred to a prestigious high school in Kraków, were he matriculated. As an eighteen year old student at the Medical Academy in Gdańsk, inspired by a conference on cybernetics, he transferred to the University of Warsaw to study philosophy, and then mathematics, the passion of his life. In Warsaw Andrzej worked in the field of geometric and algebraic topology under the direction of Karol Borsuk. Andrzej received an M.S. degree in 1966 at the University of Warsaw and a Ph.D. in 1974 at the Polish Academy of Sciences. In spite of his work in topology, his fascination with artificial intelligence persisted and was reinforced by interaction