1.

Rigid Cantor sets in R^3 with simply connected complement

We treat wild embeddings of cantor sets into the three dimensional Euclidean space. First such embedding was discovered by French mathematician Louis Antoine in 1920s. Since then topologists have been trying to find more examples with additional properties. Our paper is the first breakthrough in this area after several years. We prove that there exist uncountably many inequivalent rigid wild Cantor sets in R^3 with simply connected complement.

COBISS.SI-ID: 13945689

2.

Kreisgeometrie - gestern und heute : von der Anschauung zur Abstraktion

An integral approach to the circle geometry is discussed, including historical development, algebraic calculations and main axiomatic systems. The monograph collects results concerning geometry of the circle and the sphere, which date back to such classics as Gauss, Möbius, Riemann in Hilbert, and which had a most intensive evolution during the last century.

COBISS.SI-ID: 59021313

3.

Generating discrete Morse functions from point data

The paper is devoted to the area of applications of modern topology, namely the discrete topology. In this paper we make a substantial progress by showing an efficient procedure for extending an injective map, given on the vertices of a finite simplicial complex K, to a discrete Morse function on K. The resulting discrete Morse function on K mirrors the large-scale behaviour of the given map.

COBISS.SI-ID: 13872985

4.

On the mod p cohomology of BPU(p)

We study the mod-p cohomology of the classifying space BPU(p) of projective unitary group PU(p) of rang p. Our results corroborate a conjecture posed by J.F.Adams, conjecture posed by A.Kono and N.Yagita, and a general conjecture about cohomological uniqueness of classifying spaces. All three conjectures mentioned above are verified for a large number of compact Lie groups.

COBISS.SI-ID: 13691737

5.

Rational homology spheres and the four-ball genus of knots

In this paper we study the 4-ball genus of knots in the 3-sphere. We show how to extract the information about the 4-ball genus from the Heegaard-Floer homology d-invariants of the branched double cover of the knot. We describe an algorithm implementing the ideas and using it (along with some constructions of explicit surfaces) determine the 4-ball genus of some small Montesinos knots.

COBISS.SI-ID: 13875033