The topological complexity introduced by M. Farber and the higher topological complexity as defined by Y. Rudyak may be both viewed as instances of the fibrewise Lusternik-Schnirelmann category of suitably chosen fibrewise-pointed spaces. In order to extend methods from the classical LS-category theory to the fibrewise setting Pavešić introduced in this invited talk at the conference in Poland a construction that for a given continuous endofunctor on the category of pointed topological spaces yields a corresponding operation on the fibres of fibrewise-pointed spaces. In this way he was able to construct the Whitehead-Ganea framework for the topological complexity.

B.04 Guest lecture

COBISS.SI-ID: 16852313This was an invited talk at the international conference Spring Topology and Dynamics Conference, March 23-25, 2013, Central Connecticut State University, New Britain, CT, USA. The results were presented from the paper Dikranjan, Dikran N.,Dense subgroups of compact abelian groups.

B.04 Guest lecture

COBISS.SI-ID: 16825945So far knots have been classified up to a certain number of crossings only for a handful of spaces: the 3-dimensional Euclidean space, the projective space, and the solid torus, the latter being classifed only up to a so-called flip. In this thesis we append the infinite family of lens spaces to this modest list. As a side product, we refine the case of the solid torus by providing a complete classification of knots in it. In both cases we classify knots up to four crossings and up to five crossings with a few exceptions. We also establish which of the knots in the solid torus are amphichiral. We see that for each lens space, a subset of prime knots in the solid torus gives the classification in the lens space. Since there are very few applicable invariants of links in $L(p, q)$, a necessary condition formaking a classification in these spaces is to develop invariants of links in $L(p, q)$. The first invariant we introduce is the HOMFLYPT skeinmodule. The HOMFLYPT skein module has so far only been calculated only for $S^3$ and the solid torus. We show that the HOMFLYPT skein module of $L(p, 1)$ is a free $R$-module and we present a basis of this module for each $p ) 1$. The second invariant is the Khovanov homology of the Kauffman bracket skein module of $\mathbb{R}P^3$. Khovanov homology, an invariant of links in ${\mathbb R}^3$, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda, Przytycki, and Sikora generalized this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $\mathbb{R}P^2$, where the construction fails due to the strange behavior of links when projected to the non-orientable $\mathbb{R}P^2$. We categorify the missing case of the twisted $I$-bundle over $\mathbb{R}P^2$, $\mathbb{R}P^2 \widetilde{\times} I \approx \mathbb{R}P^3 \setminus \{\ast\}$, by redefining the differential in the Khovanov chain complex in a suitable manner. The classification, the calculations of the HOMFLYPT skein modules of the knots, and the calculations of the Kauffman bracket skein modules of the knots are done by a computer program that is available online at https://github.com/bgabrovsek/lpq-classification.

D.09 Tutoring for postgraduate students

COBISS.SI-ID: 16639833