In this paper we study the topological and metric rigidity of hypersurfaces in ${\mathbb H}^{n+1}$, the $(n+1)$-dimensional hyperbolic space of sectional curvature $-1$. We find conditions to ensure a complete connected oriented hypersurface in ${\mathbb H}^{n+1}$ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.
Pointfree formulas for three kinds of separating points for closed sets by maps are given. These formulas allow controlling the amount of factors of the target product space so that it does not exceed the weight of the embeddable space. In literature, the question of how many factors of the target product are needed for the embedding has only been considered for specific spaces. Our approach is algebraic in character and can thus be viewed as a contribution to Kuratowski's topological calculus.
For an order embedding $G\overset{h}{\rightarrow }{\rightarrow }\Gamma $ of a partly ordered group $G$ into an $l$-group $\Gamma $ a topology $\mathcal T_{\widehat{W}}$ is introduced on $\Gamma $ which is defined by a family of valuations $W$ on $G$. Some density properties of sets $h(G)$, $h(X_t)$ and $(h(X_t)\setminus \lbrace h(g_1),\dots ,h(g_n)\rbrace )$ ($X_t$ being $t$-ideals in $G$) in the topological space $(\Gamma ,\mathcal T_{\widehat{W}})$ are then investigated, each of them being equivalent to the statement that $h$ is a strong theory of quasi-divisors.
In this paper we present some formulae for topological invariants of projective complete intersection curves with isolated singularities in terms of the Milnor number, the Euler characteristic and the topological genus. We also present some conditions, involving the Milnor number and the degree of the curve, for the irreducibility of complete intersection curves.
Nobel Lecture, presented on December 8, 2016, at Aula Magna, Stockholm University. This article describes the history and background of three discoveries cited in this Nobel Prize: The "TKNN" topological formula for the integer quantum Hall effect found by David Thouless and collaborators, the Chern insulator of quantum anomalous Hall effect, and its role in the later discovery of time-reversal invariant topological insulators, and the unexpected topological spin-liquid state of the spin-1 quantum antiferromagnetic chain, which provided an initial example of topological quantum matter. This article summarizes how these early beginnings have led to the exciting, and currently extremely active, field of "topological matter"., F. Duncan M. Haldane ; přeložil Ivan Gregora ; foto A. Mahmoud, Odile Belmontová., and Obsahuje bibliografii
Nobel Lecture, presented on December 8, 2016, at Aula Magna, Stockholm University. In his lecture F. J. M. Kosterlitz described theoretical discoveries of topological phase transitions and topological phases of matter, partially on behalf of the main researcher behind those discoveries, David Thouless, who was not able to give the talk. First, the history of the collaboration between Kosterlitz and Thouless was briefly described. Then, a summary of their contribution to applications of topology to classical Berezinskii-Kosterlitz-Thouless of BKT phase transition was described., John Michael Kosterlitz ; přeložil Ivan Gregora ; foto A. Mahmoud., and Obsahuje bibliografii