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In this paper we introduce and study so-called $k^*$-metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory. By…

Given a Lie group $G$ we study the class $\M$ of proper metrizable $G$-spaces with metrizable orbit spaces, and show that any $G$-space $X \in \M$ admits a closed $G$-embedding into a convex $G$-subset $C$ of some locally convex linear…

Most of results of Bestvina and Mogilski [\textit{Characterizing certain incomplete infinite-dimensional absolute retracts}, Michigan Math. J. \textbf{33} (1986), 291--313] on strong $Z$-sets in ANR's and absorbing sets is generalized to…

Let $\pi:P\to B$ be a smooth $G$-bundle over a compact Riemannian manifold $B$ and $c$ a smooth loop in $B$ of constant seed $a(>0)$, where $G$ is compact semi-simple Lie group. In this paper, we prove that the holonomy map ${\rm…

A classical question in quantitative topology is to bound the mapping degree $\operatorname{deg}(f)$ in terms of its Lipchitz constant $\operatorname{Lip}(f)$. For a closed, oriented manifold $M$, the flexible exponent $\alpha(M)$ is the…

Let $K$ be a number field, let $X$ be a smooth integral variety over $K$, and assume that there exists a finite set of finite places $S$ of $K$ such that the $S$-integral points on $X$ are dense. Then the combined conjectures of Campana and…

The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $\Sigma(f)$. We study the parity of the number of components of $\Sigma(f)$. The image $f(\Sigma)$ of…

Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a…

Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot{S}=S \setminus \{\hat{z}_0 \}$. Then we have the composed map $\varphi\circ r$ of a map $r: T(S) \times U \rightarrow F(S)$ and the Bers isomorphism $\varphi: F(S)…

Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let $\Sigma$ be a compact orientable surface with boundary. We show that for any continuous $f: (\Sigma,…

Given a smooth free action of a compact connected Lie group $G$ on a smooth compact manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ whose automorphism group is precisely $G$ is open dense in the space of all…

We prove that: 1. If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable. 2. A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally…

For a metrizable space $X$ of density $\kappa$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is…

Every expanding map on a closed manifold is topologically conjugate to an expanding map on an infra-nilmanifold, but not every infra-nilmanifold admits an expanding map. In this article we give a complete algebraic characterization of the…

A mapping $f:X\to Y$ between metric spaces is called \emph{little Lipschitz} if the quantity $$ \operatorname{lip}(f(x)=\liminf_{r\to0}\frac{\operatorname{diam} f(B(x,r))}{r} $$ is finite for every $x\in X$. We prove that if a compact (or,…

Let $\mathcal{H}$ be Hilbert space and $(\Omega,\mu)$ a $\sigma$-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of $ L^2(\Omega, \mathcal{H})$ that are invariant under point-wise multiplication by…

A densely-defined symmetric linear map from/to a real Hilbert space extends to a self-adjoint map. Extension is expressed via Riesz representation. For a case including Friedrichs extension of a strongly monotone map, self-adjoint extension…

If a graph $G_M$ is embedded into a closed surface $S$ such that $S \backslash G_M$ is a collection of disjoint open discs, then $M=(G_M,S)$ is called a {\em map}. A {\em zigzag} in a map $M$ is a closed path which alternates choosing, at…

Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in Z=L(H), the algebra of bounded linear operators on H. We represent A as a disjoint union of subsets M of Z and, using the…

An involutive diffeomorphism $\sigma$ of a connected smooth manifold $M$ is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to…