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$p$-Adic compactifications of geometric loop and diffeomorphism groups of compact manifolds on finite-dimensional spaces over non-Archimedean fields are investigated. Weakened topology is introduced. The structure of newly constructed…

Working in the soft-element (classical) viewpoint, we introduce \emph{soft bitopological groups}: soft groups endowed with two soft topologies such that the induced topologies on the set of soft elements make the soft-element group into a…

Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds…

Let $X$ be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry $\text{Nil}$. We show that the diffeomorphism group $\text{Diff}(X)$ deformation retracts to the isometry group $\text{Isom}(X)$. Combining this with…

We construct embeddings of surface groups into the group of germs of analytic diffeomorphisms in one variable.

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of topological properties of smooth manifolds. Round fold maps were introduced as stable fold…

Let $M$ be a closed connected smooth manifold and $G=\textmd{Diff}_0(M)$ denote the connected component of the diffeomorphism group of $M$ containing the identity. The natural action of $G$ on $M$ induces the trace homomorphism on homology.…

For quantum systems described by finite matrices, linear and affine maps of matrices are shown to provide equivalent descriptions of evolution of density matrices for a subsystem caused by unitary Hamiltonian evolution in a larger system;…

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for…

This is a review with examples concerning the concepts of affine (in particular, constant and linear) vector fields and fundamental vector fields on a manifold. The affine, linear and constant vector fields on a manifold are shown to be in…

Let $f:M\to\mathbb{R}$ be a Morse-Bott function on a closed manifold $M$, so the set $\Sigma_f$ of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by $\mathcal{S}(f) = \{h \in…

A bounded automorphism of a field or a group with trivial approximate centre is definable. In an expansion of a field by a Pfaffian family F of additive endomorphisms such that algebraic closure in the expansion coincides with relative…

Let f : Y -> X be a morphism of complex projective manifolds, and let F be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a foliation. This short paper contains an elementary and very geometric…

Finite-sheeted covering mappings onto compact connected groups are studied. It is shown that a finite-sheeted covering mapping from a connected Hausdorff topological space onto a compact connected abelian group G must be a homeomorphism…

We show that for a closed Riemannian manifold the quotient of the group of projective transformations by the group of isometries contains at most two elements unless the metric has constant positive sectional curvature or every projective…

It is shown that a family of distributions on the complex space is characterized as the only family such that the orbit of one distribution under a certain group of transformations on the complex space is the same as that under the group of…

In this note we prove that every finite collection of connected algebraic subgroups of the group of triangular automorphisms of the affine space generates a connected solvable algebraic subgroup.

This note proves the geodesic completeness of any compact manifold endowed with a linear connection such that the closure of its holonomy group is compact.

Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed,…