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We prove an equivariant version of the classical Menger-Nobeling theorem regarding topological embeddings: Whenever a group $G$ acts on a finite-dimensional compact metric space $X$, a generic continuous equivariant function from $X$ into…

To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic…

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only…

We consider a class of semidirect products $G = \mathbb{R}^n \rtimes H$, with $H$ a suitably chosen abelian matrix group. The choice of $H$ ensures that there is a wavelet inversion formula, and we are looking for criteria to decide under…

Let $\GR \to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\GR^i(Y)$. These groups satisfy the usual properties of the equivariant…

We define and study the index map for families of $G$-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual…

The equivariant Hilbert series of an ideal generated by an orbit of a monomial under the action of the monoid $\mbox{Inc}(\mathbb{N})$ of strictly increasing functions is determined. This is used to find the dimension and degree of such an…

We prove the following to results: (1) A subgroup G of the isometry group of a Riemannian manifold M acts properly on M if and only if G is closed in the isometry group of M. (2) The orbits of an isometric action are closed if and only if…

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely…

For a finite group $G$, the notion of a $G$-transfer system provides homotopy theorists with a combinatorial way to study equivariant objects. In this paper, we focus on the properties of transfer systems for non-abelian groups. We…

It known from the work of Feigin-Tsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite…

A finite spin system invariant under a symmetry group G is a very illustrative example of the finite group action on a set of mappings f:X->Y. In the case of spin systems X is a set of spin carriers and Y contains 2s+1 z-components -s<=m<=s…

In this paper, we introduce a notion of categorified cyclic operad for set-based cyclic operads with symmetries. Our categorification is obtained by relaxing defining axioms of cyclic operads to isomorphisms and by formulating coherence…

We consider a finite system $\{X_1, X_2, \ldots, X_n\}$ of complete vector fields acting on smooth manifolds $M$ equipped with a smooth positive measure. We assume that the system satisfies H\"ormander's condition and generates a finite…

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…

The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian…

We show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical…

We prove that the word problem of a finitely generated group $G$ is in NP (solvable in polynomial time by a non-deterministic Turing machine) if and only if this group is a subgroup of a finitely presented group $H$ with polynomial…

An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of…

We propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first…