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In this paper we study stable finiteness of ample groupoid algebras with applications to inverse semigroup algebras and Leavitt path algebras, recovering old results and proving some new ones. In addition, we develop a theory of (faithful)…
We study the mathematical structure of the notion of measurement space, which extends aspects of noncommutative topology that are based on quantale theory. This yields a geometric model of physical measurements that provides a realist…
The notions of a {\em 2-precontact space}\/ and a {\em 2-contact space}\/ are introduced. Using them, new representation theorems for precontact and contact algebras are proved. It is shown that there are bijective correspondences between…
Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space. The proof is basedon a lemma about…
We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have…
A locally compact groupoid is said to be exact if its associated reduced crossed product functor is exact. In this paper, we establish some permanence properties of exactness, including generalizations of some known results for exact…
This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck. Such maps include all harmonic maps from…
We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be…
Although the introduction of generalised and extended geometry has been motivated mainly by the appearance of dualities upon reductions on tori, it has until now been unclear how (all) the duality transformations arise from first principles…
As a continuation of \cite{NSY:local}, we mainly discuss the global structure of two-dimensional locally compact geodesically complete metric spaces with curvature bounded above. We first obtain the result on the Lipschitz homotopy…
In the setting of C*-categories, we provide a definition of "spectrum" of a commutative full C*-category as a one-dimensional unital saturated Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gelfand…
With the aid of the theory of Jordan triple systems, we construct an explicit bi-symplectomorphism between a Hermitian symmetric space of non-compact type and $\C^n$ equipped with both the flat Kaehler-form and the Fubini-Study form. Our…
Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The…
The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that…
The article primarily surveys work that followed from the formulas discovered by Avramov and Iyengar in 2008, which permit one to compute certain Hochschild homology and cohomology modules as expressions involving dualizing complexes. One…
Like quantum groups, quantum groupoids frequently appear in pairs of mutually dual objects. We develop a general Pontrjagin duality theory for quantum groupoids in the algebraic setting that extends Van Daele's duality theory for multiplier…
We generalize the Arzel\`a-Ascoli theorem in the space of continuous maps on a compact interval with values in Euclidean N-space by providing a quantitative link between the Hausdorff measure of noncompactness in this space and a natural…
The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that…
We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of…
We study the Gromov-Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally,…