Orbifolds lie at the intersection of many different areas of mathematics, including algebraic and differential geometry, topology, algebra and string theory. Orbifolds were first introduced into topology and differential geometry by Satake [6], who called them V-manifolds. Satake described them as topological spaces generalizing smooth manifolds and generalized concepts such as de Rham cohomolo...

We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the S equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symplectic quotient.

We present a homogeneous space geometry for the manifold of symmetric positive semidefinite matrices of fixed rank. The total space is the general linear group endowed with its natural rightinvariant metric, and the metric on the homogeneous space is chosen such that the quotient space is the image of a Riemannian submersion from the total space. As a result, we obtain complete geodesics that a...

We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme. (Mathematics Subject Classification: Primary 14A20,...

We show that an analytic subset of the finite dimensional Euclidean space R is purely unrectifiable if and only if the image of any of its compact subsets under every local Lipschitz quotient function is a Lebesgue null. We also construct purely unrectifiable compact sets of Hausdorff dimension greater than 1 which are necessarily sent to finite sets by local Lipschitz quotient functions.

In this brief note, we prove that every space is a sequence-covering image of a topological sum of convergent sequences. As the application of this result, sequence-covering images of locally compact metric spaces (or, locally separable metric spaces, metric spaces) and sequentially-quotient images of a locally compact metric spaces (or, locally separable metric spaces, metric spaces) are equiv...

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

If we consider the quotient space of each tangent plane with respect to this equivalence relation, and the related quotient tangent bundle TS/∼1, a field FD : S −→ TS/∼1 maps each point on the surface to a direction. It is customary to take unit-length vectors as representatives of their equivalence classes, so that a direction field can be regarded as a vector field where the length of all vec...

Complex analyses involving multiple, dependent random quantities often lead to graphical models—a set of nodes denoting variables interest, and corresponding edges statistical interactions between nodes. To develop for data, especially towards generative modeling, one needs mathematical representations metrics matching comparing graphs, subsequent tools, such as geodesics, means, covariances. T...