Related papers: Cubulations, immersions, mappability and a problem…

We classify isometric immersions $f\colon M^{n}\to \mathbb{R}^{n+p}$, $n \geq 5$ and $2p \leq n$, with constant Moebius curvature and flat normal bundle.

Submersions with definite folds are submersions on manifolds with boundary whose restrictions to the boundary are definite fold maps. In this paper, we study the properties from the viewpoint of differential topology of manifolds with…

We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a…

In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson…

In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…

We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are…

A strategy for constructing an embedded sphere in a 4-manifold realizing a given homology class which has been successfully applied in the past is to represent the class as a first step stably by an embedded sphere, i.e. after adding…

We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the…

We show that two properly embedded compact surfaces in an orientable 4-manifold are cobordant if and only if they are $\mathbb{Z}/2$-homologous and either the 4-manifold has boundary or the surfaces have the same normal Euler number. If the…

Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold.…

This paper presents two results in the realm of conformal Kaehler submanifolds. These are conformal immersions of Kaehler manifolds into the standard flat Euclidean space. The proofs are obtained by making a rather strong use of several…

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism…

We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used…

For an integer $m\geq 1$, a combinatorial manifold $\widetilde{M}$ is defined to be a geometrical object $\widetilde{M}$ such that for $\forall p\in\widetilde{M}$, there is a local chart $(U_p,\phi_p)$ enable $\phi_p:U_p\to…

The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of…

We show that the mutants of compactified representations constructed by Franz and Puppe can be written as intersections of real quadrics involving division algebras and as generalizations of polygon spaces. We also show that these manifolds…

We study the moduli space $\widetilde{\calM}$ of marked cubic surfaces. By classical work of A.\,B. Coble, this has a compactification $\widetilde{M}$, which is linearly acted upon by the group $W(E_6)$. $\widetilde{M}$ is given as the…

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\lambda) = \exp(i(\lambda_1\psi_1 + ... + \lambda_n\psi_n)) for \lambda in R^n, gives rise to a manifold M of orthogonal projections…

We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical 4-polytope that has a…

A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow…