相关论文: The Center Conjecture for spherical buildings of t…

Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize 2-dimensional CAT(0) cube complexes and are a square analog of systolic complexes. We introduce and study…

In this paper, we prove a local rigidity of convex hypersurfaces in the spaces of constant curvature of dimension $n\ge4$. Namely, we show that two convex isometric hypersurfaces are congruent locally around their corresponding under the…

This paper gives two different proofs to a structural theorem of decreasing minimization (lexicographic optimization) on integrally convex sets. The theorem states that the set of decreasingly minimal elements of an integrally convex set…

Recently, I. Kossovskiy and R. Shafikov have settled the so-called Dimension Conjecture, which characterizes spherical hypersurfaces in ${\mathbb C}^2$ via the dimension of the algebra of infinitesimal automorphisms. In this note, we…

We construct some canonically defined central extensions of groups of symplectomorphisms. We show that this central extension is nontrivial in the case of a torus of dimension $\ge 6$ and in the case of a two-dimensional surface of genus…

A convex figures F is called uniquely constructed if it satisfies the following condition: if F equidecomposable to a convex figure G then F is congruent to G. We classify all convex uniquely constructed figures. The paper written primary…

In this paper we prove equivalence of sets of axioms for non-discrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends…

In his classical work, W. Blaschke proved that a convex body whose shadow boundaries are flat for every direction of parallel illumination must be an ellipsoid. An extension recently proposed by I. Gonzalez-Garc\'ia, J. Jer\'onimo-Castro,…

We identify as topological spheres those complete submanifolds lying with any codimension in hyperbolic space whose Ricci curvature satisfies a lower bound contingent solely upon the length of the mean curvature vector of the immersion.

A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent…

Convex geometries form a subclass of closure systems with unique criticals, or $UC$-systems. We show that the $F$-basis introduced in [1] for $UC$-systems, becomes optimum in convex geometries, in two essential parts of the basis: right…

The present thesis studies structural properties of non-crossing partitions associated to finite Coxeter groups from both algebraic and geometric perspectives. On the one hand, non-crossing partitions are lattices, and on the other hand, we…

The classification of isoparametric hypersurfaces in spheres with four or six different principal curvatures is still not complete. In this paper we develop a structural approach that may be helpful for a classification. Instead of working…

We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially…

We consider hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. Note that C deformation retracts to the union of the torus boundary and a single arc with its endpoints on the torus.…

We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.

We establish half-space type results for a class of height-dependent weighted minimal surfaces in $\mathbb{R}^3$, namely critical points of a weighted area functional whose weight depends on the height. When the weight has at most quadratic…

We describe the constructible derived category of sheaves on the $n$-sphere, stratified in a point and its complement, as a dg module category of a formal dg algebra. We prove formality by exploring two different methods: As a combinatorial…

The investigation into large families of non-opposite flags in finite spherical buildings has been a recent addition to a long line of research in extremal combinatorics, extending classical results in vector and polar spaces. This line of…

We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group…