Related papers: A unique representation of polyhedral types

It is proved that a general polyhedral perfect conducting obstacle in $\mathbb{R}^3$, possibly consisting of finitely many solid polyhedra, is uniquely determined by the far-field pattern corresponding to a single incident wave. This…

A universal tiler is a convex polyhedron whose every cross-section tiles the plane. In this paper, we introduce a certain slight-rotating operation for cross-sections of pentahedra. Based on a selected initial cross-section and by applying…

In this paper, we study the representations of integral quadratic polynomials. Particularly, it is shown that there are only finitely many equivalence classes of positive ternary universal integral quadratic polynomials, and that there are…

A polytope $P$ is circumscribed about a convex body $\Phi\subset \mathbb{R}^n$ if $\Phi\subset P$ and each facet of $P$ is contained in a support hyperplane of $\Phi$. We say that a convex body $\Phi\subset \mathbb{R}^n$ is a rotor of a…

For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs…

Associated varieties are geometric objects appearing in infinite-dimensional representations of semisimple Lie algebras (groups). By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie…

We prove that contact homeomorphisms preserve characteristic foliations on surfaces in contact $3$-manifolds. More precisely, since the characteristic foliation is a singular $1$-dimensional foliation, we show that singular points are…

The paper considers a class of Lagrangian surfaces in $\mathbb C^2$ with isolated singularities of the unfolded Whitney umbrella type. We prove that generically such a surface is locally polynomially convex near a singular point of this…

We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell…

While intersections of convex sets are convex, their unions have rather complicated behavior. Some natural contexts where they appear include duality arguments involving boundaries of convex sets and valuations, which have an Euler…

We prove that for any knot $K$, there exists a one-vertex triangulation of the $3$-sphere containing an edge forming $K$. The proof is constructive, and based on fully augmented links. We use our method to produce ``complicated'' simplicial…

A combinatorial polytope $P$ is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling property but is difficult to verify. In this paper, we merge…

We consider triangulations of surfaces with edges painted three colors so that edges of each triangle have different colors. Such structures arise as Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate pairs of…

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how…

There is a well known bijective correspondence between isomorphism classes of polystable vector bundles $E$ with $c_i(E)=0$ for $i\geq 1$ on a smooth complex projective variety and equivalence classes of unitary representations of the…

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat (2019) and the Polymath REU (2020), continues to investigate representations of convex geometries with small…

A shape of a combinatorial polytope is a convex embedding into Euclidean space. We provide necessary and sufficient conditions for a piecewise linear map between two shapes of the same polytope to be a compression (respectively a weak…

In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized…

Biconservative surfaces are surfaces with divergence-free stress-bienergy tensor. Simply connected, complete, non-$CMC$ biconservative surfaces in $3$-dimensional space forms were constructed working in extrinsic and intrinsic ways. Then,…

Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this…