Related papers: Derived subdivisions make every PL sphere polytopa…

The volume of a cyclic polytope can be obtained by forming an iterated integral along a suitable piecewise linear path running through its edges. Different choices of such a path are related by the action of a subgroup of the combinatorial…

A simplicial polytope is a polytope with all its facets being combinatorially equivalent to simplices. We deal with the edge connectivity of the graphs of simplicial polytopes. We first establish that, for any $d\ge 3$, for any $d\ge 3$,…

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…

We describe the boundary of linear subvarieties in the moduli space of multi-scale differentials. Linear subvarieties are algebraic subvarieties of strata of (possibly) meromorphic differentials that in local period coordinates are given by…

We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic…

Starting from a finite simple graph $G$, for each eigenvalue $\theta$ of its adjacency matrix one can construct a convex polytope $P_G(\theta)$, the so called $\theta$-eigenpolytop of $G$. For some polytopes this technique can be used to…

The author has already proven that the space $\Delta(\Pi_n)/G$ is homotopy equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers $n\geq 3$ and all subgroups $G\subset S_1\times S_{n-1}$. We construct an $S_1\times…

A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani in 1987, via a non-constructive proof…

Every closed oriented PL 4-manifold is a branched cover of the 4-sphere branched over a PL-surface with finitely many singularities by Piergallini [Topology 34(3):497-508, 1995]. This generalizes a long standing result by Hilden and…

In this work the confined domains for a point-like particle propagating within the boundary of an ideally reflecting paraboloid mirror are derived. Thereby it is proven that all consecutive flight parabola foci points lie on the surface of…

This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the…

A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley--Reisner ring has a linear resolution. It turns out that the Stanley--Reisner ring of the sphere which is the boundary complex of…

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a complete classification of neighborly polytopes of dimension $4$, $6$ and $7$…

The result of Padrol asserts that for every $d\geq 4$, there exist $2^{\Omega(n\log n)}$ distinct combinatorial types of $\lfloor d/2\rfloor$-neighborly simplicial $(d-1)$-spheres with $n$ vertices. We present a construction showing that…

The shellability of the boundary complex of an unbounded polyhedron is investigated. To this end, it is necessary to pass to a suitable compactification, e.g., by one point. This observation can be exploited to prove that any tropical…

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any $d\ge 3$, the graph of a cubical $d$-polytope…

Using tropical convexity Dochtermann, Fink, and Sanyal proved that regular fine mixed subdivisions of Minkowski sums of simplices support minimal cellular resolutions. They asked if the regularity condition can be removed. We give an…

We prove that any derived equivalence between derived discrete algebras is standard, i.e.\ is isomorphic to the derived tensor product by a two-sided tilting complex.

This paper is a continuation of our previous work in which we defined the notion of a polytope complex and its $K$-theory. In this paper we produce formulas for the delooping of a simplicial polytope complex and the cofiber of a morphism of…

We show that for every convex polyhedral sphere $P$ in $S^3$, there exist two canonical, non-edge-to-edge tilings of $S^{2}$ whose tiles are given by all the faces of $P$ and the dual convex polyhedral sphere $P^*$ to $P$. Under the…