arXiv research

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…

Study of Horn's problem in PU(n,1) for n≥1.

We present a constructive proof that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope $β^k$ into closed surfaces of genus $g \leq 1$, each with a transitive automorphism group given by the vertex transitive $\mathbb{Z}_{2k}$-action on $β^k$. Furthermore we show that for each $k \equiv …

The study classifies 331 specific 4D polytopes with 7 facets.

We present a constructive proof, that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope β^k into closed surfaces of genus \leq 1, each with a transitive automorphism group given by the vertex transitive Z_{2k}-action on β^k. Furthermore we show, that for each k \equiv 1,5(6) the 2-skele…

Finite volume Coxeter polytopes are quasiperfect and related to finite covolume reflection groups.

Analyzes vector fields in polytope decompositions, proving curve finiteness.

Researchers prove finiteness of integral representations on specific polytopes.

Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then $P$ is infinitesimally rigid. More generally, let $P$ be a polyhedron bounding a domain which is the union of p…

Moment polytope of toric exponential families is a projection of a simplex.

Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.

Diffeomorphisms of convex polytopes form a Lie group.

To construct flexible nonlinear predictive distributions, the paper introduces a family of softplus function based regression models that convolve, stack, or combine both operations by convolving countably infinite stacked gamma distributions, whose scales depend on the covariates. Generalizing logistic regression that…

This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…

The study broadens the concept of cyclic polytopes to Veronese polytopes.

New noncompact Coxeter polytopes found in various dimensions.

We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra …

New hyperbolic manifolds discovered that fiber algebraically up to dimension 8.

The study constructs links from polytope subgraphs and proves their hyperbolic properties.

Reconstructing polytopes with fixed facet directions from support function evaluations.

We study the structure of finite quandles in terms of subquandles. Every finite quandle $Q$ decomposes in a natural way as a union of disjoint $Q$-complemented subquandles; this decomposition coincides with the usual orbit decomposition of $Q$. Conversely, the structure of a finite quandle with a given orbit decomposit…

We propose a novel method for computing exact pointwise robustness of deep neural networks for all convex $\ell_p$ norms. Our algorithm, GeoCert, finds the largest $\ell_p$ ball centered at an input point $x_0$, within which the output class of a given neural network with ReLU nonlinearities remains unchanged. We relat…

The paper characterizes geometrically finite surfaces via geodesic covers.

New method calibrates deep models for both in-distribution and out-of-distribution samples.

This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formu…

The study explores convex unions and completions in simplicial pseudomanifolds, revealing unexpected behavior.

We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K{ä}hler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in R n , and through t…

Study of infinitesimal rigidity in hyperbolic manifolds.

We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as $H_5(M_{23},\ZZ)=\ZZ_7$ and $H_3(M_{24},\ZZ)=\ZZ_{12}$. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce…

The paper provides conditions for realizing graphs and polytopes with specified edge lengths.

Given an $L^2$-acyclic connected finite $CW$-complex, we define its universal $L^2$-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group $\operatorname{Wh}^w(G)$. We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré d…

In this paper, we construct a right-angled 5-polytope P of finite volume such that all the right-angled Coxeter groups with Fuchsian ends obtained from P are locally rigid.

We establish geometric and topological properties of the space of value functions in finite state-action Markov decision processes. Our main contribution is the characterization of the nature of its shape: a general polytope (Aigner et al., 2010). To demonstrate this result, we exhibit several properties of the structu…

The paper studies deformation spaces of Coxeter truncation polytopes.

Neural networks approximate unit spheres as polytopes.

We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.

The study classifies all compact hyperbolic polytopes with eight facets.

New method improves sampling from logconcave distributions truncated on polytopes.

Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.

Proves stability in Weyl polytopes using optimal transport.

The paper sets new limits on hyperbolic polyhedra volumes.

Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.

We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with $3$ ends and arbitrary finite genus.

Study shows limits of volume-constrained sets are finite unions of Wulff shapes.

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic $4$-space, on o…

Given an elliptic integrand of class $\mathscr{C}^{3}$, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

The paper derives a formula for Chow weights of toric blow-ups.

The paper examines deformations of simple dotted graphs made of circles.