arXiv research

Survey on constructing manifolds over simple polytopes using Lickorish's method.

A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3…

Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.

We compute the sheaf of automorphisms of a multiplicity free Hamiltonian manifold over its momentum polytope and show that its higher cohomology groups vanish. Together with a theorem of Losev, arXiv:math/0612561, this implies a conjecture of Delzant: a compact multiplicity free Hamiltonian manifold is uniquely determi…

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative …

This paper solves a problem in 3D geometry by defining a canonical partition for certain manifolds.

We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of small covers. The lifting conjecture in all cases is verified to be true. The probl…

Contact manifolds' momentum polytopes are convex.

Paper addresses optimization on Hadamard manifolds, generalizing gradient flow.

The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.

Proves stability in Weyl polytopes using optimal transport.

Generalizes moment-angle manifolds to arbitrary nice manifolds with corners.

Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.

The study of symmetries in manifolds derived from colored polytopes.

We introduce two operations named biflip and puzzle-move on simple polytopes producing polytopes with diffeomorphic moment-angle manifolds.

It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover being homeomorphic to a real Bott manifold. In addition, we…

We introduce a globally-convergent algorithm for optimizing the tree-reweighted (TRW) variational objective over the marginal polytope. The algorithm is based on the conditional gradient method (Frank-Wolfe) and moves pseudomarginals within the marginal polytope through repeated maximum a posteriori (MAP) calls. This m…

Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.

We study unbounded 2-dimensional metric polytopes such as those arising as Kähler quotients of complete Kähler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete classification of metrics on such polytopes, and as a result classify all possible m…

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

Study of real polytopes with group and symplectic involutions.

PolytopeWalk library efficiently samples high-dimensional polytopes.

Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.

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…

Smooth approximations bound dihedral angles of convex polytopes.

The article studies crystallizations of small covers over simple polytopes and finds unique crystallizations for the $n$-simplex.

For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3-manifold Y, there is a bijection between vertices of the HF^+(Y) polytope carrying the group Z and the faces of the Thurston norm unit ball that corresp…

The study proves curvature rigidity for convex polytopes.

We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural smoothness parameters of an associated Riemannian manifold. We then apply the metho…

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so…

The paper constructs submanifolds with corners in Delzant polytopes from affine subspaces.

The paper explains how to parameterize facets of moment polytopes in real symplectic geometry.

We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

Researchers compute the index of a specific operator on contact manifolds.

Proves rigidity for specific initial data sets under the dominant energy condition.

We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytop…

Study of infinitesimal rigidity in hyperbolic manifolds.

Study finds critical points of volume functionals on Sasaki manifolds.

In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulations of $\mathbb{S}^3$; namely, the boundary complexes of cyclic polyt…

Constructs examples of complex 3D shapes with specific properties.

By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.

The article provides formulas for the number of terms in connected sums of sphere products associated with dual-neighborly polytopes.

Generalizes Delzant theorem for torus-equivariantly embedded toric hypersurfaces.

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics in $\Bbb C^n$ which are invariant with respect to the natural action of the real torus $(\Bbb S^1)^n…

New method finds hyperelliptic 4-manifolds from polytope vector-colorings.

The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.

We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes in R^n specified by m inequalities. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold equipped with the metric induc…

Quantizes $b$-symplectic toric manifolds using $T$-modules.