arXiv research

Study angle structures on 3-manifolds, linking to representation theory.

We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxim…

Study of asymptotics of meromorphic 3D-index as q approaches 1.

Method selects interpretable circular coordinates from data.

The paper solves conditions for extending circle-valued Morse functions.

The paper describes orbits of circle-valued functions on a 2-torus.

New hyperbolic 4-manifolds found with special functions.

New example of hyperbolic 6-manifold with circle-valued Morse function.

An introduction to circle valued Morse theory and Novikov homology, from an algebraic point of view.

We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a …

Study stretch laminations in hyperbolic 3-manifolds via circle-valued maps.

The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction to a fundamental domain of the real-valued Morse function on the pullback infinite cyclic cover.

We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions $f, g: M\to S^1$ belong to same path-component of this space if and only if they are homotopic and have equal numbers of critical points at each index.

Novikov homology refined for 2D integer group coverings.

Let A be an essential complex hyperplane arrangement in an n-dimensional complex vector space V. Let H denote the union of the hyperplanes, and M denote the complement to H in V. We develop the real-valued and circle-valued Morse theory for M and prove, in particular, that M has the homotopy type of a space obtained fr…

The study connects lamination and orbit closures in hyperbolic manifolds.

The works of Donaldson and Mark make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map on a 3-manifold. We study these invariants using the Morse-Novikov theory and H…

Bayesian model predicts circular data with fast Gibbs sampling.

Study angle structures on pseudo 3-manifolds, proving existence for some cases.

The paper introduces a method to decorrelate circular coordinates using lattice reduction.

Let X be a compact oriented Riemannian manifold and let $φ:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $φ$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $φ$ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lin…

By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm, we propose a new state-integral model for the Teichmüller TQFT, where the circle valued state variables live on the edges of oriented leveled shaped triangulations.

This notes explores angle structures on ideally triangulated compact $3$-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic $3$-manifold with totally geodesic boundary has an ideal…

Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic 3-manifolds. We prove, by a constructive argument, that every veering triangulation admits positive angle structures, recovering a result of Hodgson, Rubins…

Proves hyperbolic 3-manifolds have angle structures under certain conditions.

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

Let M be the interior of a compact 3-manifold with non-empty boundary, and T be an ideal (topological) triangulation of M. This paper describes necessary and sufficient conditions for the existence of angle structures, semi-angle structures and generalised angle structures on (M; T) respectively in terms of a generalis…

A novel method optimizes variable-stiffness structures for better strength and weight.

For 3-dimensional hyperbolic cone structures with cone angles $θ$, local rigidity is known for $0 \leq θ\leq 2π$, but global rigidity is known only for $0 \leq θ\leq π$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $π$ do not degenerate in defo…

In this paper we characterize a function defined on the set of edges of a triangulated surface such that there is a spherical angle structure having the function as the edge invariant (or Delaunay invariant). We also characterize a function such that there is a hyperbolic angle structure having the function as the edge…

Surveying connections between graph combinatorics and algebraic right-angled Artin groups.

The paper defines angle structures on 3-manifolds using abelian groups.

We survey the role of right-angled Artin groups in the theory of diffeomorphism groups of low dimensional manifolds. We first describe some of the subgroup structure of right-angled Artin groups. We then discuss the interplay between algebraic structure, compactness, and regularity for group actions on one--dimensional…

Characterizes representations for complex projective structures with specific branch data.

Proves positive mass theorem on conical manifolds with small angles.

We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami …

The paper introduces surfaces with constant solid angle for designing shell structures.

The moment-angle complex Z_K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in…

We construct open book structures on all moment-angle manifolds and describe the topology of their leaves and bindings under certain restrictions. II. We also show, using a recent deep result about contact forms due to Borman, Eliashberg and Murphy [6], that every odd-dimensional moment-angle manifold admits a contact …

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.

We describe some properties of noncompact Euclidean cone manifolds with cone angles less than c less than 2pi and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corollary we classify those with cone angles less than 3pi/2 and those with all cone angles equal to 3pi/2…

The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). No…

Hyperbolic links in thickened torus decompose into angled tetrahedra.

Union of Subspaces (UoS) is a popular model to describe the underlying low-dimensional structure of data. The fine details of UoS structure can be described in terms of canonical angles (also known as principal angles) between subspaces, which is a well-known characterization for relative subspace positions. In this pa…

We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.

We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $π$: any first-order deformation changes either one of those angles or the conformal …

This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3-manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program …

There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both the geometry and combinatorics of the triangulation. We prove that detecting ta…