Study shows infinite dimensional zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
problem Understanding the zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
method Introduced combinatorial volume forms and a new seminorm on exact bounded cohomology to construct non-trivial classes.
result Shows infinite dimensional zero norm subspace in degree 3 bounded cohomology of acylindrically hyperbolic groups.
This research connects combinatorial Teichmüller space geometry to Weil-Petersson geometry.
problem Understanding the geometry of combinatorial Teichmüller space.
method Developed a parallel between combinatorial Teichmüller space and Weil-Petersson geometry, using measured foliations and Fenchel-Nielsen coordinates.
result Established a geometric recursion and topological recursion for mapping class group invariants.
In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both L2 combinatorial and L2 analytic torsion invariants …
Smooth symplectic manifolds can be approximated by PL symplectic manifolds.
problem Understanding the relationship between smooth and piecewise linear symplectic structures.
method Defining PL symplectic manifolds and proving approximations.
result Smooth symplectic manifolds can be C0-approximated by PL symplectic manifolds. Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.
problem Finding complete hyperbolic metrics on cusped 3-manifolds.
method Analogue of surface and compact 3-manifold flows, minimizing co-volume, extending through singularities.
result Existence of complete hyperbolic metric is equivalent to flow convergence.
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…
Survey of methods for computing volumes of moduli spaces.
problem Computing volumes of moduli spaces for Riemann surfaces with different metrics.
method Combinatorial enumeration, intersection theory, recursion relations.
result Review of key results and methods in computing both Weil-Petersson and Masur-Veech volumes.
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…
Study proves hyperbolic structures for link complements in Seifert fibered spaces.
problem Proving hyperbolic structures for link complements in Seifert fibered spaces.
method Combinatorial bounds on volume of hyperbolic structures.
result Complement of a link in a Seifert fibered space admits a hyperbolic structure of finite volume.
Completed volumes match with combinatorial classes of the double ramification cycle.
problem Computing Masur-Veech volumes for quadratic differentials.
method Describing components of the double ramification cycle and their excess intersection classes, leading to a recursion for completed volumes.
result Completed volumes agree with top intersection of tautological classes on the double ramification cycle.
We suggest a method of computing volume for a simple polytope P in three-dimensional hyperbolic space H3. This method combines the combinatorial reduction of P as a trivalent graph Γ (the 1-skeleton of P) by I−H, or Whitehead, moves (together with shrinking of triangular faces) aligned with its …
Study on random alternating link diagrams and their hyperbolic volumes.
problem Understanding the relationship between the combinatorial structure and hyperbolic volume of random links.
method Model based on random 4-valent maps, analyzing alternating and nonalternating diagrams.
result Expected hyperbolic volume is asymptotically linear in the number of crossings for random alternating diagrams.
Study measures volume of foliations on surfaces, finding integrability range.
problem Volume of combinatorial unit ball of measured foliations on bordered surfaces.
method Analyzes combinatorial moduli spaces and Kontsevich measure.
result Determines range of integrability for (BΣmcomb)s. We give a explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.
We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types …
Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he re…
The rich theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic n-manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4,5 and 6-dimensions, are given, each of very small volume, …
Proves a conjecture about the maximum tet-volume of triangulations of a 2-sphere.
problem Proving the conjectured maximum tet-volume for all triangulations of a 2-sphere.
method Simplified version of Mathieu and Thurston's combinatorial proof using more general volume notions.
result Proves the full conjecture about the maximum tet-volume for all triangulations of a 2-sphere.
New complexity defined for groups, inspired by topological spaces.
problem Complexity of finitely presented groups.
method Inspired by Karoubi-Weibel work, introduces combinatorial complexity.
result New complexity (covering type) defined and properties considered.
Piecewise flat extrinsic curvature approximations for simplicial manifolds.
problem Approximating smooth curvature on irregular meshes.
method Combinatorial constructions using hinge angles and dual tessellations.
result Approximations of extrinsic curvature are mostly mesh-independent.
We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal tria…
New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
Proof shows volumes of certain geometric representations are always integers.
problem Integrality of volumes of specific geometric representations.
method Elementary, combinatorial-geometrical proof.
result Volumes of representations are integers when n≥2. In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferent…
We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume b…
Proof confirms volume conjecture for a specific knot.
problem Verifying the volume conjecture for a specific knot.
method Using a generalized topological quantum field theory and a tetrahedral decomposition.
result Volume conjecture holds for the 73 knot in S3. We extend to a scheme-theoretic context the notion of a combinatorial differential form, due to A.Kock in the framework of synthetic differential geometry. We show that group-valued combinatorial forms on a scheme may be identified, under very general hypotheses, with traditional Lie algebra-valued differential forms, …
We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the 1-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices…
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…
Unified cosmological and Einstein polytope theories.
problem Unified understanding of cosmological and Einstein polytope theories.
method Unified combinatorial perspective of cosmological and Einstein polytope theories.
result Unified construction of cosmological and Einstein polytope theories.
In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depen…
The paper defines Ricci curvature on cell-complexes and proves a Gauss-Bonnnet theorem.
problem Defining and proving geometric theorems on cell-complexes.
method Defining differential forms and Laplacian on cell-complexes, constructing Bochner-Weitzenböck formula, and calculating Ricci curvature combinatorially.
result Established a Gauss-Bonnnet theorem for cell-complexes.
We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds wit…
We find the asymptotic expansion of Masur-Veech volumes for large genus.
problem The asymptotic behavior of Masur-Veech volumes as genus increases.
method Combination of combinatorial and algebro-geometric approaches.
result Existence and computation of a complete asymptotic expansion.
Proves a conjecture for 3D Artin groups using new combinatorial curvature.
problem Proving the K(π,1) conjecture for Artin groups of dimension 3. method Introduces new combinatorial non-positive curvature.
result Proves the K(π,1) conjecture for Artin groups of dimension 3. New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.
We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions…
Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.
problem Establishing equivalence of Morse-Bott volume forms.
method Adapting Moser's trick to Morse-Bott volume forms.
result Two Morse-Bott volume forms with the same zero set are diffeomorphic if and only if they have equal total volumes.
Develops Kähler geometry on new varieties for canonical metrics.
problem No specific problem stated; focuses on new varieties.
method Introduces new varieties, develops Kähler geometry, associates convex functions with metrics.
result Provides expression for Mabuchi functional and combinatorial sufficient condition of properness.
The study finds upper bounds and computes volumes of ideal right-angled polyhedra in Lobachevsky space.
problem Finding upper bounds and computing volumes of ideal right-angled polyhedra in Lobachevsky space.
method Analyzing a class of right-angled polyhedra with vertices on the absolute, obtaining upper bounds on volumes, computing volumes for polyhedra with up to 23 faces, and introducing the class of polyhedra with isolated triangles.
result Minimum volumes are realized on antiprisms and twisted antiprisms, and the first 248 values of volumes are presented.
Study floating bodies of polytopes, linking volume to flags.
problem Understanding floating bodies of polytopes in various spaces.
method Introducing flag simplices to connect metric and combinatorial structures.
result Weighted volume depends on complete flags of polytopes.
New formula calculates volumes and Chern-Simons invariants for closed 3-manifolds.
problem Computing volumes and Chern-Simons invariants for non-parabolic representations.
method Introducing deformed Ptolemy varieties to extend Zickert's formula.
result Volume and Chern-Simons invariants computed for closed 3-manifolds.
Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.
problem Calculating volumes of specific strata of quadratic differentials.
method Expressed volumes as a sum over stable graphs, with coefficients as intersection numbers of psi classes with combinatorial classes.
result Formula for volumes of odd strata of quadratic differentials.
Natural volume forms defined for pseudo-Finslerian manifolds with specific metrics.
problem Defining natural volume forms on pseudo-Finslerian manifolds with m-th root metrics. method Definitions depend on the parity of m, expressed in terms of Cayley hyperdeterminants. result Volume forms computation simplified by avoiding integration over the indicatrix.
Study of n-cylinder surfaces to calculate Masur-Veech volumes.
problem Calculating Masur-Veech volumes for hyperbolic surfaces.
method Combinatorial approach using metric ribbon graphs and plane trees.
result Found generating function for n-cylinder contributions. Extends potential function to non-boundary parabolic representations for computing 3-manifold invariants.
problem Computing invariants of 3-manifolds from representations.
method Extends Cho and Murakami's potential function to non-boundary parabolic representations and derives combinatorial formulas.
result Combinatorial formulas for volume and Chern-Simons invariants of 3-manifolds.
The volume conjecture for surface diffeomorphisms connects volumes to polynomial evaluations.
problem Connecting surface volumes to polynomial evaluations of quantum invariants.
method Developing combinatorial and algebraic techniques to compute isomorphisms between representations.
result Numerical evidence supports a conjecture linking surface volumes to polynomial evaluations.