This paper studies regular Poisson manifolds of compact types, revealing their geometric and algebraic properties.
problem Understanding the geometric and algebraic properties of Poisson manifolds of compact types.
method Analyzing regular Poisson manifolds, proving properties of their leaf spaces, and introducing symplectic gerbes.
result Regular Poisson manifolds of compact types have rich transverse geometry, with linearly varying cohomology classes and a distinguished polynomial function for leafwise symplectic volume.
Study shows dynamics of rank 1 orbifolds in flat surfaces.
problem Characterize dynamics of rank 1 affine invariant orbifolds.
method Analyzes M-isoperiodic foliations and their ergodic properties.
result Leaves of the isoperiodic foliation are either all closed or all dense.
Classifies non-arithmetic orbifolds in specific hyperbolic spaces.
problem Classifying non-arithmetic affine invariant orbifolds in Hodd(2, 2) and H(3, 1).
method Classification through Veech surfaces and rigidity results.
result Classification of non-arithmetic rank one orbifolds.
The paper classifies fibrations of 3-dimensional flat orbifolds.
problem Classifying fibrations of compact flat 3-orbifolds.
method Developed a theory for classifying fibrations of compact flat n-orbifolds, applying it to 3-orbifolds. result All geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.
This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.
problem Classifying fibrations of compact flat orbifolds.
method Developed theory for classifying fibrations up to affine equivalence.
result Classified fibrations of compact flat 2-orbifolds.
Develops Poisson and Dirac manifolds of compact types with applications.
problem Understanding Poisson and Dirac manifolds of compact types.
method Establishing structural results, local normal forms, canonical stratifications, and Weyl type resolutions.
result Every Poisson manifold of compact type is necessarily regular.
New theorem extends Hadamard's to transversely affine geometry.
problem Transversely affine foliations and their properties.
method Introduced and investigated novel transversely affine foliations, extending Hadamard's theorem.
result Transversely affine version of Hadamard's theorem for foliations.
This paper classifies special ends of real projective orbifolds.
problem Understanding the structures of ends of real projective orbifolds.
method Theoretical tools from affine manifolds and Riemannian foliations.
result Convex but non-properly convex and non-complete-affine radial ends are quasi-joins of horospheres and totally geodesic radial ends.
Constructs moduli spaces for complex affine and dilation surfaces.
problem Classifying and understanding moduli spaces of complex surfaces.
method Using Veech's ideas, constructs holomorphic affine bundles and covering spaces.
result Moduli spaces of dilation surfaces are orbifold K(G,1) where G is the framed mapping class group.
This paper classifies Hamiltonian actions by symplectic groupoids using Delzant subspaces.
problem Classifying Hamiltonian actions by regular proper symplectic groupoids.
method Using Delzant subspaces and cohomology groups to classify actions.
result Classifies faithful multiplicity-free Hamiltonian actions in terms of Delzant subspaces.
Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.
problem Dynamics of isoperiodic leaves in rank 1 affine invariant suborbifolds.
method Defines foliation FM and establishes density criterion.
result Establishes criterion for density of isoperiodic leaves.
Hypertoric varieties are hyperkähler analogues of toric varieties, and are constructed as abelian hyperkähler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold hypertoric varieties are intimately related to the combinatorics of hyperplane arrange…
Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi construction, that is, E^n/G is represented as the quotient of the Cartesian product …
The paper defines and analyzes configuration Lie groupoids and orbifold braid groups.
problem Understanding the structure and properties of orbifold braid groups.
method Definitions and proofs of fibration theorems, short exact sequences, and poly-virtually free structures.
result The pure orbifold braid groups have poly-virtually free structure, generalizing classical braid groups.
Study of D2 ALF manifolds via hyperkahler quotients of affine spaces.
problem Resolving flat orbifold quotients of R4. method Infinite-dimensional generalization of Kronheimer's construction, hyperkahler quotients of affine spaces, singular equivariant instantons, stability of Nahm data.
result Construction of the family of D2 ALF manifolds as a deformation of the flat orbifold (R3×S1)/Z2. New examples of Calabi-Yau 3-folds with unique properties.
problem Finding new Calabi-Yau 3-folds with specific properties.
method Constructing complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones with orbifold singularities.
result Examples of Calabi-Yau 3-folds with maximal volume growth and orbifold singularities.
In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingo…
Researchers prove finiteness of integral representations on specific polytopes.
problem Proving finiteness of integral representations on 2-perfect truncation polytopes.
method Analyzing the geometric component of the deformation space of properly convex real projective structures on Coxeter orbifolds.
result Contains only finitely many integral representations.
The paper calculates the full asymptotics of analytic torsions for compact orbifolds.
problem Analytic torsions of compact locally symmetric orbifolds.
method Using Selberg's trace formula and geometric localization, the paper evaluates the heat trace and orbital integrals.
result Explicit formula for the asymptotic Ray-Singer analytic torsion of compact orbifolds.
The paper studies Ricci flow with finite curvature integrals on manifolds.
problem Finite curvature integrals on closed manifolds.
method Ricci flow with integral curvature bounds.
result The flow converges to a smooth manifold except for orbifold singularities.
In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurabili…
This paper describes integral affine structures on compact 3-manifolds.
problem Understanding integral affine structures on compact 3-manifolds.
method Analyzing complete integral affine structures on compact 3-manifolds up to finite-sheeted coverings.
result A complete list of integral affine structures on the three-dimensional torus and compact three-dimensional nilmanifolds was obtained.
Study torus orbifolds with two fixed points and their cohomology.
problem Understanding the topological and cohomological properties of torus orbifolds with two fixed points.
method Analyze the equivariant topological type and use results from [DKS] to compute integral equivariant cohomology.
result Results on generators and relations for the cohomology of torus orbifolds with two fixed points.
Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.
problem Calculating measures on symplectic groupoid quotients.
method Using Hamiltonian groupoid actions and proper moment maps.
result Duistermaat-Heckman measure is polynomial.
We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group.…
Defines fundamental racks for braid spaces of complex reflection groups.
problem Understanding fundamental racks for braid spaces of complex reflection groups.
method Defines an augmented rack associated to the orbifold fundamental group.
result Yields representations of the orbifold fundamental group on the cohomology of the rack space.
Introduces orbifolds from charts and various topological perspectives.
problem No specific problem stated; introduces new mathematical objects.
method Charts and classical topological perspectives.
result Orbifolds defined and properties from Algebraic, Differential, and Riemannian Geometries.
Two-dimensional Yang-Mills theory defects and orbifolds studied.
problem Discrete symmetries and their defects in Yang-Mills theory.
method Path integral over twisted G-bundles, orbifold construction, defect network approach.
result Exact computation of gauge theory partition function with defects.
Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.
problem Understanding uniformization of Klt pairs under equality in Miyaoka-Yau inequality.
method Analyzing Kähler klt pairs with specific conditions and using orbifold Miyaoka-Yau inequality.
result Orbifold universal cover is either the unit ball or affine space.
In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus acti…
Proof shows volume equals integral points for certain manifolds.
problem Counting integral points on affine manifolds.
method Rational Ehrhart theory and Fourier analysis.
result Volume equals number of integral points for integral-integral affine manifolds.
We present the theory of pseudodifferential operators acting on a vector orbibundle over an orbifold, construct the zeta function of an elliptic pseudodifferential operator and show the existence of a meromorphic extension to the complex plane with at most simple poles. We give formulas for generalized densities on the…
Integral formulas for metric-affine spaces with specific distributions.
problem Finding geometrical obstructions for distributions and foliations.
method Integral formulas involving Ricci and scalar curvatures, second fundamental forms, and integrability tensors.
result Splitting of manifolds and geometrical obstructions for distributions and foliations.
The goal of this work is to generalize the Gauss-Bonnet and Poincaré-Hopf Theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake. In this case, the local data (i.e. integral of the curvature in the case of the Gauss-Bonnet Theorem and the index of the vec…
The paper introduces a new geometric capacity and proves inequalities related to it.
problem Developing a new geometric capacity and comparing it to classical quantities.
method Introducing the general p-affine capacity and proving its properties and inequalities. result Sharp geometric inequalities for the general p-affine capacity are derived. We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …
Perfect pairing for tropical cycles on integral affine manifolds.
problem Computing period integrals and versality of Calabi-Yau degenerations.
method Introducing a cap product pairing and using simplicial methods for constructible sheaves.
result The pairing is perfect in degree one for symplectic singularities.
The paper connects geodesic flows and higher-dimensional Reidemeister torsion for hyperbolic orbifolds.
problem Understanding the relationship between geodesic flows and higher-dimensional Reidemeister torsion.
method Using the integral expression of the Ruelle zeta function and the Selberg zeta function.
result The absolute value at zero of the Ruelle zeta function equals the higher-dimensional Reidemeister torsion.
Real projective structures on n-orbifolds are useful in understanding the space of representations of discrete groups into SL(n+1,R) or PGL(n+1,R). A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The …
Study non-integrable distributions with various affine connections.
problem Characterize non-integrable distributions in Riemannian manifolds with different connections.
method Obtain Gauss, Codazzi, and Ricci equations for non-integrable distributions with semi-symmetric metric, non-metric, and statistical connections.
result Find new examples of Einstein and distributions with constant scalar curvature.
Study on Einstein deformations and desingularizations, finding some 4-orbifolds cannot be limits of smooth metrics.
problem Whether every Einstein 4-orbifold is a limit of smooth Einstein 4-manifolds.
method Analysis of integrability of deformations through variations of Schoen's Pohozaev identity and introduction of preserved integral quantities.
result Spherical and hyperbolic 4-orbifolds with the simplest singularities cannot be limits of smooth Einstein 4-manifolds.
Derives new integral formula for affine connections and curvature.
problem Developing a new integral formula for affine connections.
method Establishes an integral Bochner technique to derive new Ricci curvature.
result Introduces a 2-parameter family of affine connections and derives Ricci curvature.
Builds torus fibrations over singular manifolds with specific features.
problem Creating torus fibrations over manifolds with singularities.
method Constructs a topological space X as a torus fibration over an integral affine manifold B with singularities. result The fibration has a discriminant in codimension 2.
The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
problem Understanding the structure of leaf spaces of Riemannian foliations.
method Analyzing the holonomy groupoid and using diffeological spaces.
result The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
Affine manifolds linked to integrable equations and geometric structures.
problem Understanding the geometric and algebraic properties of affine manifolds.
method Analyzing the Kahlerian tangent bundle and multi-dimensional consistency of the TED equation.
result Affine manifolds are related to self-dual Einstein spaces and Hessian structures.
The paper studies Ricci flows with bounded scalar curvature and proves convergence to orbifolds.
problem Bounding scalar curvature in Ricci flows and understanding convergence behavior.
method Integral bounds on curvature tensors, non-collapsing estimates, non-inflating estimates, Orbifold Ricci flow.
result Ricci flows with bounded scalar curvature converge to orbifolds as time approaches the singular time.
Conditions for affine connections to have linear first integrals and obstructions to Hamiltonian systems.
problem Conditions for affine connections to have linear first integrals and obstructions to Hamiltonian systems of hydrodynamic type.
method Analyzes necessary and sufficient conditions for local geodesic flows of affine connections on surfaces, using scalar invariants of differential orders 3 and 4.
result Explicit obstructions to the existence of a Hamiltonian formulation of Dubrovin--Novikov type for one-dimensional systems of hydrodynamic type.
Affine connections become simple spaces locally.
problem Understanding affine connections and their properties.
method Combinatorial and geometric argument.
result Affine connections are locally affine spaces.