arXiv research

Study shows dynamics of rank 1 orbifolds in flat surfaces.

Classifies non-arithmetic orbifolds in specific hyperbolic spaces.

Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.

The paper classifies fibrations of 3-dimensional flat orbifolds.

This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.

New theorem extends Hadamard's to transversely affine geometry.

Constructs moduli spaces for complex affine and dilation surfaces.

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…

The paper constructs Morse complexes for orbifolds and shows their homologies are orbifold invariants.

New spectral invariants distinguish Joyce orbifolds from other manifolds.

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 …

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

The paper defines and analyzes configuration Lie groupoids and orbifold braid groups.

This paper classifies Hamiltonian actions by symplectic groupoids using Delzant subspaces.

To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the "orbifold Toledo invariant." For each such orbifold, there exists an elliptic surface ov…

Study of $D_2$ ALF manifolds via hyperkahler quotients of affine spaces.

New examples of Calabi-Yau 3-folds with unique properties.

Develops orbibundle theory for complex hyperbolic geometry.

For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. I…

We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it. Moreover, we give a sufficient condition for the equality in the inequality. In order t…

Estimates Manolescu's κ-invariant using spin 4-orbifolds.

The Hodge spectra help distinguish orbifolds from manifolds with singularities.

Formula proves invariant matches for smooth and orbifold test configurations.

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…

An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.

We consider the $G$-invariant spectrum of the Laplacian on an orbit space $M/G$ where $M$ is a compact Riemannian manifold and $G$ acts by isometries. We generalize the Sunada-Pesce-Sutton technique to the $G$-invariant setting to produce pairs of isospectral non-isometric orbit spaces. One of these spaces is isometric…

Study on counting flat connections over $G_2$-orbifolds, proving moduli space compact and smooth.

We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M\times S^3/Γ$, where $M$ is a smooth manifold and $S^3/Γ$ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associ…

Study on moduli space of bi-invariant metrics in Lie groups.

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…

We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an understanding of how the components of the invariant change when we remove a curve from the …

Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.

Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invarian…

Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.

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.

We extend the Neumann's methods and give the explicit formulae for the volume and the Chern-Simons invariant for hyperbolic alternating knot orbifolds.

Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.

Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector decompositions. Particularly interesting instances of these product formulas occur for …

New Heegaard Floer homology for orbifolds with cyclic singularities.

A formula is given which computes the Seiberg-Witten invariant of a 3-orbifold from the invariant of the underlying manifold. As an application, we derive a formula for the Seiberg-Witten invariant of a non-Kähler complex surface, which was originally due to O. Biquard \cite{Biq} and S.R. Williams \cite{W} independentl…

Study shows infinitely many nonnegatively curved metric spaces on exotic 7-manifolds.

The Hodge spectra can distinguish orbifolds from manifolds, especially in low dimensions.

We establish a lower bound on the complexity orientable locally orientable geometric 3-orbifolds in terms of Delzant's T-invariants of their orbifold-fundamental groups, generalizing previously known bounds for complexity of 3-manifolds.

Develops invariants for webs and foams using Seiberg-Witten theory.

The paper explores subspaces in hyperbolic lattices and their arithmetic properties.

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.