arXiv research

This paper classifies components of meromorphic differential strata.

Classifies connected components of meromorphic differentials with residue conditions.

Classifies orbit closures in translation surface strata.

Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.

Harmonic maps to Euclidean buildings have rectifiable singular strata.

Study classifies Morse functions with 4 critical points on immersed 2-spheres.

Constructs a function to prove meromorphic differential strata don't have complete subvarieties.

The paper proves properties of strata of differentials, showing they are affine and extremal.

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata ar…

The paper constructs cohomology classes on curve strata.

The paper studies tautological rings of strata of differentials, proving cohomological stability and no relations in specific degrees.

Alternative proof for non-existence of complete curves in differential strata.

Homology of abelian differentials stabilizes with more zeros.

Survey of compactifications for differential strata.

We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.

Study linear subvarieties of meromorphic differential strata, proving toric closures and new proofs of theorems.

Non-asphericity of strata of genus-one differentials

Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the …

In this paper, we study fundamental groups of strata of the moduli space of quadratic differentials. We use certain properties of the Abel-Jacobi map, combined with local surgeries on quadratic differentials, to construct quotient groups of the fundamental groups for a particular family of strata.

Convex iso-Delaunay regions found in flat surface strata.

The paper describes a cover of strata of k-differentials with a formula for fiber cardinality.

We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic measures. These measures are supported on closed immersed manifolds with boundary th…

Classifies components of strata of k-differentials on Riemann surfaces.

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the cla…

SageMath package diffstrata calculates intersection theory on abelian differentials.

Study uses graph techniques to understand meromorphic quadratic differential strata.

Study describes splitting and filtration of Hodge bundle on quadratic differentials.

The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quad…

Study counts ergodic measures in surface lamination strata.

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show…

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of $k$-differentials on smooth curves which parameterize sections of the $k$-th power of the canonical line bund…

The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.

We describe a conjectural formula via intersection numbers for the Masur-Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to compu…

New measure defined on surface strata, invariant under scaling.

Study of polynomial strata using braid groups and translation surfaces.

In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymp…

In this paper, we study connected components of strata of the space of quadratic differentials lying over $\T_g$. We use certain general properties of sections of line bundles to put a upper bound on the number of connected components, and a generalized version of the Gauss map as an invariant to put a lower bound on t…

For g>2 we study the cohomology classes in the closure of a stratum of abelian differentials defined by the boundary strata of codimension one. As an application, we find an explicit stratification of the spin moduli space for an odd spin structure consisting of g-1 strata D_j of codimension j-1 such that D_j does not …

Paper calculates distances between strata in Teichmüller space, proving a constant separation.

Strata of $k$-differentials on smooth curves parameterize sections of the $k$-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the $κ$ and $ψ$ classes of moduli spaces of pointed smooth curves along with the tautological class $η$…

Very few results are known about the topology of the strata of the moduli space of quadratic differentials. In this paper, we prove that any connected component of such strata has only one topological end. A typical flat surface in a neighborhood of the boundary is naturally split by a collection of parallel short sadd…

This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the c…

The paper defines and computes volumes of meromorphic differentials with simple poles.

For $g\geq2$, $j=1,\dots,g$ and $n\geq g+j$ we exhibit infinitely many new rigid and extremal effective codimension $j$ cycles in $\overline{\mathcal{M}}_{g,n}$ from the strata of quadratic differentials and projections of these strata under forgetful morphisms and show the same holds for $k$-differentials with $k\geq …

Study of quasi-Kähler metrics on complex manifolds linked to c-projective metrizability.

Study shows monodromy kernels are large, failing to prove commensurability in specific strata.

Circle packings on translation surfaces are consistent across different surfaces.