Study of polynomial strata using braid groups and translation surfaces.
problem Understanding the monodromy of polynomial strata.
method Using infinite-area translation surfaces and braid groups.
result Determine the monodromy of polynomial strata in the braid group.
We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region Ω, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in …
Noninjective monodromy found in polynomial critical point tracking.
problem Tracking critical points in polynomials leads to noninjective monodromy.
method Monic squarefree complex polynomials with prescribed critical point multiplicities.
result Monodromy map is noninjective for polynomials with exactly two critical points.
Researchers map the fundamental group of polynomial strata to a braid group.
problem Understanding the fundamental group of polynomial strata.
method Analyzing the logarithmic derivative of polynomials to determine the map to a braid group.
result The map from the fundamental group of a stratum to a braid group is characterized by the geometry of the translation surface structure.
By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…
This paper classifies components of meromorphic differential strata.
problem Understanding the boundary of multi-scale compactification of meromorphic differentials.
method Classifying connected components of residueless meromorphic differentials.
result Classification of connected components of strata of residueless meromorphic differentials.
Classifies connected components of meromorphic differentials with residue conditions.
problem Understanding the structure of meromorphic differentials with residue constraints.
method Analyzes the multi-scale compactification and residue conditions.
result Classified connected components of generalized strata of meromorphic differentials.
Given an automorphism of a free group Fn, we consider the following invariants: e is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); d is the maximal degree of polynomial growth of conjugacy classes; R is the rank of the fixed subgrou…
Classifies orbit closures in translation surface strata.
problem Classifying orbit closures in translation surface strata.
method Classification of extGL(2,R) orbit closures. result Applications to joinings of certain Masur-Veech measures.
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.
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into F-connected complexes. Study classifies Morse functions with 4 critical points on immersed 2-spheres.
problem Classifying Morse functions with 4 critical points on immersed 2-spheres.
method Used dual graph of immersion and Reeb graphs to classify functions.
result Found all possible structures of the functions.
Constructs a function to prove meromorphic differential strata don't have complete subvarieties.
problem Proving meromorphic differential strata don't contain complete subvarieties.
method Explicit construction of a strictly plurisubharmonic function.
result Proves meromorphic differential strata do not contain positive-dimensional complete subvarieties.
The paper proves properties of strata of differentials, showing they are affine and extremal.
problem Properties of strata of differentials, particularly their geometry and tautological rings.
method Analyzing the tautological rings and using Teichmüller dynamics to prove properties.
result Strata of differentials are affine and their stratification is extremal.
We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on CP1 and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenki…
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.
problem Understanding cohomology classes on curve strata.
method Using geometry of the boundary stratification of moduli space of multi-scale differentials.
result Construction of non-trivial and non-tautological cohomology classes.
The paper studies tautological rings of strata of differentials, proving cohomological stability and no relations in specific degrees.
problem Understanding the structure of tautological rings of strata of differentials.
method Analyzing degrees of relations and cohomological stability for strata with different numbers of simple zeros.
result For strata with more than 4g/3 simple zeros, there are no relations in degrees less than ⌊g/3floor+1. Alternative proof for non-existence of complete curves in differential strata.
problem Non-existence of complete algebraic curves in strata of holomorphic differentials.
method Using positivity of divisor classes on moduli spaces of curves.
result Alternative proof confirming Gendron's result on non-existence.
Homology of abelian differentials stabilizes with more zeros.
problem Understanding the homology of abelian differentials with many simple zeros.
method Developed an h-principle for these strata, valid in a range of homological degrees.
result Homology stabilizes in a range where the number of simple zeros is large.
Survey of compactifications for differential strata.
problem Compactify strata of holomorphic 1-forms on Riemann surfaces.
method Discuss relations between different compactifications from a geometric perspective.
result Relations between compactifications defined from a flat geometric perspective.
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.
problem Understanding linear subvarieties in strata of meromorphic differentials.
method Investigate closures in multi-scale compactification, prove restrictions on period coordinates.
result Prove closures are locally toric varieties, generalize cylinder deformation theorem.
Non-asphericity of strata of genus-one differentials
problem Strata of genus-one differentials
method Non-asphericity
result Infinitely many counterexamples to conjectures
Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.
problem Classifying Teichmüller curves in hyperelliptic components of meromorphic strata.
method Non-existence criterion based on intersections with moduli space boundary.
result Contradiction to algebraicity of candidate Teichmüller curves outside Hurwitz covers.
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.
problem Understanding the geometry of flat surfaces.
method Analyzing triangulations and involutions in strata of translation surfaces.
result Convex iso-Delaunay regions in strata of translation surfaces, especially in hyperelliptic components.
The paper describes a cover of strata of k-differentials with a formula for fiber cardinality.
problem Understanding the ramification locus and cardinality of fibers in strata of k-differentials.
method Intersection calculations on multi-scale compactification and flat geometry.
result A formula for the cardinality of each fiber involving the k-factorial function.
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…
The paper finds a geometric explanation for coinciding Thom polynomials of cusp and corank-2 singularities.
problem Explaining the coincidence of Thom polynomials for cusp and corank-2 singularities.
method Analyzing geometrically the coincidence of Thom polynomials for Morin and corank-2 singularities.
result Found a geometric explanation for the coincidence of Thom polynomials for Morin and corank-2 singularities.
Classifies components of strata of k-differentials on Riemann surfaces.
problem Classifying connected components of strata of k-differentials.
method Developed new techniques to study connected components of strata of k-differentials for general k.
result Complete classification of connected components of the strata of quadratic differentials with arbitrary poles.
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.
problem Computing intersection theory on the boundary of strata of abelian differentials.
method Explicit combinatorial description of the boundary, implemented algorithms in SageMath.
result Computes the Euler characteristic of strata using intersection theory.
Study uses graph techniques to understand meromorphic quadratic differential strata.
problem Understanding the topology of meromorphic quadratic differential strata.
method Exchange graph techniques to study fundamental groups; generalizes relations for mixed-angulations.
result Explicit presentations of fundamental groups in genus-zero case with four singularities.
Study describes splitting and filtration of Hodge bundle on quadratic differentials.
problem Understanding the structure of Hodge bundles on quadratic differentials.
method Harder-Narasimhan filtration and splitting as direct sum of line bundles.
result Determine all Lyapunov exponents of algebraically primitive Teichmüller curves.
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.
problem Counting ergodic measures in surface lamination strata.
method Determined through analysis of geodesic laminations.
result Number of ergodic measures identified in each stratum.
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.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
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.
problem Defining an invariant measure on moduli spaces of dilation surfaces.
method Novel computation of cohomology with coefficients for mapping class group.
result SL(2,R)-invariant Lebesgue class measure on strata.
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…
Paper calculates distances between strata in Teichmüller space, proving a constant separation.
problem Measuring distances in the Weil-Petersson metric on Teichmüller space.
method Analyzes distances between strata, proving a constant separation and providing bounds.
result Proves the optimal value for minimal separation between strata is a constant δ1,1. The paper calculates volumes of moduli spaces of flat metrics on spheres with specific angles.
problem Calculating volumes of moduli spaces of flat metrics on spheres with prescribed angles.
method Recursive formula and application of Kontsevich's formula.
result The volume of moduli spaces of flat metrics on spheres is a continuous piecewise polynomial function of the angles.
Geodesics in 3D space with 1-2 analytic obstacles, proving geodesic independence.
problem Understanding geodesics in 3D space with obstacles.
method Analyzing algebraic varieties and strata, proving geodesic independence.
result Proving geodesic independence in R3 and generalizing to two intersecting obstacles.