The paper studies the index of a specific monodromy for origamis in a particular stratum.
problem Determining the index of a Kontsevich-Zorich monodromy for origamis in H(2). method Analyzing the action of the Veech group on the non-tautological part of the homology.
result The index of the Kontsevich-Zorich monodromy is either 1 or 3 for origamis in H(2). The paper studies the geometry and topology of a specific foliation on a complex surface.
problem Characterizing the geometry and topology of a specific foliation on a complex surface.
method Analyzes the isoperiodic foliation of the stratum ΩM1(1,1,−2), proving each leaf is a surface of infinite genus. result Each leaf is a surface of infinite genus homeomorphic to the Loch Ness monster surface.
Study confirms non-injective monodromy for even genus 4 translation surfaces.
problem Characterizing monodromy of translation surfaces in even genus 4.
method Analysis of orbifold classifying spaces and finite-type Artin groups.
result Monodromy of Heven(6) contains a non-abelian free group of rank 2. Paper finds surfaces where KVol is close to the surface's genus.
problem Finding the minimum KVol on translation surfaces.
method Constructing families of translation surfaces in H(2g−2). result KVol can be made arbitrarily close to the surface's genus.
The paper calculates the size of origami orbit graphs in complex surfaces.
problem Calculating the size of origami orbit graphs in complex surfaces.
method Classification of SL(2,Z)-orbits of primitive origamis and reuse of machinery for Prym eigenforms. result Diameter bounds of O(N2/3logN) for orbit graphs in H(2) and H(4), H(6). Study local models for special Kähler metrics near discriminant locus components.
problem Analyzing singularities of special Kähler metrics along discriminant locus of SL2(C) Hitchin base. method Computed Taylor expansion, defined subsystems, and analyzed asymptotics and convergence of metrics.
result Logarithmic asymptotics in transversal directions and convergence to a metric on strata.
Unique Teichmüller curve found in complex geometry.
problem Classifying Teichmüller curves in complex geometry.
method Complete classification of algebraically primitive Teichmüller curves.
result Veech 14-gon generates unique algebraically primitive Teichmüller curve.
The dynamics of holomorphic 1-forms are studied, showing ergodic foliations and connected spaces.
problem Analyzing the dynamics of absolute period foliations in strata of holomorphic 1-forms.
method Using cohomology classes and isoperiodic forms, the authors show ergodicity and connectedness of spaces.
result The absolute period foliation is ergodic on the area-1 locus and has non-dense leaves in explicit suborbifolds.
The paper calculates the Saito determinant for Coxeter discriminant strata.
problem Calculating the Saito determinant for specific geometric strata.
method Using the Saito flat metric and Lie derivatives, the paper finds the determinant of the metric restricted to Coxeter discriminant strata.
result The determinant of the Saito metric on Coxeter discriminant strata is proportional to a product of linear factors in flat coordinates.
We calculate the Masur-Veech volume of the gothic locus G in the stratum H(23) of genus four. Our method is based on the use of the formulae for the Euler characteristics of gothic Teichmüller curves to determine the number of lattice points of given area. We also use this method to recalcula…
Origamis' orbits are non-planar except for a few specific cases.
problem Determining the planarity of origamis' orbits under SL(2,Z) action.
method Analyzing 4-valent graphs from SL(2,Z) action on origamis in H(2).
result Most origamis' orbits are non-planar, with specific exceptions.
We prove that for each discriminant D≡0,1mod4,D∈{4,9}, the corresponding Prym eigenform locus discovered by McMullen in the stratum H(6) is connected. Thus, the projection of any of those loci in the moduli space is a single Teichmüller curve. Along the way, we obtain a classification …
In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic rati…
Algorithm computes fundamental classes of spin components in moduli space.
problem Computing fundamental classes of spin components in moduli space.
method Reconstruct cycles by boundary restrictions via clutching maps and solve linear equations.
result Algorithm implemented in Sage package for cycle class computation.
The traceless SU(2) character variety R(S2,{ai,bi}i=1n) of a 2n-punctured 2-sphere is the symplectic reduction of a Hamiltonian n-torus action on the SU(2) character variety of a closed surface of genus n. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimens…
Non-ergodic measures found in horocycle flow on Abelian differentials.
problem Finding non-ergodic measures in the horocycle flow on Abelian differentials.
method Analyzing weak convergence of ergodic measures to non-ergodic invariant measures.
result Existence of points with non-equidistributing horocycle flow orbits.
Origami graphs' Euler characteristics grow as origami complexity increases.
problem Proving McMullen's conjecture about origami graphs' expansion properties.
method Counting integral and orbifold points on algebraic hypersurfaces, Teichmüller curves, and pseudo-Anosov diffeomorphisms.
result The absolute values of Euler characteristics go to infinity with origami complexity.
Permutation of Weierstrass points on Veech surfaces in H(2) classified by discriminant.
problem Classifying permutations of Weierstrass points on Veech surfaces in H(2) based on discriminant. method Analyzing the permutation group induced by the affine group on Weierstrass points, considering both affine and Dehn multitwists.
result The permutation group is Dih4, Dih5, or Dih6 depending on the discriminant value. Modular curves X1(N) parametrize elliptic curves with a point of order N. They can be identified with connected components of projectivized strata PH(a,−a) of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the …
We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. O…
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.
The paper classifies adjacencies in L∞-Delaunay triangulations of abelian differentials.
problem Classifying adjacencies in L∞-Delaunay triangulations of abelian differentials. method Classification through a finite simplicial complex construction.
result A finite simplicial complex with the same homotopy type as H(κ) is constructed. In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by C(o,n) (resp. C′(o,n)) the set of (resp. primitive) conjugacy classes of pointed length at most n for a basep…
Compactifies strata of d-differentials in genus 0.
problem Compactify strata of d-differentials in genus 0.
method Specify an ideal sheaf and show isomorphism to blow-up of M0,n. result Incidence variety compactification is isomorphic to the blow-up of M0,n. We study the gradient flow lines of a Yang-Mills-type functional on the space of gauged holomorphic maps H(P,X), where P is a principal bundle on a Riemann surface Σ and X is a Kähler Hamiltonian G-manifold. For compact Σ, possibly with boundary, we prove long time existence of the gradient flow. …
The paper studies singular sets in Ricci flow limits, proving rectifiability and curvature bounds.
problem Understanding singular sets in Ricci flow limits.
method Stratification of singular sets, analysis of tangent flows, and geometric measure theory.
result Parabolic rectifiability of singular sets in certain dimensions and uniform curvature bounds.
In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum H(α) of Abelian differentials. The first is the saddle connection Siegel-Veech constant cscmi,mj(H(α)) counting saddle conne…
Study on knot types using thickness and length constraints.
problem Understanding the ideal stratum and deformation persistence of knot types.
method Ropelength-filtered spaces and admissible deformations.
result The first birth level of admissible components corresponds to the ropelength of the knot.
Totally geodesic subvarieties in moduli spaces are studied.
problem Characterizing totally geodesic subvarieties in moduli spaces.
method Analyzing the Deligne-Mumford boundary and its strata.
result Boundary loci of totally geodesic subvarieties are themselves totally geodesic.
The paper classifies constraint mappings in optimization problems.
problem Understanding generic behavior of constraint functions in optimization.
method Using singularity theory of smooth mappings and subgroup classification.
result Families of constraint mappings are a residual set with at most 4 parameters.
We consider random walks on the mapping class group whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anoso…
Connected boundaries of strata of differentials are always connected in various compactifications.
problem Understanding the connectedness of boundaries of differentials' strata in various compactifications.
method Explicit degeneration techniques, algebraic compactifications, and properties of Teichmüller curves.
result The boundaries of differentials' strata are always connected in any complete algebraic compactification.
Study saddle connections on hyperelliptic surfaces, finding growth rates.
problem Count saddle connections on hyperelliptic surfaces without interior intersections.
method Used horocycle renormalization to prove lower bound growth rate.
result Found saddle connections satisfy L(logL)d−2 growth rate. In this paper we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-Lein…
In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local co…
This paper studies connectivity of cyclic-Schottky strata in Schottky space.
problem Connectivity of cyclic-Schottky strata in Schottky space.
method Using Klein-Maskit Combination Theorems and combinatorial analysis of conjugacy classes of Schottky groups.
result Connectivity of cyclic-Schottky strata for p≥3. In this paper we are interested in the stratum H^{hyp}(4) of translation surfaces, which consists of pairs (M,ω), where M is a hyper-elliptic Riemann surface of genus 3, and ωis a holopmorphic 1-form on M having only one zero. We first show that every surface in this stratum can be decomposed into parallelograms follow…
The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmueller curves. We show that the stratum Prym(2,1,1) contains no such Teichmueller curve and the stratum Prym(2,2) at most 92 such Teichmueller curves. This complements the recent progress esta…
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…
Study of origamis in minimal stratum with single cylinders, calculating spin parities and monodromy groups.
problem Understanding the structure and properties of origamis in the minimal stratum of moduli space.
method Construction and analysis of minimal [1,1]-origamis, calculation of spin parities, and investigation of monodromy groups. result All minimal [1,1]-origamis have monodromy groups that are almost always finite simple groups. Proves regularity of harmonic maps into Teichmüller space.
problem Harmonic maps into Teichmüller space and their singularities.
method Analyzes harmonic maps from Riemannian domains to Teichmüller space with specific conditions.
result If a harmonic map intersects a stratum, it is entirely contained in that stratum.
ABae efficiently computes subset means with expensive predicates using stratified sampling.
problem Computing subset means with expensive predicates efficiently.
method Stratified sampling and proxy models.
result Mean squared error of O(N−1) when N is split evenly between stages. Fixed pseudo-Anosov homeomorphisms detect cinquefoil knot.
problem Detecting the cinquefoil knot using knot Floer homology.
method Using pseudo-Anosov homeomorphisms and Floer homology.
result The cinquefoil knot is the only genus-two L-space knot.
For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation be…
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmuell…
We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that…
Let M be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on M to a small tubular neighborhood of each connected component of its singular stratum is foliated-diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes c…
Algorithm constructs surfaces with specific Veech groups in lattice strata.
problem Finding all translation surfaces with a given lattice Veech group.
method Developed an algorithm and provided a new proof of finiteness.
result Algorithm constructs all translation surfaces with a given lattice Veech group.