Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.
problem Understanding periodic billiard paths in right triangles and orbit closures in strata of Abelian and quadratic differentials.
method Classifying orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials.
result Computed orbit closures and deduced asymptotic number of periodic billiard trajectories in right triangles.
The study connects lamination and orbit closures in hyperbolic manifolds.
problem Understanding the geometric and dynamical properties of horocycle orbit closures in Z-covers of compact hyperbolic manifolds. method Exposes connections between distance minimizing laminations and horospherical orbit closures in Z-covers of compact hyperbolic manifolds. Provides novel constructions and explicit descriptions. result Even slight perturbations to hyperbolic metrics can drastically change horocycle orbit closures.
New proof classifies orbit closures in Hodge bundle.
problem Classifying mGL+(2,R)-orbit closures in Hodge bundle. method Using deformations of flat pairs of pants.
result Short proof of absolute period foliation classification.
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.
The study bounds the number of closed geodesics in a specific orbit closure of surfaces.
problem Counting closed geodesics in a specific orbit closure of surfaces.
method Analyzes triangulations and Teichmüller geodesics to bound the number of closed geodesics.
result Obtains exponential bounds on the number of closed geodesics of length at most R.
Study horocycle orbits in Z-covers of hyperbolic surfaces.
problem Classify horocycle orbit closures in Z-covers of compact hyperbolic surfaces. method Careful analysis of distance minimizing geodesic rays in the cover.
result All non-maximal horocycle orbit closures have integer Hausdorff dimension.
The paper classifies orbit closures of symplectic Lie algebras.
problem Classifying orbit closures of symplectic Lie algebras under the action of Sp(4,R). method Analyzing the natural action of Sp(4,R) on the set of 4-dimensional Lie algebras with symplectic structures. result A complete classification of orbit closures of 4-dimensional symplectic Lie algebras.
In hyperelliptic components, GL(2,R) orbits are either closed, dense, or cover loci.
problem Understanding GL(2,R) orbits in hyperelliptic components of abelian differentials.
method Analyzing orbits in hyperelliptic components of abelian differentials.
result Orbits are either closed, dense, or cover loci.
Study stationary measures and orbit closures for non-abelian actions on surfaces.
problem Classify stationary measures and orbit closures for non-abelian action on a surface.
method Use a finite verifiable average growth condition and results from Brown and Rodriguez Hertz.
result Show that under certain conditions, the only nonatomic stationary measure is the given smooth invariant measure, and every orbit closure is either finite or dense.
The paper proves orbit closure theorems for hyperbolic manifolds with Fuchsian ends.
problem Analyzing the orbit closures of unipotent flows on hyperbolic manifolds with specific ends.
method Establishing an analogue of Ratner's orbit closure theorem for unipotent subgroups in SO(d,1).
result Proves the closure of k-horospheres and (k+1)-geodesic planes are properly immersed submanifolds. Characterizes closures of mapping class group orbits on non-orientable surfaces.
problem Understanding closures of orbits in Teichmüller spaces for non-orientable surfaces.
method Analyzes closures in ML and PML for measured laminations, projective measured laminations, and points. result Characterizes closures of weighted two-sided curves in ML. The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant subman…
The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^{odd}(4) the only GL^+(2,R) orbit closures are closed orbits, the Prym locus Q(3,-1^3), and H^{odd}(4). Together with work of Matheus-Wright, this implies that there are only f…
Classifies orbit closures of mapping class group action on affine character variety.
problem Classifying orbit closures of Mod(Σ)-action on χ(Aff(C)). method Classification up to exceptions of Mod(Σ)-action on affine character variety. result The only obstruction for a non-abelian representation to be the holonomy of a branched affine structure is to be Euclidean and not have positive volume.
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …
Generalizes Delzant theorem for torus-equivariantly embedded toric hypersurfaces.
problem Conditions for nonsingularity of complex subtorus orbits in symplectic toric manifolds.
method Clarification of Delzant theorem conditions using polytopes.
result Generalization of Delzant theorem for torus-equivariantly embedded toric hypersurfaces.
The study finds dense orbits and absolute period leaves for complex flows.
problem Existence of dense orbits for real Rel flows on holomorphic 1-forms.
method Established a density criterion for mSL(2,R)-orbit closures, verified using explicit constructions. result Found dense leaves and examples of absolute period foliation.
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a …
The horocyclic flow on geometrically infinite surfaces shows recurrent irregular orbits or non-minimal closures.
problem Complex dynamics of horocyclic flow on geometrically infinite surfaces.
method Analyzing the recurrence and minimality of irregular orbits.
result Irregular orbits are recurrent or have non-hR minimal closures. We prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of [EMi2] and a certain isolation property of closed SL(2,R)…
New topological proof for a theorem about orbits in a specific group.
problem Characterizing orbits of dense subgroups in a specific group.
method Topological approach, using dynamics of unipotent flows.
result Orbits are either finite or dense, proving a theorem.
Classifies components of k-differentials and their orbit closures.
problem Classifying components of strata of k-differentials and their orbit closures.
method Algebraic approach using multiscale compactification.
result Complete classification of components of strata of holomorphic and meromorphic k-differentials.
New surfaces with special geodesic and horocycle behaviors discovered.
problem Understanding geodesic and horocycle dynamics on hyperbolic surfaces.
method Constructing geometrically infinite hyperbolic surfaces with tailored recurrence properties.
result First examples of non-trivial minimal horocyclic orbit closures and infinite locally-finite conservative horocyclic invariant measures.
For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…
Minimal rational curves on compactified symmetric spaces are orbit-closures of 1-parameter subgroups.
problem When are minimal rational curves on equivariant compactifications of symmetric spaces orbit-closures of 1-parameter subgroups?
method Combining algebraic geometry of minimal rational curves with differential geometry of symmetric spaces, showing Gauss-nondegeneracy of VMRT.
result The Gauss-nondegeneracy of VMRT implies that minimal rational curves on equivariant compactifications of symmetric spaces are orbit-closures of 1-parameter subgroups.
Study flat metrics from right prisms, finding non-lattice surfaces with translation coverings.
problem Analyzing flat metrics from right regular prisms.
method Viewing prisms as n-differentials and analyzing unfoldings, proving translation coverings to hyperelliptic surfaces.
result Non-lattice surfaces admit translation coverings to hyperelliptic surfaces, allowing explicit computation of orbit closures and counting problems.
In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichmüller curv…
Coarse density of subspaces in moduli space of Riemann surfaces.
problem Characterizing subspaces of moduli space that are coarsely dense.
method Using Teichmüller metric and projections of orbit closures in abelian differentials.
result Projections of certain strata in abelian differentials are coarsely dense in moduli space.
Study of unitary and groupoid orbits of normal operators, focusing on manifold structures and spectral conditions.
problem Understanding the manifold structures of orbits of normal operators under different norm topologies.
method Unified treatment of unitary and groupoid orbits, using moment maps and conditional expectations.
result Differentiable structures for orbits and necessary spectral conditions for norm closure and submanifold properties.
Minimal action of mapping class group on character variety.
problem Character variety of Deroin-Tholozan representations.
method Geometric perspective using symplectic structure.
result Infinite mapping class group orbits are dense.
The paper analyzes orbits of integer tuples using braid diagrams.
problem Determining orbits of integer tuples under braid diagram actions.
method Monoid action of braid diagrams on integer tuples.
result Orbits of integer tuples under up-down action of braid diagrams.
Study periodic points on genus two surfaces, solving dynamics and geometry problems.
problem Classifying and understanding periodic points on genus two surfaces.
method Analyzing GL(2, R)-equivariant point markings and using properties of hyperelliptic involution, Weierstrass points, and golden points.
result All GL(2, R)-equivariant point markings over orbit closures arise from specific point exchanges.
Classifies horocycle flow closures in hyperbolic 3-manifolds.
problem Classifying horocycle flow closures in hyperbolic 3-manifolds.
method Classifies orbit closures of the 1-dimensional horocycle flow on the frame bundle of M.
result The closure of a horocycle in M is a properly immersed submanifold.
We prove that for certain endomorphisms of a nilmanifold N the set S of those points such that the closure of its (forward) orbit contains no periodic points is large in the sense that for any non-empty open set U, the set U\cap S is of full Hausdorff dimension. When the manifold N is a torus, this result is due to S.G…
Simplified proof of foliation closure theorem for linear foliations.
problem Proving the closure of linear foliations on Riemannian manifolds.
method Direct geometric approach, focusing on projectable foliations and compatible connections.
result Smoothness of the closure of linear foliations directly proven.
The paper proves geodesics in weakly symmetric spaces are homogeneous.
problem Characterizing geodesics in Riemannian manifolds.
method Establishing sufficient conditions for geodesics to be homogeneous and providing new proofs.
result Every weakly symmetric space is a geodesic orbit manifold.
For a given ε>0, we show that there exist two finite index subgroups of PSL2(Z) which are (1+ε)-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any ε>0 there are two finite regular covers of the Modular once punctured torus T0 (or just the Mod…
In this paper, we investigate the closure of a large class of Teichmüller discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus L of translation surfaces of genus three. We describe a systematic way to prove that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of L. The …
The paper proves that on translation surfaces, regular points are part of infinitely many geodesics with dense directions.
problem Existence and density of geodesics through regular points on translation surfaces.
method Apisa's classifications of periodic points and orbit closures, recent Eskin-Filip-Wright result.
result Regular points on translation surfaces are part of infinitely many geodesics with dense directions.
Compactifies stability space for A2 category, introducing q-deformed rational numbers.
problem Stability conditions in triangulated categories and their compactifications.
method Embedding into an infinite-dimensional projective space, using B3 braid group action. result Two orbits in the boundary correspond to q-deformed rational numbers. The study classifies Veech groups of flat surfaces with poles.
problem Classifying Veech groups of flat surfaces with poles.
method Classification through orbit closure and determination of Veech groups.
result Characterization of surfaces with closed orbits and determination of Veech groups.
New surfaces show horocyclic flow isn't always minimal.
problem Complex dynamics on infinite fineness surfaces.
method Construction of infinite hyperbolic surfaces.
result Horocyclic flow is not minimal on infinite fineness surfaces.
We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a c…
New examples of horocycle invariant measures on strata of translation surfaces.
problem Understanding new invariant measures on complex geometric structures.
method Integrating vector fields on translation surfaces to find invariant measures.
result Found new closed immersed manifolds invariant under horocycles.
In this work we show that there is a Riemannian groupoid whose orbits are the closures of the leaves of a regular Riemannian foliation on a compact manifold. This groupoid is equivalent (in a generalized sense of Haefliger) with a transformational groupoid on the basic manifold.
We prove here that given a proper isometric action K×M→M on a complete Riemannian manifold M then every continuous isometric flow on the orbit space M/K is smooth, i.e., it is the projection of an K-equivariant smooth flow on the manifold M. As a direct corollary we infer the smoothness of isometric …
We show that every surface in H^hyp(4) is either a Veech surface or a generic surface, i.e. its GL^+(2,R)-orbit is either a closed or a dense subset of H^hyp(4) . The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Recent results of Eskin-Mirzak…
This paper is devoted to the classification of GL^+(2,R)-orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of quadratic differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus. The pro…