Formula calculates equivariant LS-category of symplectic toric manifolds.
problem Estimating the number of critical points in equivariant settings.
method Localization formula for equivariant LS-category.
result Equivariant LS-category equals number of fixed points for symplectic toric manifolds.
The natural partial ordering of the orbit types of the action of the group of local gauge transformations on the space of connections in space-time dimension d<=4 is investigated. For that purpose, a description of orbit types in terms of cohomology elements of space-time, derived earlier, is used. It is shown that, on…
We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. It is key to give a recipe to determine the local orbit types of hyperbolic principal orbits by using three kind of restricted root systems and Satake diagram…
The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
problem Constructing a complex Whitney stratification of the moduli space of Higgs bundles.
method Showed that the orbit type decomposition is a complex Whitney stratification with each stratum being a complex symplectic submanifold.
result The moduli space of Higgs bundles is a stratified complex symplectic space.
Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbi…
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…
We develop a method of extending actions of compact transformation groups which is then applied to the problem of preservation of equivariant extensor property by passing to a subspace of given orbit types.
New stratification reveals intrinsic singularity types of orbit spaces.
problem Understanding the intrinsic structure of orbit spaces under Lie group actions.
method Introduced the isostabilizer decomposition and established a map to Klein strata.
result A new canonical stratification on the manifold clarifies the relationship with classical structures.
The Engel group's sub-Riemannian structure is analyzed, revealing unique geometric features.
problem Analyzing the sub-Riemannian structure on the Engel group.
method Global structure of the cut locus, Maxwell set, and caustic described.
result Cut locus has 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata.
A Lie group G naturally acts on its Lie algebra ≫, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group G2 in its Lie algebra ≫2. As results, the group G2 has four orbit types in the Lie algebra ≫2 as $$ G_2/G_2, \quad G_2/(U(1) \times U(1)), …
The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.
problem Existence and moduli space of area-minimizing surfaces with fractal singular sets.
method Proof of prevalent existence, determination of moduli space, refinement of strata.
result Sharp results on moduli space and refinement of strata, showing fractal singularities do not completely dissolve under generic perturbations.
Control data constructed for smooth weak deformation retraction of stratified spaces.
problem Construct control data for smooth weak deformation retraction of stratified spaces.
method Show smooth local triviality with conical fibers, construct control data, use fiber-wise scalar multiplications.
result Obtain neighbourhood smooth weak deformation retraction of stratified spaces.
We use analytical and numerical methods to investigate the equations for cohomogeneity one shrinking gradient Ricci solitons. We show the existence of a winding number for this system around the subvariety of phase space corresponding to Einstein solutions and obtain some estimates for it. We prove a non-existence resu…
The study confirms a conjecture about polynomials related to symmetric spaces.
problem Understanding polynomials associated with isotropy orbits of symmetric spaces.
method Identified Reiswich's polynomials as special cases of Jacobi polynomials and proved their conjecture.
result The polynomials have pairwise different real roots in the interval [0,1].
Study identifies specific subvarieties in translation surfaces with quadratic field.
problem Characterizing invariant subvarieties in translation surfaces with quadratic field.
method Analyzing algebraically primitive subvarieties in strata of translation surfaces.
result Only specific subvarieties identified: decagon, Weierstrass curves, etc.
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. Study of quasi-Kähler metrics on complex manifolds linked to c-projective metrizability.
problem Characterizing quasi-Kähler metrics on almost complex manifolds.
method Analyzing the c-projectively invariant metrizability equation and its solutions.
result New geometries induced by non-degenerate solutions with non-vanishing scalar curvature.
Minimal cones defined by rank conditions in matrix spaces.
problem Characterizing minimal cones in matrix spaces.
method Proving determinantal varieties are minimal cones with regular strata.
result Determinantal varieties are minimal cones with regular submanifolds.
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.
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.
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.
Constructs area-minimizing submanifolds with fractal singularities.
problem Area-minimizing submanifolds with fractal singular sets.
method Integral currents, mod v currents, stable stationary varifolds.
result Sharp dimensionwise solution to Almgren's conjecture.
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.
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.
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.
New criteria found for Willmore submanifolds in Lie group orbits.
problem Criteria for Willmore submanifolds in Lie group orbits.
method Criteria for Willmore submanifolds based on orbit type stratification.
result Found Willmore orbits in each stratified subset of orbit type.
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.
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.
Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic--the chordal variety--in the corresponding complex projective space; the cubic and projective space are identified in term…
Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.
problem Understanding the structure of perimeter minimizing sets in specific metric spaces.
method Established a monotonicity formula and proved rigidity for perimeter minimizers in RCD(0,N) spaces.
result Sharp Hausdorff dimension estimates for singular strata and existence of blow-down cones.
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. Study calculates volumes and constants from intersection theory on abelian differential strata.
problem Calculating volumes and constants from intersection theory on abelian differential strata.
method Intersection numbers on strata with prescribed zeros orders.
result Evaluation of large genus limits and saddle connection Siegel-Veech constants for all strata.
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.
The paper examines affine properties of differential strata on curves.
problem Characterizing affine structures in strata of differentials.
method Analyzing affine varieties and Teichmüller dynamics.
result Affine strata do not contain complete curves with certain pole orders.
The long-standing problem of the perfectness of the compactly supported equivariant homeomorphism group on a G-manifold (with one orbit type) is solved in the affirmative. The proof is based on an argument different than that for the case of diffeomorphisms. The theorem is a starting point for computing $H_1(\mathcal…
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.
Slice theorem in infinite dimensions for Lie groups.
problem Generalizing slice theorem to infinite-dimensional settings.
method Developed slice theorem for locally convex Lie groups on locally convex manifolds using advanced theorems.
result Existence of orbit type stratification under slice condition.
The tautological ring of strata of differentials is characterized based on the presence of poles.
problem Characterizing the tautological ring of strata of differentials.
method Using the κ, ψ, and η classes of moduli spaces of pointed smooth curves.
result The tautological ring is generated by η only if there are no poles of order k, otherwise by the ψ classes corresponding to the poles of order k.
Computes cohomology classes of strata of meromorphic differentials.
problem Computing cohomology classes of differentials with prescribed poles.
method Defined a space of stable meromorphic differentials, stratified by zero multiplicities, computed Poincaré-dual cohomology classes, proved tautological, and provided an algorithm.
result All cohomology classes are tautological and can be computed.
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.
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.
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 investigate the rudiments of Riemannian geometry on orbit spaces M/G for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space M/G and they can hit strata which are more singular only at the end points. This is phrased as convexity …
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.