Solves inverse problem for Calderón using microlocal normal forms.
problem Recovering unknown coefficient from boundary measurements.
method Microlocal normal forms and propagation of singularities.
result Recovery of integrals of unknown coefficient over good bicharacteristic leaves.
In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X∘,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on…
Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics …
Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…
Study solves inverse problems for real principal type operators using unique data sets and ray transforms.
problem Determining coefficients in real principal type equations from boundary data.
method Unique data sets, bicharacteristic ray transforms, and propagation of singularities.
result Global uniqueness results for determining coefficients in nonlinear real principal type equations.
Proves analyticity of quasinormal modes in Kerr and Kerr-de Sitter spacetimes.
problem Analyticity of quasinormal modes in extreme Kerr and Kerr-de Sitter spacetimes.
method Observation of stable radial point source/sink structure in bicharacteristic flow; recent microlocal analysis result by Galkowski and Zworski.
result Quasinormal modes are real analytic in subextremal Kerr and Kerr-de Sitter spacetimes.
We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …
We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-R…
Study normal operators of double fibration transforms with conjugate points.
problem Normal operators of double fibration transforms with conjugate points.
method Stable conditions on the distribution of conjugate points, splitting into elliptic and Fourier integral operators.
result Normal operator splits into an elliptic pseudodifferential operator and Fourier integral operators.
A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same con…
Study on the topology of leaves in singular Riemannian foliations.
problem Characterizing the topology of leaves in singular Riemannian foliations.
method Analyzing the fundamental groups and using nilpotent spaces.
result Leaves of singular Riemannian foliations are finitely covered by nilpotent spaces when M is simply connected. We investigate the coarse homology of leaves in foliations of compact manifolds. This is motivated by the observation that the non-leaves constructed by Schweitzer and by Zeghib all have non-finitely generated coarse homology. This led us to ask whether the coarse homology of leaves in a compact manifold always has to …
Optimizes Lasso hyperparameters using leave-one-out CV.
problem Finding optimal hyperparameters for Lasso regression.
method Develops an algorithm to compute exact or approximate leave-one-out CV.
result Algorithm finds optimal hyperparameters for Lasso.
New examples of non-homeomorphic foliation leaves found.
problem Finding non-homeomorphic foliation leaves in manifolds.
method Examples of 5-manifolds and foliations of 6-manifolds.
result Examples of non-homeomorphic foliation leaves in C1 and C∞ foliations. Proves isometric correspondence of leaves for generic rolling distributions.
problem Isometric correspondence of leaves in generic rolling distributions.
method Uses Bäcklund transformation for proof.
result Requires Bäcklund transformation for isometric correspondence of leaves.
Study shows conditions for continuity of foliated homeomorphisms action on space of leaves.
problem Conditions for continuity of foliated homeomorphisms action on space of leaves.
method Investigated sufficient conditions for continuity of the homomorphism ψ: H(X, Δ) → H(Y) induced by the action of foliated homeomorphisms on the space of leaves.
result Similar results hold for a more general class of partitions of locally compact Hausdorff spaces.
Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.
problem Characterizing Lie foliations with symmetric leaves.
method Analyzing the rigidity of Lie foliations with locally symmetric leaves.
result Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.
Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.
problem Dynamics of isoperiodic leaves in rank 1 affine invariant suborbifolds.
method Defines foliation FM and establishes density criterion.
result Establishes criterion for density of isoperiodic leaves.
New proof for solenoidal laminations with symmetric leaves in dimensions other than 4.
problem Characterizing Riemannian solenoidal laminations with symmetric leaves.
method Proving homeomorphism to inverse limits of finite covers of compact locally-symmetric manifolds.
result Each lamination is homeomorphic to a specific inverse limit structure.
For a singular Riemannian foliation whose leaves are properly embedded, we show in the first part of this article the existence of global tubular neighbourhoods, and we develop a global description of the foliation as stratification by types of leaves. The second part deals with the further restriction to a foliation w…
Extends foliation results to singular cases.
problem Understanding foliations near singular leaves.
method Proves semi-local Levi-Malcev theorem for holonomy Lie algebroid.
result Formal semi-local triviality for all 2-connected and a wide class of 1-connected leaves.
The number of Klein-bottle leaves in taut foliations is invariant under smooth deformations.
problem Invariance of Klein-bottle leaf counts in taut foliations.
method Proving invariance of the parity of Klein-bottle leaf counts under smooth deformations.
result The parity of Klein-bottle leaf counts is invariant under smooth deformations.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
Compact foliations preserve entropy if leaves are strictly convex projective.
problem Entropy rigidity for foliations by strictly convex projective manifolds.
method Analysis of foliated volume entropies and homeomorphisms.
result Equality in foliated volume entropies implies homothetic leaves.
Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center…
Investigates metric degeneracies on symplectic leaves using a generalized gradient flow.
problem Degeneracies in metrics on symplectic leaves of Poisson manifolds.
method Introduces the generalized double bracket (GDB) vector field to generalize gradient dynamics.
result Identifies admissible regions where the double bracket metric remains non-degenerate on symplectic leaves, enabling GDB as a gradient flow.
Holomorphic foliations found in ball space with unique properties.
problem Finding holomorphic foliations in the ball space.
method Proving existence of nonsingular holomorphic foliations by closed complex hypersurfaces.
result First example of a holomorphic foliation with complete and incomplete leaves.
Optimizes hyperparameter tuning for models using approximate leave-one-out cross-validation.
problem Finding optimal hyperparameters for regularized models using approximate leave-one-out cross-validation.
method Derive efficient formulas for gradient and hessian of approximate leave-one-out cross-validation, apply second-order optimization.
result Demonstrates the effectiveness of the approach on real-world data sets.
Characterizes surfaces made from strips with boundary leaves.
problem Classifying foliated surfaces formed from strips.
method Analyzes surfaces (Z,Δ) glued from open strips with boundary leaves. result Characterizes a subclass of foliated surfaces.
We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares intrinsic distances in leaves with extrinsic distances in the ambient space grows fast…
We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result is also a generalization of Conn's linearization theorem from one-point leaves to arbitrary symplectic leaves (however, we do not make use of Conn's theorem).
We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not only extends but also provides a new approach (and proofs) to the classical stabili…
ALO-CV approximates leave-one-out error in proportional regime.
problem Estimating generalization error in high-dimensional settings.
method Developed new analysis for ALO-CV, showed consistency under strong convexity.
result ALO-CV approximates leave-one-out error up to negligible error.
Paper accelerates conformal prediction by using approximate leave-one-out estimators.
problem Limited computational cost for conformal prediction.
method Incorporates approximate leave-one-out estimators to accelerate conformal prediction.
result ALO-based methods achieve comparable coverage and efficiency to exact methods but with significantly reduced runtime.
For a connected abelian Lie group T acting on a Poisson manifold (Y,π) by Poisson isomorphisms, the T-leaves of π in Y are, by definition, the orbits of the symplectic leaves of π under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves…
Constructs Riemannian foliations with exotic tori as leaves.
problem Creating Riemannian foliations with exotic tori.
method Smooth fiber bundles with exotic tori fibers and finite abelian fundamental group total space.
result Examples of Riemannian foliations with exotic tori leaves and finite abelian fundamental group total space.
The paper explores symplectic foliations and their leaves on manifolds.
problem Which manifolds can be realized as leaves of codimension-1 symplectic foliations?
method Observations and deformations of symplectic structures; examples of manifolds.
result Examples of manifolds that can be realized as leaves but not as symplectic leaves.
Let F be a non-singular foliation on the plane with all leaves being closed subsets, H+(F) be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and H0+(F) be the identity path component of H+(F). The quotient $π_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}…
Non-exact Poisson structures found on toric varieties.
problem Existence of exact Poisson structures on toric varieties.
method Geometric criterion for non-exactness of Poisson structures with finite symplectic leaves.
result Non-exactness of Poisson structures on projective toric varieties.
Uniform bounds on center leaves volume for codimension one center foliations.
problem Bounding the volume of center leaves in codimension one center foliations.
method Analyzing dynamically coherent partially hyperbolic diffeomorphisms with one-dimensional unstable bundle.
result Volume of center leaves is uniformly bounded.
Consider a singular Riemannian foliation (s.r.f for short) on a compact manifold. By successive blow-ups along the strata, we construct a regular Riemannian foliation on another compact Riemannian manifold and a desingularization map that projects leaves of the regular Riemannian foliation into leaves of the s.r.f. Thi…
Study examines conditions for quotient maps of foliated manifolds to be locally trivial.
problem Conditions for quotient maps of foliated manifolds to be locally trivial.
method Analyzes necessary and sufficient conditions for a quotient map to be a locally trivial fibration.
result Necessary and sufficient conditions for the map to be a locally trivial fibration are presented.
Develops support theorem for analytic transforms in tomography.
problem Analytic wave front set resolution for integral transforms.
method Microlocal analysis, double fibration framework, wave packet transforms.
result Uniqueness and support theorems for analytic transforms.
The paper improves ALO for ℓ1-regularized models.
problem Estimating out-of-sample error for ℓ1-regularized models. method Developed a novel theory for ℓ1-regularized problems, bounding ALO error. result For ℓ1-regularized problems, ALO error goes to zero as p goes to infinity. Classifies neighborhoods around specific leaf structures.
problem Classifying singular foliations with given leaf and transverse singular foliation.
method Analyzes the structure of singular foliations and their leaves.
result Developed a method to classify neighborhoods around specific leaf structures.
New proof of uniformization for hyperbolic foliations.
problem Uniformization of foliated spaces by surfaces of hyperbolic type.
method Laminated Ricci flow to find a conformally equivalent metric with constant curvature -1.
result Existence of a laminated Riemannian metric with leaves of constant Gaussian curvature -1.
Study scattering rigidity for Hamiltonian systems, proving lens rigidity for non-trapping Finsler manifolds.
problem Scattering rigidity for Hamiltonian systems on manifolds with boundary.
method Linearization of travel times, X-ray transform over Hamiltonian curves, Hamiltonian light ray transform.
result Prove semiglobal lens rigidity of non-trapping Finsler manifolds.
We prove that every closed, smooth n-manifold X admits a Riemannian metric together with a smooth, transversely oriented CMC foliation if and only if its Euler characteristic is zero, where by CMC foliation we mean a codimension-one, transversely oriented foliation with leaves of constant mean curvature and where t…