Paper proves stability for recovering connections from holonomy traces.
problem Recovering a connection from holonomy traces on Riemannian manifolds.
method Combination of microlocal analysis and non-Abelian approximate Livsic Theorem.
result Hölder type stability estimates for holonomy inverse problem.
The paper connects different types of Lagrangian fillings to Legendrian weaves and their sheaf quantizations.
problem Understanding and comparing different types of Lagrangian fillings of Legendrian weaves.
method Establishing new Reidemeister moves and combinatorial isotopies between Lagrangian fillings, comparing sheaf quantizations.
result Legendrian weaves generalize previously known methods to produce infinitely many distinct Lagrangian fillings.
The paper explains how microlocal analysis solves geometric inverse problems.
problem Recovering geometric information from boundary measurements.
method Microlocal analysis applied to three inverse problems.
result Microlocal techniques solve specific inverse problems in Riemannian geometry.
Let X be a C-infinity manifold. We construct a microlocalization functor μX from the derived category of bounded complexes of ind-sheaves on X to the one on the cotangent bundle of X. This functor generalizes the classical theory of microlocalization.
This is the third and last in our series of papers concerning rough solutions of the Einstein vacuum equations expressed relative to wave coordinates. In this paper we prove an important result concerning Ricci defects of microlocalized solutions, stated and used in the proof of the crucial Asymptotics Theorem in our s…
We study continuous maps between differential manifolds from a microlocal point of view. In particular, we characterize the Lipschitz continuity of these maps in terms of the microsupport of the constant sheaf on their graph. Furthermore, we give lower and upper bounds on the microsupport of the graph of a continuous m…
Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.
problem Index theory and analytic torsion of nonlinear PDEs.
method Microlocal sheaf theory, factorization algebras, Spencer hypercohomology.
result Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.
The paper explores global index formulas for one-dimensional holomorphic foliations.
problem Global index formulas for one-dimensional holomorphic foliations.
method Microlocal point of view and short proofs for existing index formulas.
result Generalizations of existing index formulas.
Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.
problem Ergodicity and mixing properties of Anosov flows and their isometric extensions.
method Microlocal analysis of Pollicott-Ruelle resonances to study isometric extensions of Anosov flows.
result Ergodicity of frame flow on negatively-curved Riemannian manifolds under specific curvature assumptions.
Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.
problem Analyzing Schrödinger operators with non-integer power-law potentials.
method Using Lie-Rinehart algebras and microlocal analysis.
result Microlocal analysis can be applied to Schrödinger operators with non-integer power-law potentials.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. The paper connects Legendrian links to cluster algebras via microlocal methods.
problem Understanding the relationship between Legendrian links and cluster algebras.
method Microlocal parallel transport of sheaf quantizations of Lagrangian fillings.
result Existence of quasi-cluster A-structures and cluster Poisson structures. Study analyzes Lévy process structure on manifolds with conjugate points.
problem Microlocal analysis of Lévy processes on manifolds with conjugate points.
method Microlocal analysis, pseudodifferential operators, Fourier integral operators.
result Generator can be expressed as sum of pseudodifferential and Fourier integral operators.
We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.
Develops a new approach to spectral asymmetry using microlocal analysis.
problem Spectral asymmetry on 3-manifolds.
method Constructs an asymmetry operator using microlocal analysis.
result The asymmetry operator generalizes the eta invariant and contains spectral asymmetry information.
We give a detailed microlocal study of X-ray transforms over geodesics-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation as…
Rectangular peg problem solved for many curves.
problem Rectangular peg problem for continuous Jordan curves.
method Microlocal sheaf theory and recent work of Greene and Lobb.
result Affirmative answer for a large class of rectifiable curves.
Anosov surfaces with same length spectrum are isometric.
problem Identifying metrics on surfaces based on their length spectrum.
method Combining microlocal tools with complex curve geometry.
result Metrics with the same length spectrum on Anosov surfaces are isometric.
Researchers extend microlocal analysis across event horizons of rotating black holes.
problem Incomplete microlocal theory of fields across black hole event horizons.
method Extended microlocal theory for extremal rotating black holes, showing null covectors form an involutive double characteristic manifold.
result Mathematical basis for asymptotic oscillatory solutions near event horizons.
The study estimates Reeb chords using sheaf theory and persistence.
problem Estimating the number of Reeb chords in geometric settings.
method Developed a duality exact triangle and used persistence structure of microlocal sheaves.
result Established lower bounds on the number of Reeb chords under specific conditions.
We complete the microlocal study of the geodesic X-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou and pursued by Guillarmou and the second author. We prove new stability estimates and clarify some properties of the operator Πm, the generalized X-ray transform. These estimates…
In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of the transform R that arises from this model. As a consequence, we sh…
Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of dimension n >= 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition [23]. Using analytic microlocal analysis, we prove a microlocal regularity theorem for ge…
This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sob…
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
Study on ion travel time on curved surfaces.
problem Mean first passage time of ion on curved surfaces.
method Layer potential argument and microlocal analysis.
result Derivation of mean first passage time and spatial average.
A systematic geometric theory for the ultradifferentiable (non-quasianalytic and quasianalytic) wavefront set similar to the well-known theory in the classic smooth and analytic setting is developed. In particular an analogue of Bony's Theorem and the invariance of the ultradifferentiable wavefront set under diffeomorp…
We study the microlocal properties of the geodesic X-ray transform X on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \math…
We propose two new approaches to the Tannakian Galois groups of holonomic D-modules on abelian varieties. The first is an interpretation in terms of principal bundles given by the Fourier-Mukai transform, which shows that they are almost connected. The second constructs a microlocalization functor relating characterist…
We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.
Proves h-principle for loose Legendrian embeddings in contact topology.
problem Existence of non-loose Legendrian embeddings.
method h-principle from microlocal sheaf theory.
result Existence of non-loose Legendrian embeddings.
New augmentations of twist knots found that can't be filled.
problem Finding augmentations of twist knots that cannot be filled by orientable Lagrangian fillings.
method Using a Floer-theoretic version of a result from microlocal sheaf theory, showing augmentations cannot be induced by algebraic tori.
result Established new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings.
Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
problem Compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
method Introduced a locally convex topology, extended compactness theorem, studied pseudo-differential operators, and applied to microlocal defect measures.
result Extended microlocal defect measures and compensated compactness theorem to Sobolev wave front set spaces.
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.
On a Weinstein manifold, we define a constructible co/sheaf of categories on the skeleton. The construction works with arbitrary coefficients, and depends only on the homotopy class of a section of the Lagrangian Grassmannian of the stable symplectic normal bundle. The definition is as follows. Take any, possibly high …
Study Legendrian surfaces using N-graphs and flag moduli.
problem Characterize and apply Legendrian surfaces in contact geometry.
method Develop diagrammatic calculus and algebraic-geometric characterization.
result Show applications in Lagrangian concordance, exact fillings, and rational point counts.
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.
Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.
problem Determining metrics from boundary measurements under magnetic perturbations.
method Runge approximation for Riemannian case, microlocal analysis for Lorentzian case.
result Metrics can be uniquely determined in both Riemannian and Lorentzian cases under specific perturbations.
Develops a new approach to study nonlinear PDEs and their singularities.
problem Understanding the propagation domains of solutions to nonlinear PDEs.
method Derived geometric machinery and sheaf theory to study nonlinear PDEs and their singular supports.
result Estimates the domains of propagation for solutions of non-linear systems.
To be prudent, the paper has been withdrawn by the authors, due an error (missing complex conjugate sign) in Equation (2.5). We are very grateful to Marco Brunella for pointed out the error.
The paper extends Yamabe flow results to non-compact manifolds with bounded geometry.
problem Yamabe flow convergence issues on manifolds with infinite volume.
method Curvature-normalized Yamabe flow for manifolds with bounded geometry.
result Long-time existence and convergence of the flow for negative scalar curvature.
Novel approach to wave equations near null infinity in flat spacetimes.
problem Analyzing regularity and decay of wave equations near null infinity in asymptotically flat spacetimes.
method Microlocal analysis in a compactified spacetime with corners, focusing on edge-type wave operators.
result Microlocal regularity propagates across null infinity via radial sets, leading to new estimates for wave equations.
Proves X-ray transform injectivity on specific manifolds.
problem Injectivity of X-ray transform on closed Anosov manifolds and spherical boundary.
method Perturbative argument of the 0-eigenvalue of elliptic operators via microlocal analysis.
result Generically injective X-ray transform on specified manifolds.
Study the spectrum of Poincaré operator in triaxial ellipsoids.
problem Spectrum of the Poincaré operator in triaxial ellipsoids.
method Microlocal analysis of partial differential equations and polynomial vector fields.
result Polynomial eigenvectors and large-degree asymptotics of the operator.
We introduce new tools for analytic microlocal analysis on Kähler manifolds. As an application, we prove that the space of Berezin-Toeplitz operators with analytic contravariant symbol is an algebra. We also give a short proof of the Bergman kernel asymptotics up to an exponentially small error.
New method reveals corners of drum shapes.
problem Determining the shape of drum corners from its sound.
method Locality principle and calculations of heat kernels.
result Corners are spectral invariants of the Laplacian.
We consider how microlocal methods developed for tomographic problems can be used to detect singularities of the Lorentzian metric of the Universe using measurements of the Cosmic Microwave Background radiation. The physical model we study is mathematically rigorous but highly idealized.
Local index formula for Lorentzian Dirac operators on spacetimes.
problem Index theory for Lorentzian Dirac operators with nontrivial dynamics.
method Local index formula based on microlocal analysis.
result Established a local index formula for Lorentzian Dirac-type operators.