arXiv research

Let X be a C-infinity manifold. We construct a microlocalization functor $μ_X$ 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.

Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.

Injectivity of X-ray transform proven for non-smooth metrics.

Develops a new approach to study nonlinear PDEs and their singularities.

Study on ion travel time on curved surfaces.

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…

The paper explains how microlocal analysis solves geometric inverse problems.

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…

The paper explores global index formulas for one-dimensional holomorphic foliations.

We prove Arnol'd's three cusps conjecture about the front of Legendrian curves in the projectivized cotangent bundle of the $2$-sphere. We use the microlocal theory of sheaves of Kashiwara and Schapira and study the derived category of sheaves on the $2$-sphere with a given smooth Lagrangian microsupport.

Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.

Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.

The paper connects Legendrian links to cluster algebras via microlocal methods.

Study analyzes Lévy process structure on manifolds with conjugate points.

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.

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.

Anosov surfaces with same length spectrum are isometric.

Researchers extend microlocal analysis across event horizons of rotating black holes.

The study estimates Reeb chords using sheaf theory and persistence.

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 $Π_m$, 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…

Study examines Hilbert area of inscribed polygons in projective geometry.

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 $\mathcal{X}$ 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…

This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasi-modes, trace formulae.Even if hypoelliptic operators have been well studied…

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.

New augmentations of twist knots found that can't be filled.

Paper proves stability for recovering connections from holonomy traces.

We establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an analogue of the Atiyah-Pato…

Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.

Develops support theorem for analytic transforms in tomography.

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.

Proves analyticity of quasinormal modes in Kerr and Kerr-de Sitter spacetimes.

The paper connects different types of Lagrangian fillings to Legendrian weaves and their sheaf quantizations.

Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.

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.

Novel approach to wave equations near null infinity in flat spacetimes.

Study the spectrum of Poincaré operator in triaxial ellipsoids.

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.