arXiv research

New tools for analyzing Kähler manifolds, proving operator algebra and asymptotic kernel.

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…

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

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

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…

Develops support theorem for analytic transforms in tomography.

The paper explains how microlocal analysis solves geometric inverse problems.

Survey of analytic and geometric results on fibred cusp spaces.

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

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

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

Study analyzes Lévy process structure on manifolds with conjugate points.

Develops a new approach to spectral asymmetry using microlocal analysis.

Let X be a closed Riemannian manifold and let H\hookrightarrow X be an embedded hypersurface. Let X=X_+ \cup_H X_- be a decomposition of X into two manifolds with boundary, with X_+ \cap X_- = H. In this expository article, surgery -- or gluing -- formulæfor several geometric and spectral invariants associated to a Dir…

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

Let $(M,g)$ be a simple Riemannian manifold. Under the assumption that the metric $g$ is real-analytic, it is shown that if the geodesic ray transform of a function $f\in L^{2}(M)$ vanishes on an appropriate open set of geodesics, then $f=0$ on the set of points lying on these geodesics. The approach is based on a micr…

Study on ion travel time on curved surfaces.

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…

New method calculates eta invariant without analytic continuation.

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…

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.

Paper proves stability for recovering connections from holonomy traces.

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 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…

Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.

In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic spaces, which was introduced recently by the author. This new method in particular constructs the analytic continuation of the resolvent for even metrics (in the sense of Guillarmou), and gives high energy estimates in strip…

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

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

Study the spectrum of Poincaré operator in triaxial ellipsoids.

Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown th…

New method reveals corners of drum shapes.

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

Local index formula for Lorentzian Dirac operators on spacetimes.

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.

The paper extends Yamabe flow results to non-compact manifolds with bounded geometry.

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.

The study estimates Reeb chords using sheaf theory and persistence.

For hyperbolic 3-manifolds, Ruelle zeta function vanishing order is 4 minus Betti number.

The paper studies stability of commutativity properties of the Dirichlet-to-Neumann map.

Study analyzes low-energy behavior of Schrödinger operators with Coulomb potentials.

Paper proves DN map determination for simple surfaces with low regularity metrics.

Study examines Hilbert area of inscribed polygons in projective geometry.

This article is the second in a series of two whose aim is to extend a recent result of Guillarmou-Lefeuvre [arXiv:1806.04218] on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to the case of manifolds with hyperbolic cusps. We deal with the nonlinear ve…

Extends compactness theory to variable-coefficient pseudo-differential operators on manifolds.