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.

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.

We study an $A_\infty$ category associated to Legendrian links in $\mathbb{R}^3$ whose objects are $n$-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a …

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

The study estimates Reeb chords using sheaf theory and persistence.

This paper is essentially made of the three preprints arXiv:1212.5818, arXiv:1311.0187, arXiv:1603.07876 gathered in a single text, with simplified proofs. We recall several results of the microlocal theory of sheaves of Kashiwara-Schapira and apply them to study the symplectic geometry of cotangent bundles. We explain…

Study Legendrian surfaces using N-graphs and flag moduli.

Proves h-principle for loose Legendrian embeddings in contact topology.

Study exact Lagrangian cobordisms in cotangent bundles, proving bounds on sheaf interleaving distance and shadow distance.

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 …

The Gromov-Eliashberg theorem says that the group of symplectomorphisms of a symplectic manifold is C^0-closed in the group of diffeomorphisms. This can be translated into a statement about the Lagrangian submanifolds which are graphs of symplectomorphisms. It is also known that such Lagrangian submanifolds are locally…

Functor connects sheaves on Lagrangian cobordisms, proving equivalence and action decreasing properties.

New dg-algebras link graph colorings to sheaves.

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…

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

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

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

We prove the existence of Lagrangian fillings for $D_n$-type Legendrian links.

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

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

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.

Study co-Higgs sheaves on toric varieties, finding explicit examples.

Contact squeezing prevented in certain prequantized balls via generating functions.

We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. We prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective alge…

Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.

Develops a new approach to spectral asymmetry using microlocal analysis.

Constructs coordinate systems from spectral curve sheaves.

New potentials found for sheaves on Calabi-Yau 4-folds.

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…

Paper constructs Chern character for coherent sheaves.

Rectangular peg problem solved for many curves.

This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic …

Anosov surfaces with same length spectrum are isometric.

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

Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.

Develops equivariant Chern characters for coherent sheaves with group actions.

We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.

The paper connects connections on sheaves to an $L_{\infty}$ morphism lifting semiregularity maps.

Defines sheaves and Čech cohomology for diffeological spaces and classifies principal bundles.

Given a smooth projective toric variety $X_Σ$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_Σ)$ into the dg derived category of constructible sheaves on a torus $Sh(T^n, Λ_Σ)$. Recently, K…

We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.

Develops a new method to study algebraic tangent cones of sheaves using valuations.

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…

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singulariti…

Unified framework for Morita invariant cohomology of Lie groupoids.