arXiv research

Developed a theory of ultradifferentiable sheafs with applications.

We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Cech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sect…

Functor connects sheaf categories of Legendrian submanifolds.

Introduces sheaf quantization, a topological approach to geometric quantization.

The study estimates Reeb chords using sheaf theory and persistence.

This paper uses sheaf theory to constrain knot types in clean intersections.

An important part of the classical theory of real or complex manifolds is the theory of (smooth, real analytic or complex analytic) vector bundles. With any vector bundle over a manifold (M,F) the sheaf of its (smooth, real analytic or complex analytic) sections is associated which is a locally free sheaf of F-modules,…

Rectangular peg problem solved for many curves.

This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent sheaf $\Cal{T}_M$, satisfying an integrability condition. $F$--manifolds and compatible fl…

Let $Λ$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $Λ$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular support at infinity is $Λ$, and such that the generating family homology is canonically isomorphic to the endomorphism algebra of this she…

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

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

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

The paper connects knot homology with sheaf theory and proves symmetry properties.

Notes describe geometric interpretations of cohomology in trisected 4-manifolds.

Paper proves direct image sheaf positivity for certain Kähler fibrations.

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

The paper explores de Rham theory for singular spaces and stacks.

The paper shows how certain complex projective varieties can be broken down into simpler types.

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…

We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.

New proof confirms operations on constructible functions match theory.

Let f : Y -> X be a morphism of complex projective manifolds, and let F be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a foliation. This short paper contains an elementary and very geometric argument to show that all obstructions to deforming the morphism f along the shea…

A new framework for knowledge graph embedding using sheaves.

The weight $θ$-sheaf $\underline{\mathbb{R}}_{X,θ}$ helps us to reinterpret Morse-Novikov cohomologies via sheaf theory. We give several theorems of Künneth and Leray-Hirsch types. As applications, we prove that the $θ$-Lefschetz number is independent of $θ$ and calculate the Morse-Novikov cohomologies of projective bu…

We prove that the ($τ$-weighted, sheaf-theoretic) SL(2,C) Casson-Lin invariant introduced by Manolescu and the first author in [CM19] is generically independent of the parameter $τ$ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions asked in [CM19]. Our arguments inv…

PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.

Extends h-principle to stratified spaces using sheaf and jet theories.

In the framework of Abstract Differential Geometry, we show that to a given principal sheaf and a representation of its stuctural sheaf in $A^n$, where A is a sheaf of associative, commutative, unital algebras (over R or C), we associate a vector sheaf. Moreover, under some natural assumptions on the compatibility of t…

Sheaf Neural Networks improve graph learning with geometric insights.

Constructs a sheaf of Bargmann-Fock modules on Kähler manifolds.

We give a local formula for the index of a transverse Dirac-type operator on a compact manifold with a Riemannian foliation, under the assumption that the Molino sheaf is a sheaf of abelian Lie algebras.

An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. F…

This paper uses sheaf theory to model virtual knots geometrically.

HSSE framework embeds single-cell RNA-seq data at multiple scales.

The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invaria…

In this paper, we study the asymptotic behavior of the Hermitian-Yang-Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder-Narasimhan-Seshadri filtration, this answers a question by Bando and Siu.

Extends quantization theory to mixed polarizations using transverse differential operators.

Local connection forms provide a very useful tool for handling connections on principal bundles, because they ignore any complexities of the total space and, essentially, involve only two fundamental features of the structure group, namely the adjoint representation and the left (logarithmic) differential. The main res…

We use the Grauert--Grothendieck complex on differentiable spaces to study basic relative forms on the inertia space of a compact Lie group action on a manifold. We prove that the sheaf complex of basic relative forms on the inertia space is a fine resolution of Bryliski's sheaf of functions on the inertia space.

Characterizes stable sheaves for equality in orbifold BG inequality.

We show that the cardinality of the transverse intersection of two compact exact Lagrangian submanifolds in a cotangent bundle is bounded from below by the dimension of the Hom space of sheaf quantizations of the Lagrangians in Tamarkin's category. Our sheaf-theoretic method can also deal with clean and degenerate Lagr…

Given a smooth $G$-vector bundle $E \to M$ with a connection $\nabla$, we propose the construction of a sheaf of vertex algebras $\mathcal{E}^{ch(E,\nabla)}$, which we call a \textit{chiral vector bundle}. $\mathcal{E}^{ch(E,\nabla)}$ contains as subsheaves the sheaf of superalgebras $Ω\otimes Γ(SE \otimes ΛE)$ and the…

We generalize work of Deligne and Gillet-Soulé on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $Γ\backslash\mathbb{H}$, for $Γ\subset PSL_{2}(\mathbb{R})$ a fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cus…

Study projective KLT varieties with projectively flat cotangent sheaves.

We study the geodesics problem in Heisenberg group H (case SR and riemannian). The sheaf of infinitesimal automorphisms of the (2n,2n+1) distribution D over H is an infinite, transitive Lie algebra sheaf.

We compare the sheaf-theoretic and singular chain versions of Poincare duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the singular intersection cohomology cup product, the hypercohomology product i…