A new framework for knowledge graph embedding using sheaves.
problem Learning representations for entities and relations in knowledge graphs.
method Using cellular sheaves to describe knowledge graph embeddings with consistency constraints.
result A generalized framework for reasoning about knowledge graph embedding models.
For each integer k≥4 we describe diagrammatically a positively graded Koszul algebra Dk such that the category of finite dimensional Dk-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type Dk or Bk−1, constructible with respect…
PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.
problem Limitations of common Neural Sheaf Diffusion implementations, including scalability and stability issues.
method Introduces Polynomial Neural Sheaf Diffusion (PolyNSD) with a degree-K polynomial propagation operator and spectral rescaling.
result PolyNSD achieves state-of-the-art results on both homophilic and heterophilic benchmarks with reduced runtime and memory requirements.
In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are re…
This article introduces descriptive cellular homology on cell complexes, which is an extension of J.H.C. Whitehead's CW topology. A main result is that a descriptive cellular complex is a topology on fibres in a fibre bundle. An application of two forms of cellular homology is given in terms of the persistence of shape…
Study co-Higgs sheaves on toric varieties, finding explicit examples.
problem Characterizing and understanding co-Higgs sheaves on toric varieties.
method Characterization and explicit computation of examples.
result Explicit examples of co-Higgs sheaves on toric varieties computed.
The paper studies equivariant sheaves on toric varieties and their quotients.
problem Understanding stability of sheaves on toric GIT quotients.
method Defining equivariant sheaves and showing stability preservation under certain conditions.
result Stability of sheaves on toric GIT quotients is related to combinatorial criteria.
The study classifies cellular pseudomanifolds and their properties.
problem Understanding the structure of cellular pseudomanifolds.
method Analyzing the combinatorial and geometric properties of cellular pseudomanifolds.
result Complete classification of cellular pseudomanifolds with excess < 2, and progress towards excess 2.
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…
The paper studies Morse theory on manifolds with boundaries, constructing cellular structures and estimating critical points.
problem Understanding Morse functions on manifolds with boundaries.
method Constructing a cellular structure and analyzing its algebraic properties.
result Estimation of the number of critical points of a Morse function with boundary conditions.
Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.
problem Understanding sheaves of Lie-Rinehart algebras and their morphisms.
method Introduced morphisms and comorphisms, proved factorization theorems, and defined higher homotopy groups and groupoids.
result Sheaves of Lie-Rinehart algebras over smooth manifolds induce partitions into orbits of the fundamental groupoid.
Characterizes tangent cones for specific connections on reflexive sheaves.
problem Analyzing tangent cones of admissible Hermitian-Yang-Mills connections over reflexive sheaves.
method Algebro-geometric characterization of analytic tangent cones.
result Complete characterization of tangent cones for admissible Hermitian-Yang-Mills connections over reflexive sheaves.
Quantum cellular automata form a homology theory.
problem Understanding the topological structure of quantum cellular automata.
method Formal properties of coarse homology theories.
result Quantum cellular automata naturally form the degree-zero part of a coarse homology theory.
Constructs coordinate systems from spectral curve sheaves.
problem Creating coordinate systems from spectral curve sheaves.
method Finite-gap integration methods for orthogonal curvilinear coordinates.
result Constructs coordinate systems over reducible spectral curves.
New potentials found for sheaves on Calabi-Yau 4-folds.
problem Understanding sheaves on Calabi-Yau 4-folds.
method Derived Quot-stacks and Lagrangian distributions.
result Globally defined −1-shifted potentials on sheaves. Paper constructs Chern character for coherent sheaves.
problem Chern character for coherent sheaves with values in Bott-Chern cohomology.
method Based on Block's fundamental construction, constructs Chern character.
result Proves Riemann-Roch-Grothendieck formula for coherent sheaves.
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 …
Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.
problem Understanding the Kobayashi-Hitchin correspondence for specific sheaves.
method Using Hermitian-Yang-Mills flow on Kähler manifolds with simple normal crossing divisors.
result Established the correspondence for saturated reflexive parabolic sheaves.
Augmentations and sheaves linked for Legendrian graphs.
problem Understanding categorical Legendrian isotopy invariants.
method Equivalence between augmentation category and DG category of sheaves.
result Proved 'augmentations are sheaves' for Legendrian graphs.
Develops equivariant Chern characters for coherent sheaves with group actions.
problem Computing Chern characters for coherent sheaves on manifolds with group actions.
method Introduces equivariant Chern characters and proves Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.
result Establishes a Riemann-Roch-Grothendieck theorem for coherent sheaves with finite group actions.
We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect…
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.
Optimizes natural frequencies of cellular composites with various microstructures.
problem Designing cellular composites with diverse microstructures for maximizing natural frequencies.
method Data-driven topology optimization with a latent-variable Gaussian process model.
result Cellular designs with multiclass microstructures achieve higher natural frequencies.
The paper connects connections on sheaves to an L∞ morphism lifting semiregularity maps.
problem Understanding connections on sheaves and their relationship to semiregularity maps.
method Proves a canonical association of a connection of type (1,0) on a sheaf to an L∞ morphism. result Establishes a connection between connections on sheaves and an L∞ morphism lifting semiregularity maps. Defines sheaves and Čech cohomology for diffeological spaces and classifies principal bundles.
problem Defining sheaves and Čech cohomology for diffeological spaces.
method Defines sheaves for diffeological spaces and constructs Čech cohomology. Uses Čech cohomology to classify principal bundles.
result First degree Čech cohomology classes classify diffeological principal G-bundles. Given a smooth projective toric variety XΣ 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(XΣ) into the dg derived category of constructible sheaves on a torus Sh(Tn,ΛΣ). 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.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
For leveled spatial graphs, we find a surface embedding that allows cellular embedding.
problem Finding a surface embedding for general spatial graphs is not always possible.
method Define leveled property, decompose graph into subgraphs, and construct surface.
result For leveled spatial graphs with a small number of levels, a surface can always be found.
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.
problem Proving Morita invariance of cohomology theories for Lie groupoids.
method Viewing cohomology as sheaves of modules on the nerve of the groupoid and establishing criteria for Morita invariance.
result Established criteria for Morita invariant cohomology theories.
Generalized Nakano positivity for certain singular cases.
problem Nakano positivity of direct image sheaves for singular cases.
method Generalization of Berndtsson's result to singular cases.
result Nakano positivity for direct image sheaves in special singular cases.
We give examples of harmonic cellular maps between negatively curved manifolds which are not diffeomorphisms but are homotopic to diffeomorphisms.
The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally …
We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit Kähler-Einstein metrics. We also show t…
Study immersions of punctured 4-manifolds for quantum automata applications.
problem Existence of immersions between specific 4-manifolds.
method Analyzing immersions of punctured 4-manifolds to establish a partial order.
result Established a partial order on closed 4-manifolds via immersions.
Sheaves on graphs link to noncommutative geometry.
problem Exploring noncommutative geometry concepts on graphs.
method Sheaf theory and simplicial sets.
result Enhanced understanding of discrete noncommutative geometry.
Geometrically computes sheaves linking HOMFLY-PT homology to Hilbert schemes.
problem Linking HOMFLY-PT homology to geometric structures on Hilbert schemes.
method Geometric sheaf theory, Hochschild homology formality, Hilbert schemes of points.
result Established formalism connecting HOMFLY-PT homology to coherent sheaves on Hilbert schemes.
Study homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
This is a large audience version of our previous work (see math.AG/0301146) in which we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of ∂ˉ-coherent sheaves. We also include here the complete proof of our main Theorem.
The paper extends Gaussian processes to model complex interactions in cellular complexes.
problem Capturing topological inductive biases in machine learning models.
method Proposes Gaussian processes on cellular complexes, introducing novel kernels.
result Derives two novel kernels for modeling interactions between cells.
In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem.
Proves Verdier duality for sheaves on stratified spaces.
problem Verdier duality for constructible sheaves on stratified spaces.
method Uses conically smooth stratified spaces and Lurie's Verdier duality.
result Shows equivalence between constructible sheaves and cosheaves.
Extends six operations to sheaves in any symmetric monoidal category.
problem Extending six operations to a broader class of sheaves.
method Develops formalism for sheaves in any closed symmetric monoidal ∞-category, proving properties of locally contractible geometric morphisms and relating pullbacks and colimits.
result Establishes the six functor formalism for a wider range of sheaves, including those with values in any closed symmetric monoidal ∞-category.
In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs th…
In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, Sesum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kahler-Einstein metrics.
Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.
We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geom…