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REVIEW 3 major objections 6 minor 1 cited by

Cellular Sheaves on Higher-Dimensional Structures

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Cellular sheaves can be defined on triangles and higher simplices, with ANM-based 0-th Laplacians recovering the ANM Hessian and higher Laplacians encoding multi-way interactions; an algebraic route via ringed spaces is also given.

desk verdict The paper's advertised ANM-Hessian recovery for higher-dimensional sheaves is contradicted by its own examples; the concrete sheaves are worth a second look, but the abstract and Theorem 2 need serious revision. read the letter →

arxiv 2505.23993 v3 pith:PEI3T6H4 submitted 2025-05-29 math.AT

Chuan-Shen Hu This is my paper
classification math.AT MSC 55N3018F2005E45
keywords cellularsheavessimplicialcomplexessheafLaplacianANMHessiantensorproductofringedspacesweightedhomologymodules
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Cellular sheaves are typically defined on graphs, but the data of a molecular system or a point cloud includes triangles, tetrahedra, and higher cells, and this paper argues that sheaf structures can be built directly on those higher-dimensional simplices. It proposes three construction methods: a geometric one extending the anisotropic network model (ANM) of protein dynamics, a tensor-product operation that combines sheaves, and an algebraic one using ringed spaces and sheaves of modules over a polynomial ring. The flagship claim is that the 0-th sheaf Laplacian of the ANM-based construction recovers the classical ANM Hessian matrix, while higher-dimensional sheaf Laplacians encode multi-way interactions among edges and faces. If correct, these constructions would move sheaf-based learning and topological data analysis beyond pairwise, graph-level models to genuine higher-order interactions on simplicial complexes.

What carries the argument

The central object is the cellular sheaf $\mathcal F:(K,\leq)\to \mathrm{vect}_{\mathbb R}$, a functor from the face poset of a simplicial complex to finite-dimensional vector spaces, together with the sheaf Laplacian $\Delta^q_{\mathcal F}=(\delta^q)^*\delta^q+\delta^{q-1}(\delta^{q-1})^*$ on cochain spaces $C^q(K;\mathcal F)=\bigoplus_{\sigma\in K^{(q)}}\mathcal F_\sigma$. The constructions are carried by the choice of stalks and restriction maps: the ANM sheaf uses vertex-to-edge maps $\frac{\gamma^{1/2}}{d^\circ_{ij}}(\mathbf r_j-\mathbf r_i)^\top$, the 2-simplex extension uses edge-to-face maps given by $v_{ijk}^\top$, and the tensor product $\mathcal F\otimes\mathcal G$ combines two sheaves stalkwise and on restriction maps. The algebraic constructions are powered by quotient sheaves $R/I_\sigma$ and localizations $R_{P_\sigma}$ attached to ideal-valued functors on the face poset, and by a module-valued functor on the opposite poset whose coboundary maps reproduce the weighted boundary operator $\partial^w_q$. What carries the argument is the commutativity of the face-poset diagrams, which forces the composition of restriction maps through a 2-simplex to be zero in Example 4 and the tensor product to preserve the sheaf axioms.

What would settle it

Compute the 0-th sheaf Laplacian of Example 4 directly from the coboundary matrix $C^0$ given in the paper: the result is an $n\times n$ scalar matrix with off-diagonal entries $-w_{ij}^2\|\mathbf r_i-\mathbf r_j\|^2$, whereas the ANM Hessian is a $3n\times 3n$ block matrix with $3\times3$ blocks; equality of the two would already fail by dimension, so the precise meaning of 'recovers the Hessian' is what the construction actually needs.

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Extended reading notes

Core claim

The paper's central claim is that cellular sheaves — functors from the face poset of a simplicial complex to finite-dimensional vector spaces — can be constructed on complexes containing triangles, tetrahedra, and higher simplices, not merely on graphs. The primary geometric construction starts from the anisotropic network model sheaf on the graph of an atomic system, in which the stalk at a vertex is $\mathbb R^3$, the stalk at an edge is $\mathbb R$, and the restriction from a vertex to an edge is the map $\frac{\gamma^{1/2}}{d^\circ_{ij}}(\mathbf r_j-\mathbf r_i)^\top$; the paper cites the result that the 0-th sheaf Laplacian of this graph sheaf equals the ANM Hessian. It then extends this data to 2-simplices by assigning edge stalks $\mathbb R^3$, face stalks $\mathbb R$, and edge-to-face restriction maps given by a face-normal vector $v_{ijk}^\top$, with the explicit consequence that the 0-th Laplacian of Example 4 is the $n\times n$ scalar weighted graph Laplacian with entries $-w_{ij}^2\|\mathbf r_i-\mathbf r_j\|^2$. A tensor-product variant, $\mathcal F\otimes\mathcal G$, produces a sheaf whose 0-th Laplacian is $L^0_{\mathcal F}\otimes I_3$, embedding the graph-scale Laplacian as a factor. In parallel, the paper defines algebraic sheaves of rings by quotienting a polynomial ring by ideals $I_\sigma$ attached to each simplex, and sheaves of modules on the opposite poset that reproduce weighted simplicial homology $H_q(K;w)$ as sheaf cohomology $H^q(K;\mathcal F)$.

Load-bearing premise

The Hessian-recovery claim is not proved in this paper; it rests wholly on the cited result from the author's earlier work that the graph-level ANM sheaf has 0-th Laplacian equal to the ANM Hessian, and on the assumption that this equality survives the tensor-product and 2-simplex extensions described here.

Editorial extensions

If this is right

  • If the constructions are correct, cellular sheaf models can attach meaningful stalks and restriction maps to triangles and tetrahedra, so sheaf Laplacians carry information about interactions among three or more vertices rather than only edge pairs.
  • The tensor-product construction yields the identity $L^0_{\mathcal F\otimes\mathcal G}=L^0_{\mathcal F}\otimes I_3$, meaning the graph-scale Laplacian is preserved as a factor in the higher-dimensional sheaf, while $L^1$ is enriched by the term $(C^1_{\mathcal F\otimes\mathcal G})^\top C^1_{\mathcal F\otimes\mathcal G}$ from edge-face relations.
  • The algebraic construction identifies weighted simplicial homology $H_q(K;w)$ with sheaf cohomology $H^q(K;\mathcal F)$ of a module-valued functor on the opposite face poset, so spectral and Hodge-theoretic tools for sheaves become available for weighted homology.
  • The ringed-space examples identify the ring of global sections of a two-vertex sheaf with the fibre product of rings, giving a concrete combinatorial model of gluing local algebraic data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An immediate test of the geometric construction is to build the sheaf of Example 4 on the alpha-shape simplicial complex of a protein and compare the 1-st Laplacian eigenvectors with molecular-dynamics fluctuations; the edge-face terms should either resolve modes the ANM Hessian misses or show up as spectral noise.
  • The quotient sheaves $S/I_\sigma$ are essentially squarefree monomial rings of the complex, so reading the 0-th sheaf cohomology of these sheaves could connect the framework to Hilbert-series invariants from combinatorial commutative algebra, a link the paper leaves implicit.
  • The tensor-product operation suggests a general lifting principle: tensor any graph-level sheaf with a sheaf encoding higher-codimension data to obtain a parametrized family of simplicial sheaf Laplacians, which could be used to interpolate between graph-only and fully simplicial models in sheaf neural networks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes three strategies for constructing cellular sheaves on simplicial complexes of dimension at least two: a direct extension of an ANM-inspired sheaf, a tensor-product extension, and an algebraic construction via ringed spaces and sheaves of modules. The main advertised claim is that certain of these sheaf structures have 0-th sheaf Laplacians equal to classical ANM Hessian matrices, with higher sheaf Laplacians capturing multi-way interactions. The paper also states a theorem identifying weighted simplicial homology with the sheaf cohomology of a canonically defined sheaf. The algebraic sections give several examples of sheaves of rings and modules based on ideals and localizations.

Significance. If the central claim were true as stated, the paper would provide a clean sheaf-theoretic interpretation of ANM Hessians and a systematic way to extend such structure to higher-dimensional simplices, which could be of interest in topological data analysis and geometric learning. The graph-level construction of Example 3 is correct: a direct calculation shows that its 0-th sheaf Laplacian has off-diagonal blocks equal to -γ/(d^0_ij)^2 (r_j-r_i)(r_j-r_i)^T, matching the ANM Hessian. The higher-dimensional constructions in Examples 4 and 5 are explicit and may be useful as building blocks, and the algebraic framework offers a clear, if elementary, perspective on weighted homology. However, the abstract's assertion about 0-th Laplacians recovering ANM Hessians does not hold for the higher-dimensional examples, which is a load-bearing issue for the paper's motivation. The paper does not include machine-checked proofs, code, or falsifiable predictions, but the computations involved are simple enough to verify by hand.

major comments (3)
  1. [Abstract; Section 4.1; Example 4; Example 5, Eq. (9)] The abstract claims that the introduced sheaf constructions have 0-th sheaf Laplacians equal to classical ANM Hessian matrices. For the graph sheaf of Example 3 this is true, but for the higher-dimensional sheaves it is not. In Example 4, the 0-th sheaf Laplacian is an n x n matrix with entries L0_ij = -w_ij^2 ||r_j-r_i||^2, i.e., a scalar weighted graph Laplacian rather than the 3n x 3n ANM Hessian. In Example 5, Eq. (9) explicitly gives L0_{F⊗G} = L0_F ⊗ I_3, again not the ANM Hessian. Hessian-like blocks appear only in C0(C0)^T, which is part of the 1-st sheaf Laplacian L1, not the 0-th. The central claim therefore must be restricted to Example 3, or reworded to say that the higher-dimensional constructions retain Hessian information in the 1-st sheaf Laplacian rather than recovering the Hessian at level 0.
  2. [Example 3, proof] The equality L0_F = H_ANM is not proved in the paper; the proof is delegated entirely to reference [38], an unpublished preprint by the same author. Since this equality is the foundation of the paper's advertised ANM connection, the paper should include the short direct computation. With the stated restriction maps F_i,[i,j] = γ^{1/2}/d^0_ij (r_j-r_i)^T, the (i,j) block of (C0)^T C0 is -γ/(d^0_ij)^2 (r_j-r_i)(r_j-r_i)^T for an edge {i,j}, which is exactly the ANM Hessian off-diagonal block; providing this computation would make the paper self-contained and remove the dependence on an unreviewed source for a load-bearing result.
  3. [Theorem 2 and Definition 3] Theorem 2 states that H_q(K; w) = H^q(K; F) for the sheaf F defined in Definition 3. As stated, this is essentially a restatement of the construction: the maps F_{τ,σ} are chosen precisely so that the weighted boundary operator ∂^w_q in Eq. (13) coincides with the sheaf coboundary. The theorem is not proven in the text, and the grading convention for 'sheaf cohomology' of a functor on (K,≤)^op is not specified carefully. The paper should either prove the identification in a few lines or present it as an observation, and it should clarify the degree indexing (q-th cohomology versus q-th homology).
minor comments (6)
  1. [Section 4.2, first paragraph] The sentence 'we focus on ANM-based sheaves defined on graphs embedded in R2' appears contradictory with the rest of the section, which uses stalks in R3 and R9; this should be R3.
  2. [Section 4.1, after Eq. for L0_ij] The phrase 'the original 3n x 3n Hessian information is, in a certain sense, encoded within the diagonal blocks of the matrix L' uses an undefined symbol L; it should refer explicitly to C0(C0)^T or to L1_F.
  3. [Definition 3, Eq. (13)] The notation F_{σ,w(σ)} in Eq. (13) is not defined; the restriction map from Definition 3 is written as F_{τ,σ} for σ ≤ τ, and Eq. (13) should use that notation or introduce a clear replacement.
  4. [Example 5] After deriving L0_{F⊗G} = L0_F ⊗ I_3, the paper should explicitly note that this is not the ANM Hessian, to avoid reinforcing the overbroad abstract claim; the Hessian information appears only in C0_{F⊗G}(C0_{F⊗G})^T.
  5. [Throughout] There are numerous typographical and notational issues that require a careful editing pass, including: 'As a part of, the n x n 0-th Laplacian matrix', 'C0F(C0F) of the 1-th sheaf Laplacian' (missing transpose), and inconsistent use of F for both a field and a sheaf. These do not affect the mathematics but should be corrected.
  6. [Sections 4.3, Examples 7–9] The algebraic constructions in Examples 7–9 are clear, but the paper should specify whether the quotient sheaves S/I are intended as sheaves of rings with the identity element preserved by the canonical maps; the current text is consistent with this, but the phrasing could be tightened.

Circularity Check

2 steps flagged · score 4.0 of 10

Theorem 2 is a definitional restatement, and the flagship L0=H_ANM identification is delegated to the author's own preprint; however, Example 3 is directly verifiable and most of the paper's constructions are not circular.

  1. self definitional [Section 4.3, Definitions 2-3 and Theorem 2]
    "Fτ,σ : R → R is defined as the R-module homomorphism r 7→ (w(τ )/w(σ)) · r ... Then, Equation (12) becomes ∂w_q |Rσ = ... which is precisely the restriction of the q-th coboundary map δq ... In other words, the weighted homology Hq(K; w) ... coincides with the sheaf cohomology H q(K; F ) ... Theorem 2 ... Hq(K; w) = H q(K; F )."

    The sheaf in Definition 3 is built so that its stalk is R and its restriction map is multiplication by w(τ)/w(σ); hence its coboundary is, by construction, exactly the weighted boundary operator ∂w. The equality of homology groups is therefore a definitional restatement of the functor, not a derived theorem. Every weighted boundary operator can be encoded this way, so no independent content is added by the identification.

  2. self citation load bearing [Section 3, Example 3 (statement and proof)]
    "In particular, as shown in the following example (see Example 3), the Hessian matrix in the ANM model ... coincides with the sheaf Laplacian associated with this construction [38]. ... Proof. To verify that the ANM Hessian matrix is exactly the sheaf Laplacian L0_F of the sheaf defined in this example, see [38] for a detailed proof."

    The paper's central advertised result—that the 0-th sheaf Laplacian recovers the ANM Hessian—is the premise on which the higher-dimensional extensions are built, yet the proof consists solely of a reference to the author's own preprint [38]. The derivation chain for the headline claim therefore terminates in a self-citation. The equality is in fact checkable from the stated vertex-to-edge maps, but the manuscript itself does not supply that computation.

full rationale

The strongest genuinely circular feature is Theorem 2, where the sheaf is defined specifically so that its coboundary operators equal the weighted boundary operators; the theorem restates the definition. The ANM-Hessian identification in Example 3 is also presented through a self-citation rather than a proof, which is load-bearing for the abstract's claim. However, the underlying computation is elementary and the stated maps do produce the correct 3x3 Hessian blocks, so this is a presentation/citation issue rather than a false or fitted result. Separately, the paper's Examples 4 and 5 do not actually realize the abstract's promise: their L0_F are a scalar weighted graph Laplacian (tensored with I3 in Example 5), and the text itself locates the Hessian-like blocks in C0(C0)^T, a component of L1_F rather than L0_F. That is an overstatement about the new constructions, not a circularity. Overall, one definitional theorem and one load-bearing self-citation justify a moderate score, but the majority of the paper's algebraic constructions are self-contained.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The only genuinely new content is a set of explicit sheaf examples. The constructions introduce no new entities; they use standard algebraic objects. The main free choice is the scaling of the restriction maps, tuned so that the Laplacian has the desired physical meaning.

free parameters (1)
  • edge restriction scaling γ^{1/2}/d^0_ij = γ and d^0_ij are inputs from the ANM model
    The vertex-to-edge restriction maps in Example 3 are chosen as γ^{1/2}/d^0_ij (r_j - r_i)^T precisely so that the 0-th sheaf Laplacian matches the ANM Hessian; this choice is the content of the claim, not derived from more basic principles.
assumptions (4)
  • standard math Cellular sheaf cohomology and the Hodge theorem H^q is isomorphic to ker Δ^q (Theorem 1)
    Invoked with a citation to [32, Theorem 3.1]; not proved in the paper.
  • standard math Weighted boundary maps satisfy ∂_{q-1} ∘ ∂_q = 0
    Assumed from weighted homology literature [24,45]; this underpins Theorem 2.
  • domain assumption Every 2-simplex with vertices embedded in R^3 has a non-zero perpendicular vector v_ijk
    Used to make edge-to-face compositions vanish in Example 4; for collinear vertices the cross product is zero and the sheaf's higher maps become trivial.
  • domain assumption The ANM Hessian equals the 0-th sheaf Laplacian of the graph sheaf from Example 3
    The paper cites [38] for this result instead of proving it; the abstract's headline claim depends on it.

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Pith. "Pith review of Cellular Sheaves on Higher-Dimensional Structures." pith.science (2026). https://pith.science/paper/PEI3T6H4

@misc{pith2026250523993,
  author       = {Pith},
  title        = {Pith review of: Cellular Sheaves on Higher-Dimensional Structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PEI3T6H4}},
  note         = {Machine review of arXiv:2505.23993}
}
read the original abstract

Defining cellular sheaves beyond graph structures, such as on simplicial complexes containing higher-dimensional simplices, is an essential and intriguing topic in topological data analysis (TDA) and the development of sheaf neural networks. In this paper, we explore methods for constructing non-trivial cellular sheaves on spaces that include structures of dimension greater than one. This extends the focus from 0- or 1-dimensional components, such as vertices and edges, to elements like triangles, tetrahedra, and other higher-dimensional simplices within a simplicial complex. We develop a unified framework that incorporates both geometric and algebraic approaches to modeling such complex systems using cellular sheaf theory. Motivated by the geometric and physical insights from anisotropic network models (ANM), we first introduce constructions that define sheaf structures whose 0-th sheaf Laplacians recover classical ANM Hessian matrices. The higher-dimensional sheaf Laplacians in this setting encode additional patterns of multi-way interactions. In parallel, we propose an algebraic framework based on commutative algebra and ringed spaces, where sheaves of ideals and modules are used to define sheaf structures in a combinatorial and algebraically grounded manner. These two perspectives -- the geometric-physical and the algebraic -- offer complementary strengths and together provide a versatile framework for encoding structural relationships and analyzing multi-scale data over simplicial complexes.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Representing Higher-Order Networks: A Survey of Graph-Based Frameworks

    cs.SI 2026-03 unverdicted novelty 4.0 of 10

    A survey organizing higher-order network formalisms into four families with a master comparison table, plus ~17 new superhypergraph-style definitions whose only supporting theorems are well-definedness checks.

Reference graph

Works this paper leans on

51 extracted references · 46 canonical work pages · cited by 1 Pith paper

  1. [38]

    C.-S. Hu, X. Liu, and K. Xia. A physics-informed sheaf model. arXiv preprint arXiv:2501.06197, 2024

    work page Pith review arXiv 2024

  2. [1]

    Alexandroff

    P. Alexandroff. Diskrete Räume. Mat. Sb, 2:501-518, 1937

    work page 1937

  3. [2]

    P. S. Alexandrov. Combinatorial Topology. Dover Publications, Inc., 31 East 2nd Street, Mineola, NY , 1947

    work page 1947

  4. [3]

    D. V . Anand, Z. Y . Meng, K. L. Xia, and Y . G. Mu. Weighted persistent homology for osmolyte molecular aggregation and hydrogen-bonding network analysis. Scientific reports, 10(1):1–17, 2020

    work page 2020

  5. [4]

    A. R. Atilgan, S. R. Durell, R. L. Jernigan, M. C. Demirel, O. Keskin, and I. Bahar. Anisotropy of fluctuation dynamics of proteins with an elastic network model. Biophysical Journal, 80(1):505–515, 2001

    work page 2001

  6. [5]

    M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra . Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802, 1969

    work page 1969

  7. [6]

    Ayzenberg, T

    A. Ayzenberg, T. Gebhart, G. Magai, and G. Solomadin. Sheaf theory: from deep geometry to deep learning. arXiv preprint arXiv:2502.15476, 2025

    arXiv 2025

  8. [7]

    Bahar, A

    I. Bahar, A. R. Atilgan, M. C. Demirel, and B. Erman. Vibrational dynamics of folded proteins: significance of slow and fast motions in relation to function and stability. Physical Review Letters, 80(12):2733–2736, 1998. 13 A PREPRINT - JULY 11, 2025

    work page 1998

Show all 51 references
  1. [8]

    Bahar, A

    I. Bahar, A. R. Atilgan, and B. Erman. Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential. Folding and Design, 2(3):173–181, 1997

    1997

  2. [9]

    Barbero, C

    F. Barbero, C. Bodnar, H. S. de Ocáriz Borde, and P. Lio. Sheaf attention networks. In NeurIPS 2022 Workshop on Symmetry and Geometry in Neural Representations, 2022

    2022

  3. [10]

    Battiloro, Z

    C. Battiloro, Z. Wang, H. Riess, P. Di Lorenzo, and A. Ribeiro. Tangent bundle filters and neural networks: From manifolds to cellular sheaves and back. In ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1–5. IEEE, 2023

    2023

  4. [11]

    G. Bell, A. Lawson, J. Martin, J. Rudzinski, and C. Smyth. Weighted persistent homology. Involve, a Journal of Mathematics, 12(5):823–837, 2019

    2019

  5. [12]

    Bodnar, F

    C. Bodnar, F. Di Giovanni, B. Chamberlain, P. Lio, and M. Bronstein. Neural sheaf diffusion: A topological perspective on heterophily and oversmoothing in GNNs. Advances in Neural Information Processing Systems, 35:18527–18541, 2022

    2022

  6. [13]

    Bodnar, F

    C. Bodnar, F. Frasca, Y . Wang, N. Otter, G. F. Montufar, P. Lio, and M. Bronstein. Weisfeiler and lehman go topological: Message passing simplicial networks. In International Conference on Machine Learning, pages 1026–1037. PMLR, 2021

    2021

  7. [14]

    Braithwaite, I

    L. Braithwaite, I. Duta, and P. Liò. Heterogeneous sheaf neural networks. arXiv preprint arXiv:2409.08036, 2024

    2024 arXiv

  8. [15]

    G. E. Bredon. Sheaf Theory, volume 170. Springer Science & Business Media, 2012

    2012

  9. [16]

    F. H. Caralt, G. Bernardez, I. Duta, E. Alarcon, and P. Lio. Joint diffusion processes as an inductive bias in sheaf neural networks. In ICML 2024 Workshop on Geometry-grounded Representation Learning and Generative Modeling, 2024

    2024

  10. [17]

    Cooperband

    Z. Cooperband. Cellular Cosheaves, Graphic Statics, and Mechanics. PhD thesis, University of Pennsylvania, 2024

    2024

  11. [18]

    Cooperband and R

    Z. Cooperband and R. Ghrist. Towards homological methods in graphic statics. Journal of the International Association for Shell and Spatial Structures, 64(4):266–277, 2023

    2023

  12. [19]

    Cooperband, R

    Z. Cooperband, R. Ghrist, and J. Hansen. A cosheaf theory of reciprocal figures: Planar and higher genus graphic statics. arXiv preprint arXiv:2311.12946, 2023

    2023 arXiv

  13. [20]

    Cooperband, M

    Z. Cooperband, M. Lopez, and B. Schulze. Equivariant cosheaves and finite group representations in graphic statics. arXiv preprint arXiv:2401.09392, 2024

    2024 arXiv

  14. [21]

    Curry, R

    J. Curry, R. Ghrist, and V . Nanda. Discrete Morse theory for computing cellular sheaf cohomology.Foundations of Computational Mathematics, 16:875–897, 2016

    2016

  15. [22]

    J. M. Curry. Sheaves, cosheaves and applications. PhD thesis, University of Pennsylvania, 2014

    2014

  16. [23]

    J. M. Curry. Topological data analysis and cosheaves. Japan Journal of Industrial and Applied Mathematics, 32:333–371, 2015

    2015

  17. [24]

    R. J. M. Dawson. Homology of weighted simplicial complexes. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 31(3):229–243, 1990

    1990

  18. [25]

    Doruker, A

    P. Doruker, A. R. Atilgan, and I. Bahar. Dynamics of proteins predicted by molecular dynamics simulations and analytical approaches: Application to α-amylase inhibitor. Proteins: Structure, Function, and Bioinformatics, 40(3):512–524, 2000

    2000

  19. [26]

    I. Duta, G. Cassarà, F. Silvestri, and P. Liò. Sheaf hypergraph networks. Advances in Neural Information Processing Systems, 36, 2024

    2024

  20. [27]

    Eyal, L.-W

    E. Eyal, L.-W. Yang, and I. Bahar. Anisotropic network model: systematic evaluation and a new web interface. Bioinformatics, 22(21):2619–2627, 2006

    2006

  21. [28]

    R. Ghrist. Network sheaf models for social information dynamics. In 2022 IEEE 8th International Conference on Collaboration and Internet Computing (CIC), pages 31–37. IEEE, 2022

    2022

  22. [29]

    Ghrist and H

    R. Ghrist and H. Riess. Cellular sheaves of lattices and the Tarski Laplacian. Homology, Homotopy and Applications, 24(1):325–345, 2022

    2022

  23. [30]

    Hajij, L

    M. Hajij, L. Bastian, S. Osentoski, H. Kabaria, J. L. Davenport, S. Dawood, B. Cherukuri, J. G. Kocheemoolayil, N. Shahmansouri, A. Lew, T. Papamarkou, and T. Birdal. Copresheaf topological neural networks: A generalized deep learning framework, 2025

    2025

  24. [31]

    Hansen and T

    J. Hansen and T. Gebhart. Sheaf neural networks. arXiv preprint arXiv:2012.06333, 2020. 14 A PREPRINT - JULY 11, 2025

    2012 arXiv

  25. [32]

    Hansen and R

    J. Hansen and R. Ghrist. Toward a spectral theory of cellular sheaves. Journal of Applied and Computational Topology, 3(4):315–358, 2019

    2019

  26. [33]

    Hansen and R

    J. Hansen and R. Ghrist. Opinion dynamics on discourse sheaves. SIAM Journal on Applied Mathematics , 81(5):2033–2060, 2021

    2021

  27. [34]

    Hartshorne

    R. Hartshorne. Algebraic Geometry, volume 52. Springer Science & Business Media, 2013

    2013

  28. [35]

    Hayes, X

    N. Hayes, X. Wei, H. Feng, E. Merkurjev, and G.-W. Wei. Persistent sheaf laplacian analysis of protein flexibility. The Journal of Physical Chemistry B, 2025

    2025

  29. [36]

    Y . He, C. Bodnar, and P. Lio. Sheaf-based positional encodings for graph neural networks. In NeurIPS 2023 Workshop on Symmetry and Geometry in Neural Representations, 2023

    2023

  30. [37]

    C.-S. Hu. A brief note for sheaf structures on posets. arXiv preprint arXiv:2010.09651, 2020

    2010 arXiv

  31. [39]

    Kashiwara and P

    M. Kashiwara and P. Schapira. Persistent homology and microlocal sheaf theory. Journal of Applied and Computational Topology, 2(1):83–113, 2018

    2018

  32. [40]

    Matsumura and M

    H. Matsumura and M. Reid. Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1989

    1989

  33. [41]

    Z. Y . Meng, D. V . Anand, Y . P. Lu, J. Wu, and K. L. Xia. Weighted persistent homology for biomolecular data analysis. Scientific Reports, 10(1):1–15, 2020

    2020

  34. [42]

    Opron, K

    K. Opron, K. L. Xia, and G. W. Wei. Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis. Journal of Chemical Physics, 140:234105, 2014

    2014

  35. [43]

    Papamarkou, T

    T. Papamarkou, T. Birdal, M. Bronstein, G. Carlsson, J. Curry, Y . Gao, M. Hajij, R. Kwitt, P. Lio, P. Di Lorenzo, et al. Position: Topological deep learning is the new frontier for relational learning. Proceedings of Machine Learning Research, 235:39529–39555, 2024

    2024

  36. [44]

    J.-K. Park, R. Jernigan, and Z. Wu. Coarse grained normal mode analysis vs. refined gaussian network model for protein residue-level structural fluctuations. Bulletin of Mathematical Biology, 75:124–160, 2013

    2013

  37. [45]

    S. Q. Ren, C. Y . Wu, and J. Wu. Weighted persistent homology. Rocky Mountain Journal of Mathematics , 48(8):2661–2687, 2018

    2018

  38. [46]

    H. Riess. Lattice theory in multi-agent systems. PhD thesis, University of Pennsylvania, 2022

    2022

  39. [47]

    Robinson

    M. Robinson. Topological Signal Processing. Mathematical Engineering. Springer Berlin Heidelberg, 2014

    2014

  40. [48]

    Wei and G.-W

    X. Wei and G.-W. Wei. Persistent sheaf laplacians. Foundations of Data Science, 2024

    2024

  41. [49]

    F. Xia, D. Tong, L. Yang, D. Wang, S. C. Hoi, P. Koehl, and L. Lu. Identifying essential pairwise interactions in elastic network model using the alpha shape theory. Journal of Computational Chemistry, 35(15):1111–1121, 2014

    2014

  42. [50]

    K. L. Xia, K. Opron, and G. W. Wei. Multiscale Gaussian network model (mGNM) and multiscale anisotropic network model (manm). The Journal of chemical physics, 143(20):204106, 2015

    2015

  43. [51]

    L. W. Yang and C. P. Chng. Coarse-grained models reveal functional dynamics–I. elastic network models–theories, comparisons and perspectives. Bioinformatics and Biology Insights, 2:25 – 45, 2008. 15

    2008

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.