Pith. sign in

REVIEW 3 major objections 5 minor 80 references

Weighted Hodge Laplacians on Manifolds with Boundary

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Adding a positive weight function to the Hodge Laplacian changes the metric but not the topology: the kernel stays isomorphic to the de Rham cohomology of the manifold, so zero eigenvalues still count holes.

desk verdict The computational framework is worth using, but the paper's advertised theorem—the boundary-case Hodge isomorphism—is asserted, not proved. read the letter →

arxiv 2608.00244 v1 pith:XPMVBETH submitted 2026-07-31 math.DG

Zhe Su , Yiying Tong , Guo-Wei Wei This is my paper · ORCID
classification math.DG MSC 58A1455N31
keywords weightedHodgeLaplaciandriftingdeRhamcohomologymanifoldswithboundarydiscreteexteriorcalculustopologicaldataanalysisproteinflexibilityBettinumbers
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

The paper establishes a weighted de Rham–Hodge theory for compact manifolds with boundary by replacing the codifferential with δ_f = e^f δ e^{-f} and redefining the inner product with density e^{-f}. The central result is that, under normal or tangential boundary conditions, the kernel of the weighted Hodge Laplacian coincides with the space of weighted harmonic forms and remains isomorphic to the ordinary relative or absolute de Rham cohomology — the weight f drops out of the cohomology entirely. So the zero eigenvalues of the weighted Laplacian continue to compute Betti numbers, while the nonzero eigenvalues become a tunable geometric fingerprint that can emphasize local regions. The paper also provides a discrete exterior calculus implementation on Cartesian grids, a weighted boundary-induced graph Laplacian, and a protein flexibility application where the zero-th Betti number plus low nonzero eigenvalues improve blind B-factor prediction.

What carries the argument

The engine of the argument is the pair (δ_f = e^f δ e^{-f}, ⋆_f = e^{-f}⋆). The weighted inner product (ω,η)_f = ∫_M ⟨ω,η⟩ e^{-f} dμ makes δ_f the adjoint of d on the boundary-adapted subspaces Ω̄^k_{n,f} = {ω : ω|_{∂M}=0, δ_f ω|_{∂M}=0} and Ω^k_t, yielding the key identity (Δ_f ω, ω)_f = (dω,dω)_f + (δ_f ω, δ_f ω)_f. This identity forces ker Δ_f = ker d ∩ ker δ_f on those subspaces. The weighted Hodge star ⋆_f = e^{-f}⋆ pairs the two boundary conditions, and the chain map T(ω)=e^{-f}ω between the δ_f and δ complexes shows the cohomology is f-independent. On the discrete side, the same structure is reproduced with projection matrices P_k for the normal support, fractional-volume Hodge stars,

What would settle it

On a small Cartesian-grid annulus or solid torus, compute the full spectrum of L_{k,f} and of the weighted BIG Laplacian for two weights, one constant and one sharply localized; if the zero-eigenvalue multiplicity changes with f or departs from β_{m-k}, the discrete cohomology-preservation claim fails. The same check can be run on a coarse grid where boundary cells are intersected by the level set.

Watch

Extended reading notes

Core claim

The paper proves that adding a smooth positive weight f to the Hodge Laplacian — by replacing the codifferential δ with δ_f = e^f δ e^{-f} and using the density e^{-f} dμ — does not change the topological content of the operator. On a compact orientable manifold with boundary, with normal (Dirichlet) or tangential (Neumann) boundary conditions, the kernel of the weighted Hodge Laplacian Δ_{n,f} or Δ_{t,f} equals the weighted harmonic space ℋ_{n,f} or ℋ_{t,f}, and this space is isomorphic to the relative cohomology H^k_{dR}(M, ∂M) or the absolute cohomology H^k_{dR}(M). The weight f therefore cancels in cohomology: the weighted de Rham cohomology H^k_{dR}(M,f) = ker δ_f / im δ_f is isomorphic

Load-bearing premise

The discrete kernel claim — that the weighted Laplacian's zero-eigenvalue count equals the Betti number for any weight, even at cells cut by the boundary — is inherited unproved from the prior unweighted discretization.

Editorial extensions

If this is right

  • Topological features are robust: zero-eigenvalue multiplicities of the weighted Hodge Laplacian still give Betti numbers under any positive weight.
  • Non-zero spectra become tunable local-geometric descriptors, so a single manifold yields many data-dependent fingerprints by varying f.
  • The weighted BIG Laplacian offers a Hodge-star-free computation with the same kernel size, making large-scale spectral analysis cheaper.
  • Spectral computation reduces to the singular spectrum of weighted discrete differentials, preserving the efficiency of unweighted methods.
  • In protein flexibility experiments, zero-th Betti number plus the first nonzero eigenvalues give blind B-factor predictions that improve on earlier models (PCC 0.862 atom-level, 0.524 protein-level).

Reading between the lines

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

  • Sweeping the weight f continuously suggests a persistence-style view: zero eigenvalues stay fixed while nonzero eigenvalues trace how local geometry changes with emphasis; the paper does not develop this, but the setup invites it.
  • The cohomology-cancellation mechanism is generic: any Laplacian built from a positive-function conjugate of d, including Witten-type operators, should inherit the same kernel-isomorphism, potentially with the standard (unweighted) inner product.
  • The atom-specific weight functions in the protein application are hand-chosen; treating them as trainable parameters could turn the weighted Laplacian into an end-to-end differentiable spectral layer.
  • Eigenvectors of weighted Laplacians, not just eigenvalues, may carry localized shape information that could sharpen structure comparison or generative modeling beyond what the paper reports.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a weighted Hodge Laplacian framework for compact Riemannian manifolds with boundary, based on the drifting (Bakry–Émery) codifferential δ_f = e^f δ e^{-f}. The main theoretical claim is that, under normal and tangential boundary conditions, the kernel of the weighted Hodge Laplacian coincides with the space of weighted harmonic fields and remains isomorphic to the relative and absolute de Rham cohomology, hence has dimension given by the Betti numbers independently of the weight f. A discrete counterpart is developed on Cartesian grids using discrete exterior calculus, including a weighted boundary-induced-graph (BIG) Laplacian, and the framework is applied to protein B-factor prediction with gradient-boosting regression, yielding improved Pearson correlation coefficients over earlier mDGL, PSL, and CAL models.

Significance. The proposed framework is a natural and potentially useful extension of Hodge Laplacians, combining topological descriptors that are weight-independent with geometric descriptors that depend on the weight. The discrete implementation and the protein flexibility experiments are concrete and reproducible in principle. However, the central mathematical claim of the paper — weight-independence of the kernel and its isomorphism to de Rham cohomology on manifolds with boundary — is not proved; it is asserted and deferred to future work. If this gap is filled, the paper would be a solid contribution to computational topology and geometric data analysis. As it stands, the significance is conditional on a nontrivial analytic result.

major comments (3)
  1. [Sec. 3.2, after Eq. (8)] The paper's central theorem — ker Δ_{n,f} = ℋ_{n,f} ≅ H^k_{dR}(M,∂M) and ker Δ_{t,f} = ℋ_{t,f} ≅ H^k_{dR}(M) — is stated with the justification that it 'follows by adapting the classical Hodge-theoretical arguments.' Remark 5 explicitly states that a proof of the weighted Hodge decomposition, and the 4- and 5-component decompositions, is 'beyond the scope of this paper.' This is not a proof. The adaptation is nontrivial because δ_f = e^f δ e^{-f} contains a first-order term in f, and the boundary trace conditions in (8) require an elliptic boundary-value analysis, not just a re-run of the classical Hodge–Morrey–Friedrichs argument. Since the weight-independence of the kernel is the basis for interpreting the zero-spectrum as Betti numbers, this gap is load-bearing for both the abstract's claim and the later discrete kernel statement. The authors should either provide a complete proof, ci
  2. [Sec. 4.2, paragraph after the definition of L_{k,f}] The statement 'Consistent with the smooth theory, the kernel of the discrete weighted Hodge Laplacian L_{k,f} is fully determined by the manifold topology, and its kernel dimension is given by the Betti number β_{m−k}, independent of f' is another load-bearing assertion. The unweighted result from [62] does not automatically cover the weighted case with boundary-intersecting cells and fractional-volume Hodge stars. The algebraic rank independence for invertible diagonal weights is plausible, but it needs to be shown for the projection-based normal support, including cells that meet ∂M. If this discrete kernel statement fails, the β0 feature used in the experiments would be miscalibrated even if the smooth theory were correct. A proof or a precise reference for the weighted discrete complex is required.
  3. [Sec. 5.5, Table 1] The experimental section uses the unproved kernel statements as part of the feature construction: the number of zero eigenvalues is reported as β0, and the remaining eigenvalues are used as geometric features. The experiments therefore do not independently validate the kernel independence claim; they only show that the resulting features are useful for B-factor regression. Since the theoretical foundation is missing, the experimental results cannot compensate for the gap. The comparison with mDGL, PSL, and CAL would be more meaningful with error bars or multiple runs, as the reported PCC improvements are modest (e.g., 0.862 vs. 0.855 at atom level). This is a secondary concern relative to the two points above.
minor comments (5)
  1. [Abstract and Sec. 3.2] The abstract states that the kernel of the weighted Hodge Laplacian coincides with the weighted harmonic space and is isomorphic to the de Rham cohomology. This is only true for the restrictions to the boundary-condition subspaces ¯Ω^k_{n,f} and ¯Ω^k_t; without those restrictions the space of harmonic fields is infinite-dimensional (as the paper itself notes in Sec. 2.2). The abstract should be phrased to avoid this ambiguity.
  2. [Sec. 3.1] The chain map T(ω)=e^{-f}ω correctly proves H^k_{dR}(M,f) ≅ H^k_{dR}(M). This is a nice observation and should be highlighted as a lemma, since it is one of the few fully proved structural statements in the paper.
  3. [Sec. 4.1] The notation f^I_k for the discretized weight is confusing: f is a function, but f^I_k is a diagonal matrix. The same applies to W_k and Remark 6. Please use a consistent notation such as F_k for the diagonal matrix of sampled values.
  4. [Sec. 5.2] The weight parameters (d=11, τ=5, η=4) are fixed by 'experiments' without a sensitivity analysis. Since the individual weight function is purported to highlight local features, reporting the dependence on these parameters would strengthen the claim that the method isolates local geometry.
  5. [General] There are several typos, e.g., 'codiffernetial' in Remark 1, and inconsistent capitalization in 'Witten–Hodge' vs. 'Witten Laplacian.' These should be corrected in a final version.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the weighted operator is defined, not fitted; the main gaps are an unproved weighted Hodge-Morrey step and a self-citation for the discrete kernel claim.

full rationale

Walking the derivation chain, the central smooth claim is not circular. The weighted codifferential δ_f = e^f δ e^{-f}, the weighted inner product (5), and the weighted Hodge Laplacian (7) are explicit definitions; identity (8) is an integration-by-parts identity; and the independence of the weighted absolute cohomology from f is proved by the chain map T(ω)=e^{-f}ω. The subsequent identification of weighted harmonic fields with de Rham cohomology is asserted, not derived: Sec. 3.2 says it "follows by adapting the classical Hodge-theoretical arguments," and Remark 5 explicitly states "A proof of these decompositions is beyond the scope of this paper." That is an omitted proof and a correctness risk, but it is not a reduction of the conclusion to the premise. The experimental section fits k, τ, η, and GBR hyperparameters to maximize PCC, but these choices do not enter the definition of Δ_f or the kernel=cohomology statement; B-factor labels are external data, so no fitted quantity is renamed as a first-principles prediction. The main self-citation load is in the discretization: the normal-support complex, the nilpotence "D_{k+1}D_k=0 remains valid [62]," and the unweighted spectral analysis are taken from [62] by the same authors, and the discrete Betti-dimension assertion in Sec. 4.2 is stated as "Consistent with the smooth theory" without proof here. This is a load-bearing citation for the numerical half, but it is not an equivalence by construction, and the weighted kernel-independence claim is algebraically checkable independently of the fitted protein data. Overall, the paper's mathematical core is derived from its own definitions and classical machinery, so there is no significant circularity; the appropriate score is 2 due to the minor-to-moderate self-citation and the unproved decomposition step.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The mathematical core rests on standard Hodge theory plus one unproved weighted generalization; the numerical/application parts introduce several hand-chosen parameters. No exotic physical entities are postulated.

free parameters (5)
  • Manifold density scale τ and isovalue c = τ=1, c=−0.1
    Chosen in Sec. 5.1 to produce smooth manifolds with stable topology; not derived from theory or data.
  • Atom-specific weight parameters cutoff d, τ, η = d=11, τ=5, η=4
    Chosen in Sec. 5.2; d is 'confirmed by our experiments', τ/η are chosen to include wider local interactions and avoid overemphasizing near neighbors.
  • Cartesian grid spacing l = 1
    Fixed in Sec. 5.3; asserted to give sufficient resolution, with no convergence study.
  • Number of WHL features k = 120 (WHL protein), 90 (consensus protein), 150 (WHL atom), 70 (consensus atom)
    Selected by 10-fold CV in Sec. 5.5; reported PCCs are the best over this search.
  • GBR hyperparameters = n_estimators=1000, max_depth=7, min_samples_split=5, learning_rate=0.002, subsample=0.8, max_features=sqrt
    Fixed by hand in Sec. 5.4 without tuning analysis; they influence reported PCCs.
assumptions (4)
  • standard math Classical Hodge theory for manifolds with boundary (Hodge-Morrey decomposition, harmonic-field isomorphisms to relative/absolute de Rham cohomology) holds as stated in Sec. 2.
    The paper relies on [27, 46, 60] for unweighted results; these are background, not proved in the paper.
  • ad hoc to paper The weighted Hodge isomorphism ℋ^k_{n,f} ≅ H^k_{dR}(M,∂M) and ℋ^k_{t,f} ≅ H^k_{dR}(M) follows by adapting classical arguments to the weighted boundary setting.
    Sec. 3.2 states this without proof; it is the central mathematical assertion and is not reduced to a cited theorem in the boundary case.
  • domain assumption The discrete projection/normal-support construction on the Cartesian grid preserves the de Rham cohomology and the kernel dimension of the weighted Laplacian, independent of f.
    Sec. 4.2 asserts 'Consistent with the smooth theory...'; carried over from [62] without proof here.
  • domain assumption Sublevel sets of the Gaussian density (12) with τ=1, c=−0.1 are smooth compact manifolds with stable topology representing each protein.
    Sec. 5.1 chooses these parameters to avoid numerical issues; no validation of topological stability is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Weighted Hodge Laplacians on Manifolds with Boundary." pith.science (2026). https://pith.science/paper/XPMVBETH

@misc{pith2026260800244,
  author       = {Pith},
  title        = {Pith review of: Weighted Hodge Laplacians on Manifolds with Boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XPMVBETH}},
  note         = {Machine review of arXiv:2608.00244}
}
read the original abstract

The spectrum of the Hodge Laplacian on differential manifolds encodes rich topological and geometric information and thus provides a powerful tool for analyzing data on manifolds. However, the classical unweighted formulation is restricted in its ability to study data with varying local features. To address this limitation, we propose a weighted Hodge Laplacian framework for manifolds with boundary, both in theory and in computation, by incorporating a weight function on the manifold. Under appropriate boundary conditions, we formulate the corresponding weighted de Rham-Hodge theory, in which the kernel of the weighted Hodge Laplacian coincides with the weighted harmonic space, and remains isomorphic to the de Rham cohomology of the underlying manifold. The harmonic spectrum of the weighted Hodge Laplacian captures the global topological information, while its non-harmonic spectrum encodes the local geometric property induced by the weight. The proposed framework therefore enables the study of topological and geometric features of data on manifolds across varying weights, and in addition, allows local structure to be highlighted by choosing weights that emphasize regions of interest. We demonstrate the effectiveness of the proposed method through proof-of-principle experiments in protein flexibility analysis, and the results show its promise.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

80 extracted references · 2 linked inside Pith

  1. [62]

    AIMS mathe- matics9(10), 27438 (2024)

    Su, Z., Tong, Y., Wei, G.-W.: Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning. AIMS mathe- matics9(10), 27438 (2024)

    2024

  2. [1]

    In: Borrowing Strength: Theory Powering Applications–a Festschrift for Lawrence D

    Adler, R.J., Bobrowski, O., Borman, M.S., Subag, E., Weinberger, S.: Persistent homology for random fields and complexes. In: Borrowing Strength: Theory Powering Applications–a Festschrift for Lawrence D. Brown vol. 6, pp. 124–144. Institute of Mathematical Statistics, (2010)

    2010

  3. [2]

    Differential Geometry and its Applications30(2), 179–194 (2012)

    Al-Zamil, Q.S., Montaldi, J.: Witten–Hodge theory for manifolds with boundary and equivariant cohomology. Differential Geometry and its Applications30(2), 179–194 (2012)

    2012

  4. [3]

    Bulletin of the American mathematical society47(2), 281–354 (2010)

    Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus: from hodge theory to numerical stability. Bulletin of the American mathematical society47(2), 281–354 (2010)

    2010

  5. [4]

    SIAM, (2018)

    Arnold, D.N.: Finite Element Exterior Calculus. SIAM, (2018)

    2018

  6. [5]

    Biophysical journal80(1), 505–515 (2001) 24 CONTENTS

    Atilgan, A.R., Durell, S., Jernigan, R.L., Demirel, M.C., Keskin, O., Bahar, I.: Anisotropy of fluctuation dynamics of proteins with an elastic network model. Biophysical journal80(1), 505–515 (2001) 24 CONTENTS

    2001

  7. [6]

    Publications Math´ ematiques de l’IH´ES68, 175–186 (1988)

    Atiyah, M.F.: Topological quantum field theory. Publications Math´ ematiques de l’IH´ES68, 175–186 (1988)

    1988

  8. [7]

    Folding and Design2(3), 173–181 (1997)

    Bahar, I., Atilgan, A.R., Erman, B.: Direct evaluation of thermal fluctu- ations in proteins using a single-parameter harmonic potential. Folding and Design2(3), 173–181 (1997)

    1997

Show all 80 references
  1. [8]

    Physical review letters80(12), 2733 (1998)

    Bahar, I., Atilgan, A.R., Demirel, M.C., Erman, B.: Vibrational dynamics of folded proteins: significance of slow and fast motions in relation to function and stability. Physical review letters80(12), 2733 (1998)

    1998

  2. [9]

    In: S´ eminaire de Probabilit´ es XIX 1983/84: Proceedings, pp

    Bakry, D., ´Emery, M.: Diffusions hypercontractives. In: S´ eminaire de Probabilit´ es XIX 1983/84: Proceedings, pp. 177–206. Springer, (2006)

    1983

  3. [10]

    Results in Mathematics79(5), 187 (2024)

    Branding, V., Habib, G.: Eigenvalue estimates on weighted manifolds. Results in Mathematics79(5), 187 (2024)

    2024

  4. [11]

    Journal of computational chemistry 4(2), 187–217 (1983)

    Brooks, B.R., Bruccoleri, R.E., Olafson, B.D., States, D.J., Swaminathan, S.a., Karplus, M.: Charmm: a program for macromolecular energy, mini- mization, and dynamics calculations. Journal of computational chemistry 4(2), 187–217 (1983)

    1983

  5. [12]

    Transactions of the American Mathematical Society351(2), 683–713 (1999)

    Bueler, E.: The heat kernel weighted Hodge Laplacian on noncompact manifolds. Transactions of the American Mathematical Society351(2), 683–713 (1999)

    1999

  6. [13]

    PLoS computational biology13(7), 1005690 (2017)

    Cang, Z., Wei, G.-W.: Topologynet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions. PLoS computational biology13(7), 1005690 (2017)

    2017

  7. [14]

    Inter- national journal for numerical methods in biomedical engineering34(2), 2914 (2018)

    Cang, Z., Wei, G.-W.: Integration of element specific persistent homology and machine learning for protein-ligand binding affinity prediction. Inter- national journal for numerical methods in biomedical engineering34(2), 2914 (2018)

    2018

  8. [15]

    The American Mathematical Monthly109(5), 409– 442 (2002)

    Cantarella, J., DeTurck, D., Gluck, H.: Vector calculus and the topology of domains in 3-space. The American Mathematical Monthly109(5), 409– 442 (2002)

    2002

  9. [16]

    Molecular pharmacology57(2), 213–218 (2000)

    Carlson, H.A., McCammon, J.A.: Accommodating protein flexibility in computational drug design1, 2. Molecular pharmacology57(2), 213–218 (2000)

    2000

  10. [17]

    Drug discovery today14(7-8), 394–400 (2009) CONTENTS 25

    Chandrika, B.-R., Subramanian, J., Sharma, S.D.: Managing protein flex- ibility in docking and its applications. Drug discovery today14(7-8), 394–400 (2009) CONTENTS 25

    2009

  11. [18]

    Foundations of Data Science5, 558–588 (2023)

    Chen, D., Liu, J., Wu, J., Wei, G.-W.: Persistent hyperdigraph homology and persistent hyperdigraph Laplacians. Foundations of Data Science5, 558–588 (2023)

    2023

  12. [19]

    Discrete and continuous dynamical systems

    Chen, J., Zhao, R., Tong, Y., Wei, G.-W.: Evolutionary de Rham-Hodge method. Discrete and continuous dynamical systems. Series B26(7), 3785 (2021)

    2021

  13. [20]

    4 and BA

    Chen, J., Qiu, Y., Wang, R., Wei, G.-W.: Persistent Laplacian projected Omicron BA. 4 and BA. 5 to become new dominating variants. Computers in Biology and Medicine151, 106262 (2022)

    2022

  14. [21]

    In: Computer Graphics Forum, vol

    De Goes, F., Goldenstein, S., Velho, L.: A hierarchical segmentation of articulated bodies. In: Computer Graphics Forum, vol. 27, pp. 1349–1356 (2008). Wiley Online Library

    2008

  15. [22]

    In: ACM SIGGRAPH 2006 Courses, pp

    Desbrun, M., Kanso, E., Tong, Y.: Discrete differential forms for com- putational modeling. In: ACM SIGGRAPH 2006 Courses, pp. 39–54 (2006)

    2006

  16. [23]

    Prx quantum5(3), 037001 (2024)

    Di Meglio, A., Jansen, K., Tavernelli, I., Alexandrou, C., Arunachalam, S., Bauer, C.W., Borras, K., Carrazza, S., Crippa, A., Croft, V.,et al.: Quan- tum computing for high-energy physics: State of the art and challenges. Prx quantum5(3), 037001 (2024)

    2024

  17. [24]

    Commentarii Mathematici Helvetici17(1), 240–255 (1944)

    Eckmann, B.: Harmonische funktionen und randwertaufgaben in einem komplex. Commentarii Mathematici Helvetici17(1), 240–255 (1944)

    1944

  18. [25]

    Discrete & computational geometry28, 511–533 (2002)

    Edelsbrunner, Letscher, Zomorodian: Topological persistence and simpli- fication. Discrete & computational geometry28, 511–533 (2002)

    2002

  19. [26]

    Journal of Computational Chemistry46(7), 70073 (2025)

    Feng, H., Zhao, J.Y., Wei, G.-W.: Multiscale differential geometry learning for protein flexibility analysis. Journal of Computational Chemistry46(7), 70073 (2025)

    2025

  20. [27]

    Communi- cations on Pure and Applied Mathematics8(4), 551–590 (1955)

    Friedrichs, K.O.: Differential forms on Riemannian manifolds. Communi- cations on Pure and Applied Mathematics8(4), 551–590 (1955)

    1955

  21. [28]

    Proceedings of the National Academy of Sciences80(12), 3696–3700 (1983)

    Go, N., Noguti, T., Nishikawa, T.: Dynamics of a small globular protein in terms of low-frequency vibrational modes. Proceedings of the National Academy of Sciences80(12), 3696–3700 (1983)

    1983

  22. [29]

    The Journal of Physical Chemistry B129(17), 4169–4178 (2025) 26 CONTENTS

    Hayes, N., Wei, X., Feng, H., Merkurjev, E., Wei, G.-W.: Persistent sheaf Laplacian analysis of protein flexibility. The Journal of Physical Chemistry B129(17), 4169–4178 (2025) 26 CONTENTS

    2025

  23. [30]

    Nucleic acids research32(suppl 2), 500–502 (2004)

    Heinig, M., Frishman, D.: STRIDE: a web server for secondary struc- ture assignment from known atomic coordinates of proteins. Nucleic acids research32(suppl 2), 500–502 (2004)

    2004

  24. [31]

    Annals of Global Analysis and Geometry43(2), 177–207 (2013)

    Horak, D., Jost, J.: Interlacing inequalities for eigenvalues of discrete Laplace operators. Annals of Global Analysis and Geometry43(2), 177–207 (2013)

    2013

  25. [32]

    Foundations of data science (Springfield, Mo.)7(3), 737 (2025)

    Jones, B., Wei, G.-W.: Persistent directed flag Laplacian. Foundations of data science (Springfield, Mo.)7(3), 737 (2025)

    2025

  26. [33]

    Annalen der Physik148(12), 497–508 (1847)

    Kirchhoff, G.: Ueber die Aufl¨ osung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Str¨ ome gef¨ uhrt wird. Annalen der Physik148(12), 497–508 (1847)

  27. [34]

    Journal of molec- ular biology181(3), 423–447 (1985)

    Levitt, M., Sander, C., Stern, P.S.: Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme. Journal of molec- ular biology181(3), 423–447 (1985)

    1985

  28. [35]

    In: Eurographics Workshop on 3d Object Retrieval (3DOR) (2011)

    Lian, Z., Godil, A., Bustos, B., Daoudi, M., Hermans, J., Kawamura, S., Kurita, Y., Lavoua, G., Suetens, P.D.,et al.: Shape retrieval on non-rigid 3D watertight meshes. In: Eurographics Workshop on 3d Object Retrieval (3DOR) (2011)

    2011

  29. [36]

    https://project.inria

    Lieutier, A.: Talk: Persistent harmonic forms (2014). https://project.inria. fr/gudhi/files/2014/10/Persistent-Harmonic-Forms.pdf

    2014

  30. [37]

    Siam Review62(3), 685–715 (2020)

    Lim, L.-H.: Hodge Laplacians on graphs. Siam Review62(3), 685–715 (2020)

    2020

  31. [38]

    Briefings in Bioinformatics22(5), 127 (2021)

    Liu, X., Feng, H., Wu, J., Xia, K.: Persistent spectral hypergraph based machine learning (PSH-ML) for protein-ligand binding affinity prediction. Briefings in Bioinformatics22(5), 127 (2021)

    2021

  32. [39]

    Accounts of chemical research50(2), 302–309 (2017)

    Liu, Z., Su, M., Han, L., Liu, J., Yang, Q., Li, Y., Wang, R.: Forging the basis for developing protein–ligand interaction scoring functions. Accounts of chemical research50(2), 302–309 (2017)

    2017

  33. [40]

    Mathematical Research Letters19(3), 627–648 (2012)

    Lu, Z., Rowlett, J.: Eigenvalues of collapsing domains and drift Lapla- cians. Mathematical Research Letters19(3), 627–648 (2012)

    2012

  34. [41]

    Comptes Rendus

    Ma, L., Du, S.-H.: Extension of Reilly formula with applications to eigen- value estimates for drifting Laplacians. Comptes Rendus. Math´ ematique 348(21-22), 1203–1206 (2010)

    2010

  35. [42]

    Pacific journal of mathematics240(2), 343–361 CONTENTS 27 (2009)

    Ma, L., Liu, B.: Convex eigenfunction of a drifting Laplacian operator and the fundamental gap. Pacific journal of mathematics240(2), 343–361 CONTENTS 27 (2009)

    2009

  36. [43]

    New York J

    Ma, L., Liu, B.: Convexity of the first eigenfunction of the drifting Lapla- cian operator and its applications. New York J. Math14(2008), 393–401 (2008)

    2008

  37. [44]

    nature267(5612), 585–590 (1977)

    McCammon, J.A., Gelin, B.R., Karplus, M.: Dynamics of folded proteins. nature267(5612), 585–590 (1977)

    1977

  38. [45]

    Science advances7(19), 5329 (2021)

    Meng, Z., Xia, K.: Persistent spectral–based machine learning (PerSpect ML) for protein-ligand binding affinity prediction. Science advances7(19), 5329 (2021)

    2021

  39. [46]

    American Journal of Mathematics78(1), 137–170 (1956)

    Morrey, C.B.: A variational method in the theory of harmonic integrals, ii. American Journal of Mathematics78(1), 137–170 (1956)

    1956

  40. [47]

    In: Computer Graphics Forum, vol

    Mullen, P., Tong, Y., Alliez, P., Desbrun, M.: Spectral conformal parame- terization. In: Computer Graphics Forum, vol. 27, pp. 1487–1494 (2008). Wiley Online Library

    2008

  41. [48]

    International journal for numerical methods in biomedical engineering35(3), 3179 (2019)

    Nguyen, D.D., Wei, G.-W.: Dg-gl: Differential geometry-based geomet- ric learning of molecular datasets. International journal for numerical methods in biomedical engineering35(3), 3179 (2019)

    2019

  42. [49]

    The Journal of chemical physics140(23) (2014)

    Opron, K., Xia, K., Wei, G.-W.: Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis. The Journal of chemical physics140(23) (2014)

    2014

  43. [50]

    The Journal of chemical physics142(21) (2015)

    Opron, K., Xia, K., Wei, G.-W.: Communication: Capturing protein mul- tiscale thermal fluctuations. The Journal of chemical physics142(21) (2015)

    2015

  44. [51]

    PhD thesis (2017)

    Poelke, K.: Hodge-type decompositions for piecewise constant vector fields on simplicial surfaces and solids with boundary. PhD thesis (2017)

    2017

  45. [52]

    Computer-Aided Design78, 126–136 (2016)

    Poelke, K., Polthier, K.: Boundary-aware hodge decompositions for piecewise constant vector fields. Computer-Aided Design78, 126–136 (2016)

    2016

  46. [53]

    Nature Computational Science3(2), 149–163 (2023)

    Qiu, Y., Wei, G.-W.: Persistent spectral theory-guided protein engineer- ing. Nature Computational Science3(2), 149–163 (2023)

    2023

  47. [54]

    arXiv preprint arXiv:1911.12173 (2019)

    Razafindrazaka, F.H., Poelke, K., Polthier, K., Goubergrits, L.: A consis- tent discrete 3D Hodge-type decomposition: implementation and practical evaluation. arXiv preprint arXiv:1911.12173 (2019)

    1911 arXiv

  48. [55]

    International Journal 28 CONTENTS of Computer Vision89(2), 287–308 (2010)

    Reuter, M.: Hierarchical shape segmentation and registration via topo- logical features of Laplace-Beltrami eigenfunctions. International Journal 28 CONTENTS of Computer Vision89(2), 287–308 (2010)

    2010

  49. [56]

    Computer-Aided Design38(4), 342–366 (2006)

    Reuter, M., Wolter, F.-E., Peinecke, N.: Laplace-Beltrami spectra as ‘Shape-DNA’ of surfaces and solids. Computer-Aided Design38(4), 342–366 (2006)

    2006

  50. [57]

    SIAM Review 66(3), 575–601 (2024)

    Ribando-Gros, E., Wang, R., Chen, J., Tong, Y., Wei, G.-W.: Combina- torial and Hodge Laplacians: similarities and differences. SIAM Review 66(3), 575–601 (2024)

    2024

  51. [58]

    In: ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp

    Roddenberry, T.M., Schaub, M.T., Hajij, M.: Signal processing on cell complexes. In: ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 8852–8856 (2022). IEEE

    2022

  52. [59]

    In: Symposium on Geometry Processing, vol

    Rustamov, R.M.: Laplace-Beltrami eigenfunctions for deformation invari- ant shape representation. In: Symposium on Geometry Processing, vol. 257, pp. 225–233 (2007)

    2007

  53. [60]

    Springer (2006)

    Schwarz, G.: Hodge decomposition - A method for solving boundary value problems. Springer (2006)

    2006

  54. [61]

    PhD thesis, University of Pennsylvania (2009)

    Shonkwiler, C.: Poincar´ e duality angles on Riemannian manifolds with boundary. PhD thesis, University of Pennsylvania (2009)

    2009

  55. [63]

    Beijing Journal of Pure and Applied Mathematics2(2), 619–657 (2025)

    Su, Z., Tong, Y., Wei, G.-W.: Topology-preserving Hodge decomposi- tion in the Eulerian representation. Beijing Journal of Pure and Applied Mathematics2(2), 619–657 (2025)

    2025

  56. [64]

    In: SIGGRAPH Asia 2024 Conference Papers, pp

    Su, Z., Tong, Y., Wei, G.: Hodge decomposition of vector fields in Cartesian grids. In: SIGGRAPH Asia 2024 Conference Papers, pp. 1–10 (2024)

    2024

  57. [65]

    Artificial Intelligence Review (2025)

    Su, Z., Liu, X., Hamdan, L.B., Maroulas, V., Wu, J., Carlsson, G., Wei, G.-W.: Topological data analysis and topological deep learning beyond persistent homology: a review. Artificial Intelligence Review (2025)

    2025

  58. [66]

    Biopolymers: Original Research on Biomolecules21(3), 711–714 (1982)

    Tasumi, M., Takeuchi, H., Ataka, S., Dwivedi, A.M., Krimm, S.: Nor- mal vibrations of proteins: Glucagon. Biopolymers: Original Research on Biomolecules21(3), 711–714 (1982)

    1982

  59. [67]

    Nature reviews Drug discovery2(7), 527–541 (2003) CONTENTS 29

    Teague, S.J.: Implications of protein flexibility for drug discovery. Nature reviews Drug discovery2(7), 527–541 (2003) CONTENTS 29

    2003

  60. [68]

    Physical review letters77(9), 1905 (1996)

    Tirion, M.M.: Large amplitude elastic motions in proteins from a single- parameter, atomic analysis. Physical review letters77(9), 1905 (1996)

    1905

  61. [69]

    Foundations of Data Science5(1), 26–55 (2023)

    Wang, R., Wei, G.-W.: Persistent path Laplacian. Foundations of Data Science5(1), 26–55 (2023)

    2023

  62. [70]

    Interna- tional Journal for Numerical Methods in Biomedical Engineering36(9), 3376 (2020)

    Wang, R., Nguyen, D.D., Wei, G.-W.: Persistent spectral graph. Interna- tional Journal for Numerical Methods in Biomedical Engineering36(9), 3376 (2020)

    2020

  63. [71]

    Briefings in Bioinformatics 23(2) (2022)

    Wee, J., Xia, K.: Persistent spectral based ensemble learning (perspect-el) for protein–protein binding affinity prediction. Briefings in Bioinformatics 23(2) (2022)

    2022

  64. [72]

    Foundations of Data Science (Springfield, Mo.)7(2), 446–463 (2025)

    Wei, X., Wei, G.: Persistent sheaf Laplacians. Foundations of Data Science (Springfield, Mo.)7(2), 446–463 (2025)

    2025

  65. [73]

    Mathe- matics13(2), 208 (2025)

    Wei, X., Wei, G.-W.: Persistent topological laplacians—a survey. Mathe- matics13(2), 208 (2025)

    2025

  66. [74]

    Journal of differential geometry17(4), 661–692 (1982)

    Witten, E.: Supersymmetry and Morse theory. Journal of differential geometry17(4), 661–692 (1982)

    1982

  67. [75]

    Annals of Global Analysis and Geometry45(3), 155–166 (2014)

    Xia, C., Xu, H.: Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds. Annals of Global Analysis and Geometry45(3), 155–166 (2014)

    2014

  68. [76]

    The Journal of chemical physics139(19) (2013)

    Xia, K., Opron, K., Wei, G.-W.: Multiscale multiphysics and multidomain models—flexibility and rigidity. The Journal of chemical physics139(19) (2013)

    2013

  69. [77]

    arXiv preprint arXiv:2607.00879 (2026)

    Zhang, H., Feng, H.: Commutative algebra learning for protein flexibility analysis. arXiv preprint arXiv:2607.00879 (2026)

    2026 arXiv

  70. [78]

    ACM Transactions on Graphics (TOG) 38(6), 1–13 (2019)

    Zhao, R., Desbrun, M., Wei, G.-W., Tong, Y.: 3D Hodge decompositions of edge-and face-based vector fields. ACM Transactions on Graphics (TOG) 38(6), 1–13 (2019)

    2019

  71. [79]

    Bulletin of mathematical biology 82(8), 1–38 (2020)

    Zhao, R., Wang, M., Chen, J., Tong, Y., Wei, G.-W.: The de Rham–Hodge analysis and modeling of biomolecules. Bulletin of mathematical biology 82(8), 1–38 (2020)

    2020

  72. [80]

    Discrete & Computational Geometry33(2), 249–274 (2005)

    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete & Computational Geometry33(2), 249–274 (2005)

    2005

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.