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Seven Research Directions in Differential Geometry and Differential Topology Enabled by AI

Primarily AI-generated textHuman understanding: all partsmath.DG — Differential Geometrymath.GT — Geometric Topologymath.SG — Symplectic Geometry

Contributed by Jianqiao Shang ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

This article presents seven AI-generated research directions in differential geometry and differential topology, each centered on a proposed research object or formulation. For each direction, we provide basic definitions and theorems, discuss connections with existing mathematical problems and theories, and outline questions for further study. The definitions introduced in these seven directions are relatively new. The article is intended as a collection of AI-generated research guides rather than a conventional research paper.

Tools used

OpenAI
ChatGPTVersion 6 Astra

References

  1. A.-L. Mare, Equivariant cohomology of real flag manifolds , preprint, 2004, revised 2005. arXiv:math/0404369 (https://arxiv.org/abs/math/0404369).arXiv
  2. R. Bott, Nondegenerate critical manifolds , Ann. of Math. (2) 60 (1954), 248--261. doi:10.2307/1969631 (https://doi.org/10.2307/1969631).DOI
  3. J. E. Humphreys, Reflection Groups and Coxeter Groups , Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, 1990.
  4. J. F. Adams, On the non-existence of elements of Hopf invariant one , Ann. of Math. (2) 72 (1960), 20--104.
  5. A. Hatcher, Vector Bundles and K-Theory , version 2.2, November 2017. Author's online text (https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf).link
  6. C. Chevalley, Invariants of finite groups generated by reflections , Amer. J. Math. 77 (1955), 778--782. doi:10.2307/2372597 (https://doi.org/10.2307/2372597).DOI
  7. D. Sullivan, Infinitesimal computations in topology , Publ. Math. IHÉS 47 (1977), 269--331. doi:10.1007/BF02684341 (https://doi.org/10.1007/BF02684341).DOI
  8. W.-Y. Hsiang, R. S. Palais, and C.-L. Terng, The topology of isoparametric submanifolds , J. Differential Geom. 27 (1988), 423--460. doi:10.4310/jdg/1214442003 (https://doi.org/10.4310/jdg/1214442003).DOI
  9. K. Igusa, Higher complex torsion and the framing principle , Mem. Amer. Math. Soc. 177 (2005), no. 835. doi:10.1090/memo/0835 (https://doi.org/10.1090/memo/0835). Preprint: arXiv:math/0303047 (https://arxiv.org/abs/math/0303047).arXivDOI
  10. A. Derdzinski and T. Januszkiewicz, Immersions of surfaces in spinc\mathrm{spin}^c-manifolds with Higgs fields , preprint (2002), arXiv:math/0210253 (https://arxiv.org/abs/math/0210253), especially §16.arXiv
Show all 56 referencesShow fewer references
  1. N. Kitazawa and O. Saeki, Round fold maps of nn-dimensional manifolds into (n−1)(n-1)-dimensional Euclidean space , J. Singul. 26 (2023), 1--12. doi:10.5427/jsing.2023.26a (https://doi.org/10.5427/jsing.2023.26a).DOI
  2. E. Fadell and L. Neuwirth, Configuration spaces , Math. Scand. 10 (1962), 111--118. doi:10.7146/math.scand.a-10517 (https://doi.org/10.7146/math.scand.a-10517).DOI
  3. K. Takao, Two Morse functions and singularities of the product map , Comm. Anal. Geom. 24 (2016), no. 3, 645--671. doi:10.4310/CAG.2016.v24.n3.a8 (https://doi.org/10.4310/CAG.2016.v24.n3.a8).DOI
  4. N. Kitazawa and O. Saeki, Round fold maps on 3-manifolds , Algebr. Geom. Topol. 23 (2023), no. 8, 3745--3762. doi:10.2140/agt.2023.23.3745 (https://doi.org/10.2140/agt.2023.23.3745).DOI
  5. M. Usher, Linking and the Morse complex , Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 1, 25--94. doi:10.5802/afst.1397 (https://doi.org/10.5802/afst.1397).DOI
  6. L. Kühne and G. Yashfe, Representability of matroids by cc-arrangements is undecidable , Israel J. Math. 252 (2022), 95--147. arXiv:1912.06123.arXiv
  7. G. Bande and D. Kotschick, The geometry of symplectic pairs , Trans. Amer. Math. Soc. 358 (2006), 1643--1655. arXiv:math/0407441.arXiv
  8. C. Blacker, Polysymplectic reduction and the moduli space of flat connections , J. Phys. A: Math. Theor. 52 (2019), 335201. arXiv:1810.04924.arXiv
  9. M. Nguiffo Boyom, Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology , Entropy 18 (2016), 433. doi:10.3390/e18120433.DOI
  10. M. Ludewig, Strong short-time asymptotics and convolution approximation of the heat kernel , Ann. Global Anal. Geom. 55 (2019), 371--394. doi:10.1007/s10455-018-9630-4.DOI
  11. A. Singer and H.-T. Wu, Vector diffusion maps and the connection Laplacian , Comm. Pure Appl. Math. 65 (2012), no. 8, 1067--1144. doi:10.1002/cpa.21395.DOI
  12. N. Sharp, Y. Soliman, and K. Crane, The vector heat method , ACM Trans. Graph. 38 (2019), no. 3. arXiv:1805.09170.arXiv
  13. S. Berceanu and M. Schlichenmaier, Coherent state embeddings, polar divisors and Cauchy formulas , J. Geom. Phys. 34 (2000), nos. 3--4, 336--358. arXiv:math/9903105.arXiv
  14. F. Knöppel, Random zero currents of sections of Hermitian line bundles over compact Riemannian manifolds , preprint, 2023. arXiv:2312.01493.arXiv
  15. M. Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a ``Weltkonstante''---or how Ramanujan split temperatures , Ramanujan J. 54 (2021), 1--27. doi:10.1007/s11139-019-00224-2.DOI
  16. J. W. Milnor and J. D. Stasheff, Characteristic Classes , Annals of Mathematics Studies 76, Princeton University Press, 1974.
  17. F. Gay-Balmaz and C. Vizman, Principal bundles of embeddings and nonlinear Grassmannians , Annals of Global Analysis and Geometry 46 (2014), no. 3, 293--312. DOI: 10.1007/s10455-014-9424-2 .DOI
  18. J. C. Becker and D. H. Gottlieb, Transfer maps for fibrations and duality , Compositio Mathematica 33 (1976), no. 2, 107--133.
  19. R. Benedetti and P. Lisca, Framing 3-manifolds with bare hands , L'Enseignement Mathématique 64 (2018), no. 3/4, 395--413. DOI: 10.4171/LEM/64-3/4-9 .DOI
  20. W. Ambrose and I. M. Singer, A theorem on holonomy , Transactions of the American Mathematical Society 75 (1953), no. 3, 428--443.
  21. A. Hatcher, Algebraic Topology , Cambridge University Press, 2002. See §4.2 and §4.B for homotopy sequences and the Hopf invariant.
  22. A. Hatcher, Spaces of knots (https://arxiv.org/abs/math/9909095), arXiv:math/9909095 (1999); expanded draft Topological moduli spaces of knots (https://pi.math.cornell.edu/~Hatcher/Papers/knotspaces.pdf) (2002), especially pp. 1--2 and the torus/hyperbolic-knot arguments.arXiv
  23. A. Dold and H. Whitney, https://webhomes.maths.ed.ac.uk/~v1ranick/papers/doldwhit.pdf Classification of oriented sphere bundles over a 44-complex , Ann. of Math. (2) 69 (1959), 667--677.link
  24. F. Hirzebruch and H. Hopf, https://hirzebruch.mpim-bonn.mpg.de/id/eprint/161/ Felder von Flächenelementen in 44-dimensionalen Mannigfaltigkeiten , Math. Ann. 136 (1958), 156--172.link
  25. F. van der Blij, https://webhomes.maths.ed.ac.uk/~v1ranick/papers/vdb.pdf An invariant of quadratic forms mod 88 , Indag. Math. 21 (1959), 291--293.link
  26. M. Furuta, https://doi.org/10.4310/MRL.2001.v8.n3.a5 Monopole equation and the 11/811/8-conjecture , Math. Res. Lett. 8 (2001), 279--291, Theorem 1.DOI
  27. M. Atiyah and C. LeBrun, Curvature, cones and characteristic numbers (https://doi.org/10.1017/S0305004113000169), Math. Proc. Cambridge Philos. Soc. 155 (2013), 13--37; arXiv:1203.6389 (https://arxiv.org/abs/1203.6389), Section 2.arXivDOI
  28. M. J. Hopkins, J. Lin, X. D. Shi and Z. Xu, https://doi.org/10.1090/cams/4 Intersection forms of spin 44-manifolds and the Pin⁡(2)\operatorname{Pin}(2)-equivariant Mahowald invariant , Comm. Amer. Math. Soc. 2 (2022), 22--132; arXiv:1812.04052 (https://arxiv.org/abs/1812.04052).arXivDOI
  29. R. Budney, https://doi.org/10.1112/plms/pdp058 Topology of knot spaces in dimension 33 , Proc. Lond. Math. Soc. (3) 101 (2010), 477--496; arXiv:math/0506524 (https://arxiv.org/abs/math/0506524).arXivDOI
  30. A. Havens and R. Koytcheff, https://arxiv.org/abs/2008.03192 Spaces of knots in the solid torus, knots in the thickened torus, and links in the 33-sphere , arXiv:2008.03192.arXiv
  31. J. F. Adams, Vector fields on spheres , Annals of Mathematics 75 (1962), 603--632. doi:10.2307/1970213 (https://doi.org/10.2307/1970213).DOI
  32. P. Patel, On a theorem of Peter Scott , preprint, revised 2014. arXiv:1204.5135 (https://arxiv.org/abs/1204.5135).arXiv
  33. S. Alvarez, J. Brum, M. Martínez and R. Potrie, with an appendix written with M. Wolff, Topology of leaves for minimal laminations by hyperbolic surfaces , Journal of Topology 15 (2022), 302--346; Theorem C and Appendix A. doi:10.1112/topo.12222 (https://doi.org/10.1112/topo.12222); arXiv:1906.10029 (https://arxiv.org/abs/1906.10029).arXivDOI
  34. J.-P. Otal, Le spectre marqué des longueurs des surfaces à courbure négative , Annals of Mathematics 131 (1990), 151--162. doi:10.2307/1971511 (https://doi.org/10.2307/1971511).DOI
  35. C. Guillarmou, T. Lefeuvre and G. P. Paternain, Marked length spectrum rigidity for Anosov surfaces , preprint, revised 2024. arXiv:2303.12007 (https://arxiv.org/abs/2303.12007).arXiv
  36. J. Kubarski and N. Teleman, Linear direct connections , Banach Center Publications 76 (2007), 425--436. doi:10.4064/bc76-0-20 (https://doi.org/10.4064/bc76-0-20).DOI
  37. J. Fernández, M. Juchani and M. Zuccalli, Discrete connections on principal bundles: abelian group case (2021). arXiv:2109.08928 (https://arxiv.org/abs/2109.08928).arXiv
  38. R. Tinarrage, Computing persistent Stiefel--Whitney classes of line bundles , Journal of Applied and Computational Topology (2021). doi:10.1007/s41468-021-00080-4 (https://doi.org/10.1007/s41468-021-00080-4).DOI
  39. E. Macías-Virgós and D. Mosquera-Lois, Homotopic distance between maps . arXiv:1810.12591 (https://arxiv.org/abs/1810.12591); doi:10.1017/S0305004121000116 (https://doi.org/10.1017/S0305004121000116).arXivDOI
  40. D. Kotschick, On products of harmonic forms , Duke Math. J. 107 (2001), no. 3, 521--531. arXiv:math/0004009 (https://arxiv.org/abs/math/0004009).arXiv
  41. K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups , Ann. of Math. (2) 59 (1954), no. 3, 531--538. doi:10.2307/1969716 (https://doi.org/10.2307/1969716).DOI
  42. P.-A. Nagy and C. Vernicos, The length of harmonic forms on a compact Riemannian manifold , Trans. Amer. Math. Soc. 356 (2004), no. 6, 2501--2513. arXiv:math/0301369 (https://arxiv.org/abs/math/0301369).arXiv
  43. L. Ornea and M. Pilca, Remarks on the product of harmonic forms , Pacific J. Math. 250 (2011), no. 2, 353--363. doi:10.2140/pjm.2011.250.353 (https://doi.org/10.2140/pjm.2011.250.353).DOI
  44. I. Kangasniemi, Conformally formal manifolds and the uniformly quasiregular non-ellipticity of (S2×S2)#(S2×S2)(S^2\times S^2)\#(S^2\times S^2) , Adv. Math. 393 (2021), article 108103. arXiv:2008.10669 (https://arxiv.org/abs/2008.10669).arXiv
  45. P. Hájek, Hodge decompositions and differential Poincaré duality models , J. Homotopy Relat. Struct. 21 (2026), 175--210; published online 2025. doi:10.1007/s40062-025-00389-2 (https://doi.org/10.1007/s40062-025-00389-2).DOI
  46. A. Milivojević and J. Stelzig, Bigraded notions of formality and Aeppli--Bott--Chern--Massey products , Comm. Anal. Geom. (2024). arXiv:2202.08617v2 (https://arxiv.org/abs/2202.08617v2).arXiv

Version history

  1. v1 · 56 pagesSubmitted by Jianqiao ShangInitial depositCurrentNov 09, 2026