T2: Moduli spaces in arithmetic and geometry

The term “moduli space” was coined by Riemann for the space Mg parametrizing all one-dimensional complex manifolds of genus g. Variants of this appear in several mathematical disciplines. In algebraic geometry, Mg is constructed using geometric invariant theory and has a well-studied compactification introduced by Deligne and  Mumford. Through Teichmüller theory, these spaces are seen as objects of differential geometry, leading to useful cell decompositions and stratifications for computing intersection numbers, with interpretations in quantum field theory, as shown by Witten and Kontsevich. Moreover, Mg  can be viewed as the classifying space of the diffeomorphism group of a surface or as a space of surfaces, a perspective used by Madsen and Weiss in proving the Mumford conjecture.

Moduli spaces that are vast generalisations of Riemann’s concept are central for many research directions. In arithmetic geometry, Shimura varieties or moduli spaces of shtukas play an important role in the realisation of Langlands correspondences. Diffeomorphism groups of high-dimensional manifolds and moduli spaces of manifolds and of metrics of positive scalar curvature are studied in differential topology. Moduli spaces are also one of the central topics in our research in mathematical physics, where we study moduli spaces of stable curves and of Strebel differentials.

  • Mathematical fields

  • Collaborations with other Topics

  • Selected publications and preprints

    since 2019

    ∙ Johannes Ebert and Michael Wiemeler. On the homotopy type of the space of metrics of positive scalar curvature. Journal of the European Mathematical Society, 26(9):3327–3363, July 2024. doi:10.4171/JEMS/1333.

    ∙ Eugen Hellmann, Valentin Hernandez, and Benjamin Schraen. Patching and multiplicities of p-adic eigenforms. arXiv e-prints, June 2024. arXiv:2406.01129.

    ∙ Eva Viehmann. On Newton strata in the B+dR-Grassmannian. Duke Mathematical Journal, 173(1):177–225, January 2024. doi:10.1215/00127094-2024-0005.

    ∙ Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann, and Changjian Su. Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells. Duke Mathematical Journal, 172(17):3257 – 3320, November 2023. doi:10.1215/00127094-2022-0101.

    ∙ Johannes Anschütz, João Lourenço, Zhiyou Wu, and Jize Yu. Gaitsgory's central functor and the Arkhipov-Bezrukavnikov equivalence in mixed characteristic. arXiv e-prints, November 2023. arXiv:2311.04043.

    ∙ Urs Hartl and Yujie Xu. Uniformizing the moduli stacks of global G-shtukas II. arXiv e-prints, September 2023. arXiv:2309.17441.

    ∙ Jörg Schürmann and Raimar Wulkenhaar. An algebraic approach to a quartic analogue of the Kontsevich model. Mathematical Proceedings of the Cambridge Philosophical Society, 174(3):471–495, May 2023. doi:10.1017/S0305004122000366.

    ∙ Yifei Zhao. Spectral decomposition of genuine cusp forms over global function fields. arXiv e-prints, February 2023. arXiv:2302.13023.

    ∙ Matthew Emerton, Toby Gee, and Eugen Hellmann. An introduction to the categorical p-adic Langlands program. arXiv e-prints, October 2022. arXiv:2210.01404.

    ∙ Johannes Ebert and Oscar Randal-Williams. The positive scalar curvature cobordism category. Duke Math. J., 171(11):2275–2406, August 2022. doi:10.1215/00127094-2022-0023.

    ∙ Johannes Branahl, Alexander Hock, and Raimar Wulkenhaar. Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures. Commun. Math. Phys., 393(3):1529–1582, August 2022. doi:10.1007/s00220-022-04392-z.

    ∙ Esmail Arasteh Rad and Urs Hartl. Uniformizing the moduli stacks of global G-shtukas. International Mathematics Research Notices, pages 16121–16192, November 2021. doi:10.1093/imrn/rnz223.

    ∙ Christophe Breuil, Eugen Hellmann, and Benjamin Schraen. A local model for the trianguline variety and applications. Publ. Math. IHES, 130(1):299–412, August 2019. doi:10.1007/s10240-019-00111-y.

    ∙ Arthur Bartels and Mladen Bestvina. The Farrell-Jones conjecture for mapping class groups. Invent. Math., 215(2):651–712, January 2019. doi:10.1007/s00222-018-0834-9.

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  • Recent publications and preprints

    since 2023

    ∙ Jörg Schürmann, Connor Simpson, and Botong Wang. A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells. Compositio Mathematica, 161(1):1–12, January 2025. doi:10.1112/s0010437x24007462.

    ∙ Thomas J. Haines, João Lourenço, and Timo Richarz. On the normality of Schubert varieties: remaining cases in positive characteristic. Annales Scientifiques de l’École Normale Supérieure, 57(3):895–959, July 2024. doi:10.24033/asens.2584.

    ∙ Johannes Ebert and Michael Wiemeler. On the homotopy type of the space of metrics of positive scalar curvature. Journal of the European Mathematical Society, 26(9):3327–3363, July 2024. doi:10.4171/JEMS/1333.

    ∙ Eugen Hellmann, Valentin Hernandez, and Benjamin Schraen. Patching and multiplicities of p-adic eigenforms. arXiv e-prints, June 2024. arXiv:2406.01129.

    ∙ Ian Gleason and João Lourenço. Tubular neighborhoods of local models. Duke Mathematical Journal, March 2024. doi:10.1215/00127094-2023-0028.

    ∙ Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann, and Changjian Su. From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes. In A glimpse into geometric representation theory, volume 804 of Contemp. Math., pages 1–52. American Mathematical Society, 2024. doi:10.1090/conm/804/16110.

    ∙ Eva Viehmann. On Newton strata in the B+dR-Grassmannian. Duke Mathematical Journal, 173(1):177–225, January 2024. doi:10.1215/00127094-2024-0005.

    ∙ Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann, and Changjian Su. Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells. Duke Mathematical Journal, 172(17):3257 – 3320, November 2023. doi:10.1215/00127094-2022-0101.

    ∙ Johannes Anschütz, João Lourenço, Zhiyou Wu, and Jize Yu. Gaitsgory's central functor and the Arkhipov-Bezrukavnikov equivalence in mixed characteristic. arXiv e-prints, November 2023. arXiv:2311.04043.

    ∙ Urs Hartl and Yujie Xu. Uniformizing the moduli stacks of global G-shtukas II. arXiv e-prints, September 2023. arXiv:2309.17441.

    ∙ Yifei Zhao. Quantum parameters of the geometric Langlands theory. Selecta Mathematica, 29(4):66, August 2023. doi:10.1007/s00029-023-00868-5.

    ∙ Arthur Bartels and Wolfgang Lueck. Almost equivariant maps for td-groups. arXiv e-prints, June 2023. arXiv:2306.00727.

    ∙ Jörg Schürmann and Raimar Wulkenhaar. An algebraic approach to a quartic analogue of the Kontsevich model. Mathematical Proceedings of the Cambridge Philosophical Society, 174(3):471–495, May 2023. doi:10.1017/S0305004122000366.

    ∙ Kieu Hieu Nguyen and Eva Viehmann. A Harder-Narasimhan stratification of the B+dR-Grassmannian. Compositio Mathematica, 159(4):711–745, March 2023. doi:10.1112/S0010437X23007066.

    ∙ Yifei Zhao. Spectral decomposition of genuine cusp forms over global function fields. arXiv e-prints, February 2023. arXiv:2302.13023.

    ∙ João Lourenço. Grassmanniennes affines tordues sur les entiers. Forum of Mathematics, Sigma, 11:e12, January 2023. doi:10.1017/fms.2023.4.

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    further publications

Back to research programme

Construction and existence

© MM/vl

Investigators: Hartl, Hellmann, Hille, Viehmann

In this unit we want to find and prove existence of moduli spaces that are important within the Langlands programme. Here, several kinds of moduli spaces play central roles. On the one hand one studies Shimura varieties and moduli spaces of local and global G-shtukas.  On the other side of the Langlands correspondence one considers deformation spaces of Galois representations.

Compactification and stratifications

© MM/vl

Investigators: Bartels, Hellmann, Schürmann, Viehmann, Wulkenhaar, Zhao

Here we study geometric questions. Compactifications of moduli spaces and their boundaries and singularities reveal interesting degeneration properties of the studied families, for example of Galois representations. Natural stratifications, a theme present in arithmetic geometry as well as in topology and mathematical physics, allow for a better understanding of the geometry of the moduli space itself as well as for the parametrized objects.

Invariants and cohomology

© MM/vl

Investigators: Ebert, Hartl, Lourenço, Schürmann, Wiemeler, Wulkenhaar, Zeidler

In this unit we study cohomological invariants of moduli spaces, including questions on homotopy, intersection theory and the (derived) category of sheaves on moduli spaces. In the Langlands program, the cohomology of suitable moduli spaces is conjectured to realize correspondences between suitable sets of representations. Nowadays this is refined to a conjectured equivalence between certain derived categories of sheaves. In homotopy theory, we study the space of Riemannian metrics of positive scalar curvature on a closed high-dimensional manifold. In mathematical physics, we study and generalize topological recursion, a far-reaching generalisation of Kontsevich's solution of the Witten conjecture.

Research examples

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  • BunG and adic Newton Strata
    © MM
  • Moduli of curves and QFT
    © MM
  • Moduli of Galois representations
    © MM
  • Moduli spaces of metrics of positive scalar curvature and index theory
    © MM