T5: Curvature, shape and global analysis

Riemannian manifolds or geodesic metric spaces of finite or infinite dimension occur in many areas of mathematics. We are interested in the interplay between their local geometry and global topological and analytical properties, which in general are strongly intertwined. For instance, it is well known that certain positivity assumptions on the curvature tensor (a local geometric object) imply topological obstructions of the underlying manifold. Curvature bounds also determine certain properties of elliptic operators (or other analytic objects), which sometimes can be related to topological properties of the underlying manifold via index-theoretic methods. The curvature or the local geometry is determined by geodesics, whose analytic properties are thus in close relation to analytical (and topological) properties of the underlying space. However, on infinite-dimensional (shape) spaces, existence of geodesics is non-trivial since they are governed by PDEs (and not ODEs as in finite dimensions).

  • Mathematical fields

    • Arithmetic geometry and representation theory
    • Topology
    • Differential geometry
    • Applied analysis and theory of PDEs
    • Stochastic analysis
    • Optimisation and calculus of variation
    • Numerical analysis, machine learning and scientific computing

  • Collaborations with other Topics

  • Selected publications and preprints

    since 2019

    $\bullet $ Christopher Deninger, Theo Grundhöfer, and Linus Kramer. Weil tensors, strongly regular graphs, multiplicative characters, and a quadratic matrix equation. J. Algebra, 656:170–195, October 2024. doi:10.1016/j.jalgebra.2023.08.028.

    $\bullet $ 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.

    $\bullet $ Christoph Böhm, Timothy Buttsworth, and Brian Clarke. Scalar curvature along Ebin geodesics. Journal für die reine und angewandte Mathematik (Crelles Journal), 2024(813):159–196, June 2024. doi:10.1515/crelle-2024-0033.

    $\bullet $ Simone Cecchini and Rudolf Zeidler. Scalar and mean curvature comparison via the Dirac operator. Geometry and Topology, 28:1167–1212, May 2024. doi:10.2140/gt.2024.28.1167.

    $\bullet $ Christoph Böhm and Ramiro A. Lafuente. Non-compact Einstein manifolds with symmetry. J. Amer. Math. Soc., 36(3):591–651, February 2023. doi:10.1090/jams/1022.

    $\bullet $ Lee Kennard, Michael Wiemeler, and Burkhard Wilking. Positive curvature, torus symmetry, and matroids. arXiv e-prints, December 2022. arXiv:2212.08152.

    $\bullet $ Hanne Hardering and Benedikt Wirth. Quartic $L^p$-convergence of cubic Riemannian splines. IMA J. Numer. Anal., 42(4):3360–3385, October 2022. doi:10.1093/imanum/drab077.

    $\bullet $ Rudolf Zeidler. Band width estimates via the Dirac operator. J. Differ. Geom., September 2022. doi:10.4310/jdg/1668186790.

    $\bullet $ 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.

    $\bullet $ Alexander Effland, Behrend Heeren, Martin Rumpf, and Benedikt Wirth. Consistent curvature approximation on Riemannian shape spaces. IMA J. Numer. Anal., 42(1):78–106, January 2022. doi:10.1093/imanum/draa092.

    $\bullet $ Zhizhang Xie, Guoliang Yu, and Rudolf Zeidler. On the range of the relative higher index and the higher rho-invariant for positive scalar curvature. Adv. Math., 390:Paper No. 107897, 24, October 2021. doi:10.1016/j.aim.2021.107897.

    $\bullet $ Lee Kennard, Michael Wiemeler, and Burkhard Wilking. Splitting of torus representations and applications in the Grove symmetry program. arXiv e-prints, June 2021. arXiv:2106.14723.

    $\bullet $ Julio Backhoff-Veraguas, Mathias Beiglböck, Martin Huesmann, and Sigrid Källblad. Martingale Benamou-Brenier: a probabilistic perspective. Ann. Probab., 48(5):2258–2289, September 2020. doi:10.1214/20-AOP1422.

    $\bullet $ Richard Bamler, Esther Cabezas-Rivas, and Burkhard Wilking. The Ricci flow under almost non-negative curvature conditions. Invent. Math., 217:95–126, July 2019. doi:10.1007/s00222-019-00864-7.

    $\bullet $ Johannes Ebert and Oscar Randal-Williams. Infinite loop spaces and positive scalar curvature in the presence of a fundamental group. Geom. Topol., 23(3):1549–1610, May 2019. doi:10.2140/gt.2019.23.1549.

    $\bullet $ Behrend Heeren, Martin Rumpf, and Benedikt Wirth. Variational time discretization of Riemannian splines. IMA J. Numer. Anal., 39(1):61–104, January 2019. doi:10.1093/imanum/drx077.

    $\bullet $ Johannes Ebert. Index theory in spaces of manifolds. Math. Ann., 374(1-2):931–962, January 2019. doi:10.1007/s00208-019-01809-4.

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  • Recent publications and preprints

    since 2023

    $\bullet $ Christopher Deninger, Theo Grundhöfer, and Linus Kramer. Weil tensors, strongly regular graphs, multiplicative characters, and a quadratic matrix equation. J. Algebra, 656:170–195, October 2024. doi:10.1016/j.jalgebra.2023.08.028.

    $\bullet $ Simone Cecchini, Georg Frenck, and Rudolf Zeidler. Positive scalar curvature with point singularities. arXiv e-prints, July 2024. arXiv:2407.20163.

    $\bullet $ 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.

    $\bullet $ Michael Wiemeler. On a conjecture of Stolz in the toric case. Proceedings of the American Mathematical Society, 152(8):3617–3621, June 2024. doi:10.1090/proc/16823.

    $\bullet $ Simone Cecchini and Rudolf Zeidler. The positive mass theorem and distance estimates in the spin setting. Transactions of the American Mathematical Society, June 2024. doi:10.1090/tran/8942.

    $\bullet $ Christoph Böhm, Timothy Buttsworth, and Brian Clarke. Scalar curvature along Ebin geodesics. Journal für die reine und angewandte Mathematik (Crelles Journal), 2024(813):159–196, June 2024. doi:10.1515/crelle-2024-0033.

    $\bullet $ Simone Cecchini and Rudolf Zeidler. Scalar and mean curvature comparison via the Dirac operator. Geometry and Topology, 28:1167–1212, May 2024. doi:10.2140/gt.2024.28.1167.

    $\bullet $ Simone Cecchini, Sven Hirsch, and Rudolf Zeidler. Rigidity of spin fill-ins with non-negative scalar curvature. arXiv e-prints, April 2024. arXiv:2404.17533.

    $\bullet $ Simone Cecchini, Martin Lesourd, and Rudolf Zeidler. Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields. International Mathematics Research Notices, 2024(9):7870–7890, January 2024. doi:10.1093/imrn/rnad315.

    $\bullet $ Anusha M. Krishnan and Michael Wiemeler. $10$-dimensional positively curved manifolds with $t^3$-symmetry. arXiv e-prints, October 2023. arXiv:2310.12689.

    $\bullet $ Simone Cecchini, Daniel Räde, and Rudolf Zeidler. Nonnegative scalar curvature on manifolds with at least two ends. Journal of Topology, 16(3):855–876, June 2023. doi:10.1112/topo.12303.

    $\bullet $ Simone Cecchini and Rudolf Zeidler. Scalar curvature and generalized Callias operators. In Perspectives in Scalar Curvature, pages 515–542. World Scientific, March 2023. doi:10.1142/9789811273223_0002.

    $\bullet $ Boris Botvinnik and Johannes Ebert. Positive scalar curvature and homotopy theory. In Perspectives in Scalar Curvature, pages 83–157. World Scientific, March 2023. doi:10.1142/9789811273230_0003.

    $\bullet $ Christoph Böhm and Ramiro A. Lafuente. Non-compact Einstein manifolds with symmetry. J. Amer. Math. Soc., 36(3):591–651, February 2023. doi:10.1090/jams/1022.

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    further publications

Back to research programme

Curvature and geometric flows

© MM/vl

Investigators: Böhm, Huesmann, Wiemeler, Wilking

In this unit we investigate Riemannian manifolds with positive sectional curvature, Ricci flow (an evolution equation on the space of Riemannian metrics of a smooth manifold), and infinite-dimensional metric measure spaces and their synthetic curvature properties.

Geodesics

© MM/vl

Investigators: Ohlberger, Rave, Santoro, Seis, Wilking, Wirth

This unit focusses on (shortest) geodesics, which determine the Riemannian distance metric. We investigate the existence, regularity and qualitative behaviour of geodesics in infinite-dimensional shape spaces. Computing these numerically is a very difficult problem with applications in computer graphics, data science, and inverse problems, which we tackle by developing and analysing (numerical) approximation schemes.

Index theory

© MM/vl

Investigators: Deninger, Ebert, Zeidler

In this unit we investigate comparison results for Riemannian manifolds with lower scalar curvature bounds, using index-theoretic methods. A common theme is the interplay between local quantities (e.g., curvature) and global properties (the topology, analytic or geometric estimates). Under symmetry assumptions (e.g., isometric group actions) this is typically easier to study. However, while in finite dimensions a Lie group (global) and its Lie algebra (local) are intimately related, this is far from true in infinite dimensions.