T6: Singularities and partial differential equations

In this topic, our goal is to utilise and further develop the theory of non-linear PDEs to understand singular phenomena arising in geometry and in the description of the physical world. Particular emphasis is put on the interplay of geometry and partial differential equations and also on the connection with theoretical physics.

The concrete research projects have been organised in three units and -- to give just a few examples -- range from problems originating in geometric analysis such as understanding the type of singularities developing along a sequence of four-dimensional Einstein manifolds, to problems in evolutionary PDEs, such as the Einstein equations of general relativity or the Euler equations of fluid mechanics, where one would like to understand the formation and dynamics (in time) of singularities. The third unit is concerned with regularity estimates, which play a crucial role in the recent breakthroughs in theory of stochastic PDEs allowing the formulation of PDEs in extremely singular settings.

  • Mathematical fields

    • Differential geometry
    • Applied analysis and theory of PDEs
    • Stochastic analysis
    • Optimisation and calculus of variation

  • Collaborations with other Topics

  • Selected publications and preprints

    since 2019

    $\bullet $ Hans-Joachim Hein, Man-Chun Lee, and Valentino Tosatti. Collapsing immortal kähler-ricci flows. arXiv e-prints, May 2024. arXiv:2405.04208.

    $\bullet $ Xin Fu, Hans-Joachim Hein, and Xumin Jiang. A continuous cusp closing process for negative Kähler-Einstein metrics. arXiv e-prints, January 2024. arXiv:2401.11468.

    $\bullet $ Olivier Biquard and Hans-Joachim Hein. The renormalized volume of a 4-dimensional Ricci-flat ALE space. Journal of Differential Geometry, 123(3):411–429, March 2023. doi:10.4310/jdg/1683307004.

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

    $\bullet $ Joachim Lohkamp. The secret hyperbolic life of positive scalar curvature. In Perspectives in Scalar Curvature, volume 1, chapter 5, pages 611–642. World Scientific, January 2023. doi:10.1142/9789811273223_0005.

    $\bullet $ Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski, and Martin Taylor. Quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes. arXiv e-prints, December 2022. arXiv:2212.14093.

    $\bullet $ Stefano Bruno, Benjamin Gess, and Hendrik Weber. Optimal regularity in time and space for stochastic porous medium equations. Ann. Probab., 50(6):2288–2343, November 2022. doi:10.1214/22-aop1583.

    $\bullet $ Angela Stevens and Michael Winkler. Taxis-driven persistent localization in a degenerate Keller-Segel system. Commun. Part. Diff. Eq., 47(12):2341–2362, October 2022. doi:10.1080/03605302.2022.2122836.

    $\bullet $ Hans-Joachim Hein, Song Sun, Jeff Viaclovsky, and Ruobing Zhang. Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface. J. Am. Math. Soc., 35(1):123–209, January 2022. doi:10.1090/jams/978.

    $\bullet $ Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski, and Martin Taylor. The non-linear stability of the Schwarzschild family of black holes. arXiv e-prints, April 2021. arXiv:2104.08222.

    $\bullet $ Hans-Joachim Hein and Valentino Tosatti. Higher-order estimates for collapsing Calabi-Yau metrics. Camb. J. Math., 8:683–773, December 2020. doi:10.4310/cjm.2020.v8.n4.a1.

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  • Recent publications and preprints

    since 2023

    $\bullet $ David Meyer and Christian Seis. Steady ring-shaped vortex sheets. arXiv e-prints, September 2024. arXiv:2409.08220.

    $\bullet $ Hans-Joachim Hein, Man-Chun Lee, and Valentino Tosatti. Collapsing immortal kähler-ricci flows. arXiv e-prints, May 2024. arXiv:2405.04208.

    $\bullet $ Julian Fischer, Sebastian Hensel, Tim Laux, and Theresa M. Simon. A weak-strong uniqueness principle for the Mullins-Sekerka equation. arXiv e-prints, April 2024. arXiv:2404.02682.

    $\bullet $ Ronan J. Conlon and Hans-Joachim Hein. Classification of asymptotically conical Calabi-Yau manifolds. Duke Mathematical Journal, April 2024. doi:10.1215/00127094-2023-0030.

    $\bullet $ Nicola de Nitti, David Meyer, and Christian Seis. Optimal regularity for the 2d Euler equations in the Yudovich class. arXiv e-prints, March 2024. arXiv:2403.13691.

    $\bullet $ Ajay Chandra, Guilherme de Lima Feltes, and Hendrik Weber. A priori bounds for 2-d generalised parabolic Anderson model. arXiv e-prints, February 2024. arXiv:2402.05544.

    $\bullet $ Xin Fu, Hans-Joachim Hein, and Xumin Jiang. A continuous cusp closing process for negative Kähler-Einstein metrics. arXiv e-prints, January 2024. arXiv:2401.11468.

    $\bullet $ X. Fu, H.-J. Hein, and X. Jiang. Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps. Calculus of Variations and Partial Differential Equations, November 2023. doi:10.1007/s00526-023-02613-4.

    $\bullet $ Ajay Chandra, Augustin Moinat, and Hendrik Weber. A priori bounds for the $\Phi ^4$ equation in the full sub-critical regime. Archive for Rational Mechanics and Analysis, 247(3):48, May 2023. doi:10.1007/s00205-023-01876-7.

    $\bullet $ Olivier Biquard and Hans-Joachim Hein. The renormalized volume of a 4-dimensional Ricci-flat ALE space. Journal of Differential Geometry, 123(3):411–429, March 2023. doi:10.4310/jdg/1683307004.

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

    $\bullet $ Joachim Lohkamp. The secret hyperbolic life of positive scalar curvature. In Perspectives in Scalar Curvature, volume 1, chapter 5, pages 611–642. World Scientific, January 2023. doi:10.1142/9789811273223_0005.

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    further publications

Back to research programme

Geometric analysis

Investigators: Böhm, Hein, Holzegel, Lohkamp, Santoro

This unit focusses on geometric questions: In what way does the assumption of symmetries make Einstein manifolds rigid? Given a sequence of 4-dimensional Einstein manifolds, what types of singularities can form in the limit? Can we better understand the potential singularities of minimal hypersurfaces? Many of the problems we intend to study here have intimate connections with theoretical physics, for instance through the study of gravitational instantons and general relativity.

Dynamics of evolutionary PDEs

Investigators: Holzegel, Seis, Simon, Stevens

In this unit we will investigate singularity formation for evolution equations, i.e., PDE problems of a hyperbolic or parabolic character. Prominent examples on the  hyperbolic side are non-linear wave equations and the stability of black hole solutions of the Einstein equations, both of which are strongly connected to the previous unit through the underlying Lorentzian geometry. Understanding the formation and evolution of singularities is also a classical problem in the study of the Euler equations, which constitute the fundamental equations of fluid dynamics. Here we are mainly interested in the two-phase Euler equations with singular vorticity at an interface, and we will investigate the existence and stability of bubble rings. On the parabolic side, this unit studies the dichotomy between singularity formation and global existence for cross-diffusion systems.

Regularity

Investigators: Hein, Seis, Weber

A central issue in the analysis of PDEs are regularity estimates and their breakdown, which form the theme of this unit. Recent breakthroughs in the theory of stochastic PDEs (regularity structures, paracontrolled calculus) have permitted researchers to develop a solution theory for many PDEs of mathematical physics in extremely singular settings. Our goal is to advance this further to quasilinear equations by studying the stochastic porous medium equation. From the deterministic point of view, this unit also encompasses regularity estimates for the complex Monge–Ampère equation and for the thin film equation.