Webseite von David Fajman

Research

I am interested in the interplay of time, space and matter. Einstein’s field equations describe how the geometry of spacetime is affected by the presence of matter, how gravitational waves evolve and how the dynamics of matter fields play out in curved spacetime. These phenomena have implications for our understanding of the formation and evolution of structures in the universe, both on astrophysical and on cosmological scales, currently and in past and future eras of the cosmological evolution. On large scales all these scenarios are governed by gravitation and the macroscopic features of matter. In consequence, the answers to all questions about the dynamics of gravity and matter on large scales are hidden within the Einstein equations complemented by those of the matter fields, such as the relativistic Euler equations or kinetic transport equations. It is our goal to reveal them. The Einstein equations describe the dispersion of gravitational waves, regions of extreme curvature such as black holes or the big bang as well as the Universe as a whole. A universal understanding of their solutions is hence an ambitious task. To achieve tangible results we focus on particular regimes and try to fully understand the dynamics of spacetime and matter in particular conditions, such as near homogeneous states. Einstein equations are geometric partial differential equations - coupled wave equations in curved space, where the curvature itself is part of the solution. The tools that enable us to solve these equations are being developed in the mathematical discipline of geometric analysis. This area of research is located at the forefront of modern mathematical physics, where rigorous mathematical methods enable us to sharply analyze the nonlinear nature of Einstein’s equations. I am particularly interested in the implications of these results for the actual understanding and reconciliation of general relativity with observations, particular in the cosmological regime.

Publications

1. Comment on ”Resolving isospectral ’drums’ by counting nodal domains” with J. Brüning and C. Puhle, J. Phys. A: Math. Theor. 40, (2007) 15143 – 15147

2. Inverse Nodal Problems, J. Phys. A: Math. Theor. 42, (2009) 175209 – 175219

3. On the nodal count for flat tori with J. Brüning, Comm. Math. Phys. 313, (3), (2012) 791 – 813

4. Nodal domains of a non-separable problem - the right angled isosceles triangle with A. Aronovitch, R. Band and S. Gnutzmann, J. Phys. A: Math. Theor. 45, (2012) 085209 – 085226; Featured Article

5. Area inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theory, with W. Simon, Adv. Theor. Math. Phys. 8, (3), (2014) 685 – 704

6. Static solutions for the Einstein-Vlasov system with non-vanishing cosmological constant, with H. Andréasson and M. Thaller, SIAM J. Math. Anal. 47, (4), (2015) 2657 – 2688

7. Topological Properties of Neumann Domains, with R. Band, Ann. Henri Poincaré 17, (9), (2016) 2379 – 2407

8. Future asymptotic behavior of 3-dimensional spacetimes with massive particles, Letter, Class. and Quantum Grav. 33, 11, (2016), 11LT01

9. Local well-posedness for the Einstein-Vlasov system, SIAM J. Math. Anal. 48, (5), (2016) 3270 – 3321

10. Models for Static Self-Gravitating Photon Shells and Geons, with H. Andréasson and M. Thaller, Ann. Henri Poincaré 18, (2), (2017), 681 – 705

11. The Einstein-Λ flow on product manifolds, with K. Kröncke, Class. and Quantum Grav. 33, (23), (2016), 235018

12. Courant–sharp eigenvalues of Neumann 2–rep–tiles, with R. Band and M. Bersudsky, Lett. Math. Phys. 107, (2017), 821–859

13. The nonvacuum Einstein flow on surfaces of negative curvature and nonlinear stability, Comm. Math. Phys. 353, (2), (2017), 905–961

14. Topology and incompleteness for 2+1-dimensional cosmological spacetimes, Lett. Math. Phys. 107, (2017), 1157–1176

15. A note on future complete spacetimes with massless outgoing particles, with C. Schaman, Class. Quant. Grav. 34, (23), (2017), 077002

16. A vector field method for relativistic transport equations with applications, with J. Joudioux and J. Smulevici, Analysis and PDE 10, (7), (2017)

17. The nonvacuum Einstein flow on surfaces of nonnegative curvature, Comm. PDE 43, (3), (2018)

18. Stable fixed points of the Einstein flow with positive cosmological constant, with K. Kröncke, Commun. Anal. Geom. 28, (7), (2020) 1533–1576

19. On the massive-massless Einstein-Vlasov system in spherical symmetry with P. Eigenschink, J. Joudioux, Phys. Rev. D 98, 044002, (2018)

20. On the CMC-Einstein-Λ flow, with K. Kröncke, Class. Quant. Grav. 35, (19), (2018), 195005

21. Stable cosmological Kaluza-Klein spacetimes, with K. Kröncke, V. Branding, Comm. Math. Phys. 368, (3), (2018), 1087–1120

22. Kantowski-Sachs cosmology with Vlasov matter with G. Heißel, Class. Quant. Grav. 36, (13), (2019), 135002

23. The stability of the Minkowski space for the Einstein-Vlasov system, with J. Joudioux, J. Smulevici, Analysis and PDE 14, (2), (2021), 425–531

24. Isotropization of slowly expanding spacetimes with H. Barzegar, G. Heißel, Phys. Rev. D 101, (2020), 044046

25. Nonlinear Stability of the Milne model with matter, with L. Andersson, Comm. Math. Phys. 378, (2020), 261–298

26. On the Oscillations and Future Asymptotics of locally rotationally symmetric Bianchi type iii cosmologies with a massive scalar field with G. Heißel, M. Maliborski, Class. Quant. Grav. 37, (2020), 135009

27. Future Attractors in 2+1 Dimensional Λ Gravity, Phys. Rev. Lett. 125, 121102 (2020)

28. Attractors of the Einstein-Klein-Gordon system with Z. Wyatt, Comm. PDE 46, 1, (2020), 1–30

29. Stabilizing relativistic fluids on spacetimes with non-accelerated expansion with T. Oliynyk, Z. Wyatt, Comm. Math. Phys. 383, (2021), 401–426

30. Asymptotic Stability of Minkowski spacetime with non-compactly supported massless Vlasov matter with L. Bigorgne, J. Joudioux, J. Smulevici, M. Thaller, Arch. Rat. Mech. Anal. 242, 1, (2021), 1–147

31. Averaging with a time-dependent perturbation parameter with G. Heißel, J. W. Jang, Class. Quant. Grav. 38, 8, (2020), 085005

32. Stable cosmologies with collisionless charged matter with H. Barzegar, J. Hyp. Diff. Eq. 19, 4, (2022), 587–634

33. Blow-up of waves on singular spacetimes with generic spatial metrics with L. Urban, Lett. Math. Phys. 112, 42, (2022)

34. Recollapsing spacetimes with Λ < 0 with M. Kraft, Class. Quant. Grav. 40, 14, (2023), 145007

35. The Stability of Relativistic Fluids in Linearly Expanding Cosmologies with M. Ofner, T. Oliynyk, Z. Wyatt, Int. Math. Res. Not. 2024, 5, (2024), 4328–4383

36. Cosmic Censorship near FLRW spacetimes with negative spatial curvature with L. Urban, Analysis and PDE 18-7, (2025), 1615–1713

37. Slowly expanding stable dust spacetimes with M. Ofner, Z. Wyatt, Arch. Rat. Mech. Anal. 248, 5, (2024)

38. Phase transition between shock formation and stability in cosmological fluids with M. Maliborski, M. Ofner, T. Oliynyk, Z. Wyatt, Letter, Class. Quant. Grav. 42, 14, (2025), 14LT01

Preprints

39. On the past maximal development of near-FLRW data for the Einstein scalar-field Vlasov system with L. Urban, (2024), arXiv:2402.08544
40. Stability of fluids in spacetimes with decelerated expansion with M. Ofner, T. Oliynyk, Z. Wyatt, (2025), arXiv:2501.12798
41. Charged black holes surrounded by shells of charged particles with H. Seer, (2025), arXiv:2506.21293

David Fajman

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