Organizers: Stefanie Winkelmann, Maximilian Engel

Abstract:

Interacting particle systems appear in physics, biology, and social dynamics, often leading to high-dimensional models. This minisymposium focuses on recent advances in model reduction techniques that enable efficient simulation, analysis, and interpretation of such systems. Approaches include mean-field models and stochastic PDEs, transfer-operator frameworks and Markov state models, hybrid multiscale models, coherent structure analysis, as well as data-driven techniques. The session aims to bring together theoretical and applied perspectives on efficient representations of complex, many-body systems.

Speakers:

11:30h: Christian Kuehn (TU München)

Towards Geometric Singular Perturbation Theory for PDEs

Geometric singular perturbation theory (GSPT) has been very successfully used in the context of spatial dynamics to understand patterns arising in partial differential equations (PDEs). Yet, for many pattern-forming systems, one cannot rely on spatial dynamics and has to work more directly with the underlying partial differential equation. In this talk I am going to explain some recent progress trying to lift slow manifold and blow-up methods directly to PDEs. In particular, I shall explain recent extensions of slow manifold techniques to PDEs and highlight novel developments in the blow-up method for various classes of reaction-diffusion systems. In the context of the blow.up method an interesting observation is that hidden transport terms can be made visible via blow-up. This is joint work with several co-authors and the talk wil present some highlights that have emerged during a longer phase to built GSPT for PDEs [1-10].

[1] Jelbart, S., Kuehn, C., & Sánchez, A. M. (2024). Characterising exchange of stability in scalar reaction-diffusion equations via geometric blow-up. arXiv preprint arXiv:2411.13679.

[2] Kuehn, C., & Sulzbach, J. E. (2026). Thin Domains, Reduction, and Slow Manifolds. arXiv preprint arXiv:2606.05398.

[3] Engel, M., Hummel, F., Kuehn, C., Popović, N., Ptashnyk, M., & Zacharis, T. (2024). Geometric analysis of fast-slow PDEs with fold singularities via Galerkin discretisation. Nonlinearity, 37(11), 115017.

[4] Kuehn, C., & Sulzbach, J. E. (2025). Approximate slow manifolds in the Fokker-Planck equation. Quarterly of Applied Mathematics.

[5] Desvillettes, L., Kuehn, C., Sulzbach, J. E., Tang, B. Q., & Tran, B. N. (2025). Slow manifolds for pde with fast reactions and small cross diffusion. arXiv preprint arXiv:2501.16775.

[6] Kuehn, C., & Sulzbach, J. E. (2025). Fast reactions and slow manifolds. Nonlinear Differential Equations and Applications NoDEA, 32(4), 72.

[7] Jelbart, S., & Kuehn, C. (2024). A formal geometric blow-up method for pattern forming systems. Topics in Multiple Time Scale Dynamics, M. Engel, H. Jardón-Kojakhmetov, and C. Soresina, eds, 806, 49-86.

[8] Engel, M., & Kuehn, C. (2024). Geometric analysis of a truncated galerkin discretization of fast-slow PDEs with transcritical singularities. SIAM Journal on Applied Dynamical Systems, 23(4), 2853-2898.

[9] Hummel, F., & Kuehn, C. (2022). Slow manifolds for infinite-dimensional evolution equations. Commentarii Mathematici Helvetici, 97(1), 61-132.

[10] Engel, M., Hummel, F., & Kuehn, C. (2021). Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions. Proceedings of the American Mathematical Society, Series B, 8(21), 252-266.

12:00h: Grigorios Pavliotis (Imperial College London)

On the Diffusive-Mean Field limit of Kinetic Interacting Particle Systems

We study the joint diffusive-mean field limit for a system of weakly interacting kinetic Langevin dynamics. We show that, in the absence of phase transitions, the two limits commute, and we calculate the covariance matrix of the limiting Brownian motion using the Green-Kubo/Kipnis-Varadhan formula. However, at low temperatures, and in the presence of phase transitions, the two limits may not commute. We demonstrate our findings by providing a detailed analysis of the diffusive-mean field limit for the O(2) model in a magnetic field. Our analysis is based on the systematic use of recently developed hypocoercivity techniques, together with an appropriate linearization of the mean field McKean-Vlasov-Fokker-Planck PDE.

12:30h: Sebastian Wieczorek (University College Cork)

Singular Basins of Attraction in Multiscale Systems

Real-world systems often evolve on different timescales and possess multiple coexisting stable states. Whether or not a system returns to a given stable state after being perturbed away from it depends on the shape and extent of its basin of attraction. We show that basins of attraction in multiscale systems can exhibit special geometric properties in the form of singular funnels. Although singular funnels are narrow, they can extend to different regions of the phase space and, unexpectedly, impact the system's resilience to perturbations. Consequently, singular funnels may prevent common dimensionality reductions in the limit of large timescale separation, such as the quasi-static approximation, adiabatic elimination and time-averaging of the fast variables. We refer to basins of attraction with singular funnels as singular basins. We show that singular basins are universal and occur robustly in a range of multiscale systems