The theory of nonautonomous and random dynamical systems (NDSs and RDSs) provides an appropriate mathematical setting for studying the effects of time-dependent forcing, both natural and anthropogenic, upon a climate system characterized by intrinsic variability [1,2]. In this theory, the forward attractors of autonomous dynamical systems are replaced by pullback and random attractors (PBAs and RAs) and classical bifurcations by “tipping points.” Over the last two decades, these relatively novel concepts have been applied to a number of simple climate models, atmospheric, oceanic and coupled [3]. Important insights into the study of PBAs and RAs arising from climate dynamics have been provided by novel tools from algebraic topology [4,5]. These tools have led to the definition and study of topological tipping points (TTPs), which we present and apply here to simple models of the mid-latitude atmosphere [6,7] and of the wind-driven double-gyre ocean circulation [8,9]. The atmospheric model is the Lorenz model for seasonal variability [6], while the oceanic one is a low-order approximation of a spectral quasigeostrophic model for the subtropical and subpolar gyres of the North Atlantic or North Pacific ocean basin, subject to time varying zonal winds [9]. The recent tools from algebraic topology applied to the latter are Branched Manifold Analysis through Homologies (BraMAH) and the Templex, which combines the cell complex underlying BraMAH with a directed graph that captures the flow in the dynamical system’s phase space [5]. The talk will be based on joint work with N. Bodnariuk & G.D. Char´o (CIMA & IFAECI, Buenos Aires, AR), H.A. Dijkstra & B. Maraldi (IMAU, Utrecht, NL), S. Pierini (U. Parthenope, Napoli, IT), D. Sciamarella (CIMA & IFAECI, Buenos Aires, AR), & S. Speich (ENS & LMD, Paris, FR).

References 1. M. Ghil, M. D. Chekroun, and E. Simonnet, 2008: Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237, 2111–2126, doi:10.1016/j.physd.2008.03.036. 2. M. D. Chekroun, E. Simonnet, and M. Ghil, 2011: Stochastic climate dynamics: Random attractors and time-dependent invariant measures, Physica D, 240(21), 1685–1700. doi :10.1016/j.physd.2011.06.005. 3. M. Ghil and V. Lucarini, 2020: The physics of climate variability and climate change, Rev. Mod. Phys., 92(3), 035002. doi: 10.1103/ RevModPhys.92.035002. 4. G. D. Char´o, M. D. Chekroun, D. Sciamarella, and M. Ghil, 2021: Noise-driven topological changes in chaotic dynamics, Chaos, 31(10). doi:10.1063/5.0059461. 5. M. Ghil and D. Sciamarella, 2023: Review article: Dynamical systems, algebraic topology, and climate sciences, Nonlin. Processes Geophys., 30(4), 399–434, doi:10.5194/npg-30-399-2023. 6. E. N. Lorenz, 1984: Irregularity: a fundamental property of the atmosphere. Tellus A, 36(2), 98–110. 7. B. Maraldi, 2024: Intraseasonal atmospheric variability under different climate trends, M.Sc. Thesis, IMAU, U. of Utrecht, supervised by H.A. Dijkstra and M. Ghil. 8. Dijkstra, H. A., and M. Ghil, 2005: Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach, Rev. Geophys., 43, RG3002, doi:10.1029/2002RG000122. 9. S. Pierini and M. Ghil, 2021: Tipping points induced by parameter drift in an excitable ocean model, Scientific Reports, 11, 11126. https://rdcu.be/clp5V.

The ongoing energy transition reshapes the dynamics of power grids by introducing new categories of actors. An important example are grid-forming inverters (GFIs) that are employed to enhance grid stability.

This talk delves into the urgent and complex task of understanding the collective behavior and stability of future grids that are characterized by a heterogeneous mix of dynamics.

Recent advancements have significantly improved our ability to describe modern power grid dynamics. Firstly, the development of the normal form for grid-forming actors offers a technology-neutral framework to describe the dynamics of GFIs. Secondly, the concept of complex frequency facilitates a seamless depiction of the impact of the nodal dynamics on power flows within the grid.

This presentation's primary focus is on demonstrating the synergy of the normal form and complex frequency in unraveling the inherent adaptive nature of power grids. We unveil a simple yet universal equation governing the collective dynamics. Remarkably, this equation is expressed solely through a matrix of complex couplings and is devoid of the network topology. These complex couplings enable a novel adaptive network formulation of power grids.

Finally, we present recent validation results of the normal form through system identification and show its accuracy in modeling a broad range of GFIs. These validations encompass laboratory measurements and simulation data. This success underscores the success of our adaptive approach in power grid modeling.

In diesem Vortrag wird ein Überblick über moderne Entwicklungen in der Topologie gegeben, insbesondere im Bereich der sogenannten "Unendlichen Geometrien". Diese Gebiete verbinden klassische topologische Konzepte mit modernen Anwendungen in der Mathematik und darüber hinaus. Ein zentrales Thema wird die Untersuchung von topologischen Invarianten in unendlichen Dimensionen sein. Besonders interessant ist dabei die Berechnung der Homotopiegruppen von Sphären, eine klassische Fragestellung, die uns zur Betrachtung der stabilen Homotopiegruppe führt: \[\pi_{n+k}(S^n) \simeq \pi_k^s\]

Diese Gleichung drückt aus, dass die Homotopiegruppe einer \(n\)-dimensionalen Sphäre in der \(n+k\)-ten Dimension äquivalent zur \(k\)-ten stabilen Homotopiegruppe ist, was uns tiefe Einblicke in die Struktur topologischer Räume ermöglicht. Anhand ausgewählter Beispiele aus der Differentialtopologie und Kategorientheorie werden wir illustrieren, wie diese Konzepte in verschiedenen mathematischen Disziplinen miteinander verknüpft sind. Der Vortrag richtet sich sowohl an Studierende als auch an Forschende, die an einem tiefgreifenden Verständnis topologischer Strukturen interessiert sind.

Hinweis: Dieser Vortrag ist ein fiktives Beispiel und dient lediglich zur Veranschaulichung.

We consider a problem of stochastic optimal control with separable drift uncertainty in strong formulation on a finite horizon. The drift coefficient of the state process is multiplicatively influenced by an unobservable random variable, while admissible controls are required to be adapted to the observation filtration. Choosing a control actively influences the state and information acquisition simultaneously and comes with a learning effect. The problem, initially non-Markovian, is embedded into a higher-dimensional Markovian, full information control problem with control-dependent filtration and noise. To that problem, we apply the stochastic Perron method to characterize the value function as the unique viscosity solution to the HJB equation, explicitly construct ε-optimal controls and show that the values of strong and weak formulations agree. Numerical illustrations show a significant difference between the adaptive control and the certainty equivalence control

Recently, the problem of predicting a response variable from a set of covariates on a data set that differs in distribution from the training data has received more attention. We propose a sequence of causal-like models from in-sample data that provide out-of-sample risk guarantees when predicting a target variable from a set of covariates. Whereas ordinary least squares provides the best in-sample risk with limited out-of-sample guarantees, causal models have the best out-of-sample guarantees by sacrificing in-sample risk performance. We introduce causal regularization by defining a trade-off between these properties. As the regularization increases, causal regularization provides estimators whose risk is more stable at the cost of increasing their overall in-sample risk. The increased risk stability is shown to result in out-of-sample risk guarantees. We provide finite sample risk bounds for all models and prove the adequacy of cross-validation for attaining these bounds.