Causal reasoning is usually formalized through structural causal models (SCMs) or potential outcomes. These frameworks have been enormously successful for modeling, identification, and inference, but they are not primarily designed as axiomatic foundations analogous to probability spaces in probability theory. This tutorial introduces causal spaces, a measure-theoretic framework in which interventions are represented by primitive causal kernels satisfying two minimal axioms: doing nothing leaves everything unchanged, and intervened coordinates take their prescribed values. The tutorial will explain the motivation for causal spaces, present the basic definition and semantics, and work through examples linking back to familiar causal models. It will then present some further development of basic causal space theory, such as causal effects, sources, identifiability, and counterfactual spaces, and touch upon advanced topics such as targeted interventions and continuous-time stochastic processes that are more difficult to express in existing frameworks.

"The talk addresses the quantitative study of resilience in dynamical systems by analyzing a range of resilience indicators within a unified mathematical and computational framework. Emphasis is placed on ecological resilience and on extending existing approaches from autonomous systems to settings with time-dependent perturbations, leading to nonautonomous dynamics. To address the lack of classical invariant structures in this context, characteristic performance ranges are used as practical reference states. The developed framework is applied to case studies, demonstrating how time-dependent disturbances and rate-induced effects influence resilience and tipping behavior."

Stirring accelerates the decay of a diffusing tracer: advection generates fine gradients on which diffusion acts efficiently, so that the two mechanisms together dissipate the tracer faster than diffusion alone. This effect, known as enhanced dissipation, underlies transport phenomena from laboratory mixing to atmospheric and oceanic flows, and its prototypical model is the advection-diffusion equation for a passive scalar.

Obtaining quantitative decay rates for a given flow is analytically delicate, as the enhancement is invisible to standard energy estimates, and has developed into an active area of PDE analysis. For stationary shear flows the theory is now well developed, and the decay rate is controlled by the local structure of the velocity profile at its critical points. This talk asks what becomes of this local mechanism when the flow depends on time. In earlier work we showed that for profiles with a time-modulated amplitude, decay is governed by the accumulated modulation. Building on this, we consider a profile translating at constant speed, with critical points sweeping through the domain, and establish a quantitative inviscid mixing estimate together with a corresponding enhanced dissipation result: translation at intermediate speeds accelerates decay, while fast translation averages the enhancement away. Numerical computations confirm the predicted scalings and motivate ongoing work on flows whose critical point structure itself changes in time, for instance through the collision and annihilation of critical points.

The talk is based on joint work with Camilla Nobili (University of Surrey) and Giuseppe M. Coclite (Politecnico di Bari).