Many efficient algorithms have been developed over the years for planar graphs and more general graphs such as low genus graphs. Intersection graphs of geometric objects (in low dimensions) with some additional properties, such as fatness or low density, provide yet another family of graphs for which one can design better algorithms. This family is a vast extension of planar graphs, and yet is still algorithmically tractable for many problems. In this talk, we will survey this class of graphs, and some algorithms and intractability results known for such graphs, and outline open problems for further research.

Functional inequalities such as the Poincaré and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains (and more generally Markov processes) by linking properties of the generator to variance and entropy decay. However, in certain applications, such as coarse-graining problems arising in molecular dynamics, it becomes necessary to study entropy decay with respect to a reference measure that is not the steady state. In such settings, the dynamics are typically non-reversible (i.e. not of gradient-flow type), and the classical functional inequality framework tied to equilibrium does not directly apply.

In this talk, I will introduce a generalisation of the log-Sobolev inequality with respect to arbitrary probability measures on a finite state space. This generalisation retains key features of the classical inequality while exhibiting properties relevant for coarse-graining applications, including continuity with respect to the reference measure and explicitly computable lower bounds. As an application, we derive quantitative error bounds for coarse-graining of finite Markov chains.

This talks is based on joint work with Bastian Hilder and Patrick van Meurs.

ICA-based causal discovery methods such as LiNGAM have been highly successful under the assumption that noise variables become independent after an appropriate causal ordering. However, this assumption is often violated in the presence of confounding. \[ \] In this talk, I present a Bayesian and information-theoretic formulation of ICA for causal order estimation that explicitly allows for confounding. Rather than enforcing independence, we quantify residual dependence among noise variables using multivariate mutual information and evaluate causal orders via Bayesian marginal likelihoods. \[ \] This approach provides a principled ranking of causal orders under confounding and recovers classical LiNGAM-type methods as special cases when confounding is absent. I will focus on the conceptual framework and discuss connections to existing ICA-based methods, as well as open questions.

I am going to review mesh-based Eulerian discretizations of transient and stationary linear advection-diffusion boundary value problems (BVPs) for differential forms on bounded domains in Rn. Advection is passive, assuming the underlying velocity vector field to be given and sufficiently regular. BVPs of this type pervade continuum modeling in the form of transport equations (0-forms), continuity equations (n-forms), but also govern the behavior of electromagnetic fields in conducting fluids (1-forms). The main challenge for accurate numerical simulation is the singularly perturbed case of dominant advection.

Though this subject has been exhaustively studied for 0-forms and n-forms, the development of numerical methods for other degrees of forms started only about 15 years ago, guided by the principle that numerical methods designed for the case of 0/n-forms can usually be generalized to ℓ-forms, 0 < ℓ < n, with suitable adaptations. As in the case of 0/n-forms an essential element of discretization is stabilization by taking into account the direction of the flow.

The presentation will cover the adaptation to ℓ-forms of the following stabilization techniques: • Discontinuous Galerkin (DG) approaches with suitable upwind fluxes • Conforming Galerkin finite element methods with upwind quadrature • Exponentially (simplex/edge) averaged stabilized finite element methods

The exposure of sovereigns to climate risks is priced and can affect credit ratings and debt servicing costs. I will discuss how integrated assessment models (IAMs) can be linked with stochastic debt sustainability analysis to inform our understanding of climate risks to sovereign debt dynamics and assess the available fiscal space to finance climate policies. The narrative scenario architecture developed within the IPCC and the transition scenarios of NGFS can be adopted to bring structure and transparency to the analysis. The analysis is complicated by deep uncertainty —risks, ambiguity, and mis-specifications— of climate change. Using scenario trees, narrative scenarios, and model ensembles, respectively, we can address these three challenges. I will discuss two applications combining well-known climate models with a high-realism debt sustainability analysis model, and discuss (i) sovereign debt effects of the transition to low-carbon economies on a large panel of countries, and (ii) effects of damages and adaptation. Results show that adaptation pays, but raises the question of who will pay for it.