Weekly Bulletin

Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that meet the criteria. This lecture offers an invitation to the field of matrix concentration and its multifarious applications.

Multiscale, parameter-dependent partial differential equations (PDEs) pose severe computational challenges due to strong coefficient heterogeneity and high-dimensional parameter spaces. We develop a geometric interpolation approach within the multiscale generalized finite element method (MS-GFEM) that targets the most expensive component: computing parameter-dependent optimal local approximation spaces. Leveraging the spatial localization of MS-GFEM and assuming local parameter dependence, we decompose the global problem into parametrically low-dimensional local subproblems. The optimal subspaces for each parameter are identified as points on a Grassmann manifold and approximated via Grassmann interpolation on sparse grids, which preserves the geometric structure of these spaces while efficiently handling high-dimensional parameter spaces. The resulting localized model reduction method inherits the nearly exponential spatial convergence of MS-GFEM and the parametric convergence rates of sparse grids. Numerical experiments for elliptic problems confirm the theoretical convergence results.

Understanding whether a treatment works differently across patient subgroups is a central question in drug development. However, answering it reliably is notoriously difficult. Clinical trials are typically powered for overall treatment effects, leaving subgroup analyses underpowered and vulnerable to both false positives and false negatives, as illustrated by multiple real cases where subgroup findings couldn't be confirmed in follow-up trials. Despite these severe limitations however questions around treatment effect heterogeneity are of great interest to sponsors, regulators, and clinicians alike and need to be addressed in drug development. In this talk, I will present WATCH (Workflow to Assess Treatment EffeCt Heterogeneity) for exploratory assessment of TEH in randomized clinical trials. The challenges resulting from insufficient data-based information on the question of interest (TEH) are approached via (i) providing a structured and pre-planned analysis approach for this exploratory question, providing more standardization and reducing analyst degrees of freedom, (ii) documenting a-priori, external evidence on potential effect modifiers upfront and as part of the workflow and (iii) utilizing flexible, multivariate analysis methods based on statistical learning methods. I will present the workflow steps, and discuss current analysis methods used to implement the workflow, as well as open methodological questions.

<div class="elementToProof"> <p class="elementToProof">In his celebrated proof of the weak cosmic censorship conjecture for the spherically symmetric Einstein-scalar field system, Christodoulou exploited the following property of that specific matter model: Naked singularities, when they arise, exhibit infinite blue-shift along the null geodesics terminating at the singularity. This behaviour is consistent with self-similarity: Even for more general spherically symmetric matter models, it can be shown that self-similar naked singularities must exhibit infinite blue-shift. Whether, for these more general models, all naked singularities have the infinite blue shift property (and hence are potentially subject to an instability mechanism analogous to that introduced by Christodoulou) still remains an open question.</p> </div> <div class="elementToProof"> <p class="elementToProof"> </p> </div> <div class="elementToProof"> <p class="elementToProof">In this talk, we will present the construction of a spherically symmetric solution to the Einstein-massless Vlasov system which contains a locally naked singularity with finite total blue-shift along its past null cone. The initial data giving rise to this solution have limited differentiability, but belong to a regularity class above the scale invariant threshold.</p> </div>

We develop a comprehensive mathematical finance framework for propagator models with transient linear price impact. These models lead to infinite-dimensional, non-Markovian control problems and fall outside the scope of classical arbitrage and duality theory. We establish a fundamental theorem of asset pricing, a superreplication theorem with liquidity-adjusted risk measures, and a full convex-duality approach to utility maximisation. Despite the non-linearity of preferences and the path-dependent impact structure, we show that optimal strategies can be obtained from an equivalent frictionless optimisation problem under a suitably constructed shadow price.