Events

Proper scoring rules are fundamental for evaluating the quality of probabilistic forecasts. A canonical property of these scores is their decomposition into calibration and refinement errors. However, reliably estimating these two components is notoriously difficult in practice. Standard approaches are often tied to the specific setting of binary classification, and suffer from well-documented finite-sample pathologies that can lead to systematically over- or under-estimating calibration error. In this talk, I will present a variational formulation of the calibration-refinement decomposition that holds for any proper scoring rule and any forecasting space. This formulation gives a natural interpretation: refinement error is the best achievable risk obtained by optimally post-processing the forecaster's predictions, while calibration error measures how much risk is reduced by this optimal post-processing. Beyond the theory, this framework motivates a robust, cross-validation-based estimation strategy. This recipe applies across proper scores and forecast spaces, naturally avoids the estimation pathologies of prior methods, and yields lower bounds on calibration error.

Tensors, or multilinear maps, are well-known classically, but relatively isolated from modern algebraic geometry. This is hopefully about to change. In the talk I will illustrate this new bridge and its benefits also for alg.geo. Some numbers, such as 588224, 161914176, or 20636043486 may appear. This is a joint work with Jakub Jagiella.

Many scientific computing problems involve families of related operators arising from variations in physical parameters, geometries, or governing equations. Multiple operator learning aims to approximate such families within a unified model. This talk develops mathematical and computational foundations for this setting from two complementary perspectives. The first is based on Multiple Neural Operators, a deep-learning framework for which we derive scaling laws for minimax approximation rates and generalization bounds. The second uses kernel methods within a general encoder-decoder framework, providing closed-form training, rigorous approximation guarantees for multi-input, multi-output operator learning, and substantially reduced computational costs. Numerical experiments on several families of parametric PDEs illustrate the accuracy and computational trade-offs of the different approaches.