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TITLE: Learning Physical Dynamics on Curved Spaces: Differential Geometry Meets Stochastic Flow

ABSTRACT: Learning and generalizing physical laws beyond conventional flat Euclidean domains has become increasingly important in scientific simulation with generative modelling. This seminar presents a concise overview of recent advances at the intersection of differential geometry and stochastic learning. By integrating geometric perspectives with probabilistic frameworks, it highlights how curvature, gauge invariances, and conservation laws can be explicitly preserved, rather than merely approximated. We will explore the conceptual shift from modelling dynamics in ambient coordinate systems toward intrinsically defined geometric reference frames, clarifying how this approach naturally reconciles stochasticity with geometric coherence. Key applications include modelling phenomena across scales such as molecular dynamics and demonstrate how respecting underlying symmetries and invariants enables stronger generalization and enhanced physical realism. The seminar provides a clear conceptual foundation for developing symmetry-aware algorithms, a critical step toward building physically consistent AI models in scientific domains.