Computing through the Lens of Geometry

While numerical methods are often treated as low-level exercises in floating-point operations and index bookkeeping, looking at computation through the lens of geometry is frequently the key to predictive modeling. This talk offers a fast-paced survey of how respecting the geometry underlying various shape modeling and physical simulation tasks results in efficient, versatile tools for both research and industry. We begin by exploring how discrete connections provide a rigorous framework for both the grooming of virtual characters and high-dimensional data analytics. We then pivot to the geometric foundations of calculus, illustrating how mimicking the structure of exterior calculus ensures that computational operators remain faithful to their fundamental continuous definitions. The discussion shifts to the geometric nature of mechanics, where we show that preserving physical constraints—such as symmetry and conservation laws—is an effective way to ensure long-term simulation stability. We conclude with a candid look at the current state of machine learning for geometry, weighing the undeniable flexibility of neural methods against the risks of losing the structural guarantees provided by classical geometric methods. Throughout this talk, the recurring theme is that drawing inspiration from differential geometry in the discrete setting is rarely just about mathematical purity; it is a practical necessity for building a robust and general-purpose computational toolbox.