Abstract: Advances in simulation and scientific machine learning have made computation an engine of discovery. For increasingly complex physical systems, however, computational scale and model expressiveness alone may not suffice for faithful simulation, exploration, and inverse design. General-purpose representations often describe a space much larger than the set of physically admissible states. Computing in this larger space can require substantial effort to ensure that solutions respect the underlying geometry, topology, and conservation laws. A complementary path is to make this structure explicit. Through examples ranging from bending rods and slithering snakes to fluid flows and magnetic fields in plasmas, this talk shows how geometric insight can make difficult problems computationally tractable, how the resulting structure can be preserved and encoded in algorithms, and how the exchange among theory, discretization, and application yields new mathematics alongside more effective computational tools.


Contact: Ian Tobasco