Seminar

We report on recent progress on some nonlocal Monge--Ampere equations. Our results include the interior Harnack inequality for the fractional linearized Monge--Ampere equation (with Diego Maldonado from Kansas State University) and the regularity for the obstacle problem for the Caffarelli--Charro fractional Monge-Ampere equation (with Yash Jhaveri from Institute of Advanced Study).

The distribution of energy across space and time scales is quite different between the atmosphere and ocean. Therefore, codes for air-sea interaction are constructed by coupling together atmosphere and ocean components, each highly optimized internally using different numerical methods, by passing fluxes of conserved quantities between the components in the form of boundary conditions at the air-sea interface. For efficiency, the fluxes are usually asynchronous, meaning they are calculated using data extrapolated from previous times for at least one component.

Convolutional Neural networks are the current state-of-the-art in image processing, far surpassing other popular strategies in terms of performance. These network models are built on heuristics, though, and have a limited foundation of mathematical guarantees. This talk will cover the basics of modern Neural network architectures and provide an overview of notable cases where they fail in unexpected, unpredictable ways.