January 31st (1 pm CT) - Owen Lewis, Department of Mathematics, Florida State University, Electrodiffusion Mediated Maintenance of the Gastric Mucus Layer.
Abstract. Diffusion of charged particles, or electrodiffusion, plays an important role in many physiological systems including the human stomach. The gastric mucus layer is widely recognized to serve a protective function, shielding your stomach wall from the extreme acidity and digestive enzymes present in the stomach. However, there is still much debate regarding the control of electrodiffusive transport through the mucus layer. In this talk, I will discuss a mathematical description of electrodiffusion within a two-phase gel model of gastric mucus and the challenges associated with its analysis and numerical simulation. This model is used to investigate physiological hypotheses regarding gastric layer maintenance that are beyond current experimental techniques.
January 22nd (1 pm CT) - Greg Ongie, Department of Statistics, University of Chicago, Rethinking regularization in modern machine learning and computational imaging.
Abstract. Optimization is central to both supervised machine learning and inverse problems in computational imaging. These problems are often ill-posed and some form of regularization is necessary to obtain a useful solution. However, new paradigms in machine learning and computational imaging necessitate rethinking the role of regularization, as I will illustrate with two examples. First, in the context of supervised learning with shallow neural networks, I will show how a commonly used form of regularization has a surprising reinterpretation as a convex regularizer in function space. This yields novel insights into the role of overparameterization and depth in learning with neural networks having ReLU activations. Second, I will discuss a novel network architecture for solving linear inverse problems in computational imaging called a Neumann network. Rather than using a pre-specified regularizer, Neumann networks effectively learn a regularizer from training data, outperforming classical techniques. Beyond these two examples, I will show how many open problems in the mathematical foundations of deep learning and computational imaging relate to understanding regularization in its many forms. (more information)
January 21st (1 pm CT) - Scott Hottovy, Department of Mathematics, United States Naval Academy, A simple stochastic model of tropical atmospheric waves.
As tropical storms go, you have probably heard of Hurricanes, Tropical Cyclones, El Niño, and La Niña. But you probably haven't heard of the Madden-Julian Oscillation (MJO). It is the major contributor to rainfall in tropical regions and influences the climate in Wisconsin regularly. Unlike Hurricanes and El Niño, the MJO is still not well understood. In an effort to understand the mechanisms of the MJO, I will describe a model building from a dynamically stationary "background" tropical rainfall model and coupling that to a tropical wave model. These models use Stochastic Differential Equations (SDE) and Stochastic Partial Differential Equations (SPDE) as the building blocks. In the "background" model, an SDE model is used which leads to characteristics of criticality and phase transitions. For the full model with waves, we use a continuous one-dimensional SPDE. Because of the simplicity of the models, we are able to solve many statistics exactly, or run fast numerical experiments. (more information)