Free-boundary problems are partial differential equations in which the unknown function and its domain must be simultaneously determined. They arise ubiquitously as mathematical models for phenomena in many fields, most notably in physics, biology and finance. Free boundary problems are typically highly nonlinear and nonlocal in nature, making their analysis challenging. I will discuss two fundamental…

A central question in machine learning theory is whether an algorithm enjoys good generalization, which is the ability to correctly predict new examples from prior observations. While classical wisdom advocates for methods with fewer parameters than data points in order to avoid overfitting, modern machine learning algorithms are severely over-parameterized and perfectly fit training data.…

Thin liquid films flowing down vertical fibers exhibit complex and interesting interfacial dynamics, including droplet formation and traveling wave patterns. Such dynamics play a crucial role in the design of heat and mass exchangers for many engineering applications, including cooling and desalination systems. Recent experiments present a wealth of new dynamics that illustrate the need…

Computerized tomography techniques, which are used for visualizing the interior structure of an object of interest in a non-invasive manner, have played a central role in medical imaging, industrial non-destructive testing, geophysics, astronomy, and other fields. Yet, the quest for more sensitive, reliable, robust, safer, and cheaper imaging methods is ongoing and has intensified in…

We focus on analysis and data-driven algorithms for rare events such as essential conformational transitions in biochemical reactions which are modeled by Langevin dynamics on manifolds. We first reinterpret the observed transition paths from the stochastic optimal control viewpoint, which realizes the transitions almost surely. Then based on collected high dimensional point clouds and nonlinear…

Energy-driven pattern formation is ubiquitous in nature; the character and dynamics of such patterns is selected as local minimizers and gradient flows, respectively, of non-convex, and often, non-local energies with multiple spatio-temporal scales. Analysis of such patterns sheds valuable insight upon their origins, and from the viewpoint of applications, is necessary for their control. In this talk, after introducing a…