Recently, [1] showed that several machine learning algorithms, such as Lasso, Support Vector Machines, and regularized logistic regression, and many others can be represented exactly as distributionally robust optimization (DRO) problems.

The semiclassical Schrodinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems.

With the development of financial markets, more and more complicated and advanced derivatives are designed to fit the market’s interests. There exotics products are often linked with multi-assets and under untraditional risk, e.g. correlation, credit-default-risk, etc.

As shown by Aronson in the 1960's, the fundamental solution of parabolic partial differential operators in divergence form can be bounded from above and below by the Gaussian heat kernel, i.e., the fundamental solution of the classical heat equation.

We will investigate various notions of hyperbolicity in Riemannian and complex geometries, with a focus on some conjectures (due to Hopf, Singer, etc.) on Euler characteristic of compact aspherical manifolds.