In this talk, I'd like to discuss the intertwining importance and connections of three principles of data science in the title in data-driven decisions. The ultimate importance of prediction lies in the fact that future holds the unique and possibly the only purpose of all human activities, in business, education, research, and government alike.
Making prediction as its central task and embracing computation as its core, machine learning has enabled wide-ranging data-driven successes. Prediction is a useful way to check with reality. Good prediction implicitly assumes stability between past and future. Stability (relative to data and model perturbations) is also a minimum requirement for interpretability and reproducibility of data driven results. It is closely related to uncertainty assessment. Obviously, both prediction and stability principles can not be employed without feasible computational algorithms, hence the importance of computability. The three principles will be demonstrated through analytical connections, and in the context of two on-going neuroscience projects, for which "data wisdom" is also indispensable. Specifically, the first project interprets a predictive model used for reconstruction
of movies from fMRI brain signals; the second project employs deep learning networks (CNNs) to understand pattern selectivities of neurons in the difficult visual cortex V4.

The origins of Carleman estimates lie with the pioneering 1939 work by the Swedish mathematician T. Carleman, concerned with the unique continuation property for solutions for linear elliptic partial differential equations with smooth coefficients in dimension two. The fundamental new idea introduced by Carleman, which consists of establishing a priori energy estimates involving an exponential weight, has permeated essentially all the subsequent work in the subject. More recently, Carleman estimates have found numerous striking applications beyond the original domain of unique continuation, from control theory to spectral theory to the analysis of inverse problems. The purpose of this talk is to provide a broad introduction to the subject and to attempt to illustrate some of its inner workings.

In manifolds with special holonomy, it is interesting to
study calibrated submanifolds, which are volume minimizer in their
homology classes. We study the calibrated submanifolds and mean
curvature flow in several famous local models of manifolds with
special holonomy. These model spaces are all total spaces of some
vector bundles, and the zero section is a calibrated submanifold. We
show that the zero section is the only compact minimal submanifold,
and is dynamically stable under the mean curvature flow. This is a
joint work with Mu-Tao Wang.

The additive group of integers is a well-studied example of a stable group, whose definable sets can be easily and explicitly described. However, until recently, very little has been known about stable expansions of this group. In this talk, we examine the relationship between model-theoretic stability of expansions of the form (Z,+,0,A), where A is a subset of the natural numbers, and the number theoretic behavior of A with respect to sumsets, asymptotic density, and arithmetic progressions.