After presenting a short survey of Markov loops, we will introduce related random fields and study their properties.

Geometric Quantization is a program of assigning to
classical mechanical systems (symplectic manifolds and the associated
Poisson algebras of C-infinity functions) their quantizations ---
algebras of operators on Hilbert spaces.  Geometric Quantization has
had many applications in Mathematics and Physics.   Nevertheless the
main proposition at the heart of the theory, invariance of
polarization, though verified in many examples, is still not proved in
any generality.  This causes numerous conceptual difficulties:  For
example, it makes it very difficult to understand the functoriality of
theory.

Nevertheless, during the past 20 years, powerful topological and
geometric techniques have clarified at least some of the features of
the program.

In 1995 Kontsevich showed that formal deformation quantization can be
extended to Poisson manifolds.  This naturally raises the question as
to what one can say about Geometric Quantization in this context.  In
recent work with Victor Guillemin and Eva Miranda, we explored this
question in the context of Poisson manifolds which are "not too far"
from being symplectic - the so called b-symplectic or b-Poisson
manifolds - in the presence of an Abelian symmetry group.

In this talk we review Geometric Quantization in various contexts, and
discuss these developments, which end with a surprise.

We show how the Galois representation of an elliptic curve over a number field can be used to determine the structure of the (topological) group of adelic points  of that elliptic curve.

As a consequence, we find that for "almost all" elliptic curves over a number field K,  the adelic point group is a universal topological group depending only on the degree  of K. Still, we can construct infinitely many pairwise non-isomorphic elliptic curves  over K that have an adelic point group not isomorphic to this universal group.

This generalizes work of my student Athanasios Angelakis (PhD Leiden, 2015).

 In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) > 1.363.... In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simpler approximate models to gain insights.

I will begin by Whitham's shallow water equation and the modulational instability index for small amplitude and periodic traveling waves, the effects of surface tension and vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is partly based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), and Ashish Pandey (Illinois).

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.