We briefly describe some ideas and techniques that lead to solutions to certain problems in number theory, such as the bounded gaps between prime numbers, and others. This talk will be made understandable to general math audiences; technical details will be avoided.

Over the past century an effort to understand dimension and structure of the harmonic measure spanned many spectacular developments in Analysis and in Geometric Measure Theory. Uniform rectifiability emerged as a natural geometric condition, necessary and sufficient for classical estimates in harmonic analysis, boundedness of the harmonic Riesz transform in L^2, and, in the presence of some background topological assumptions, for suitable scale invariant estimates on harmonic functions. While many of geometric and analytic problems remain relevant in sets of higher co-dimension (e.g., a curve in $\RR^3$), the concept of the harmonic measure is notoriously missing. In this talk, we introduce a new notion of a "harmonic" measure, associated to a linear PDE, which serves the higher co-dimensional sets. We discuss its basic properties and give large strokes of the argument to prove that our measure is absolutely continuous with respect to the Hausdorff measure on Lipschitz graphs with small Lipschitz constant.

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.

A central research challenge for the
mathematical sciences in the $21^{st}$ century is
the development of principled methodologies for the
seamless integration of (often vast) data sets
with (often sophisticated) mathematical models.
Such data sets are becoming routinely available
in almost all areas of engineering, science and technology,
whilst mathematical models describing phenomena of
interest are often built on decades, or even centuries, of
human knowledge creation. Ignoring either the data or the models
is clearly unwise and so the issue of combining them
is of paramount importance. In this talk we will give
a historical perspective on the subject, highlight some of
the current research directions that it leads to, and
describe some of the underlying mathematical frameworks
begin deployed and developed. The ideas will be illustrated
by problems arising in the geophysical, biomedical and
social sciences.

The self-avoiding walk (SAW) is a model for polymers that assigns equal probability to all paths that do not return to places they have already been. The lattice version of this problem, while elementary to define, has proved to be notoriously difficult and is still open. It is initially more challenging to construct a continuous limit of the lattice model which is a random fractal. However, in two dimensions this has been done and the continuous model (Schram-Loewner evolution) can be analyzed rigorously and  used to understand the nonrigorous predictions about SAWs.  I will survey some results in this area and then discuss some recent work on this ``continuous SAW''.