Zeckendorf proved that every positive integer has a unique representation as
a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's
natural to ask how many summands are needed. Using a continued fraction
approach, Lekkerkerker proved that the average number of such summands needed
for integers in $[F_n, F_{n+1})$ is $n / (\varphi^2 + 1) + O(1)$, where
$\varphi = \frac{1+\sqrt{5}}2$ is the golden mean. Surprisingly, no one appears
to have investigated the distribution of the number of summands; our main
result is that this converges to a Gaussian as $n\to\infty$.

We provide effective bounds on the family of all elliptic curves and
one-parameter families of elliptic curves modulo p (for p prime tending to
infinity) obeying the Sato-Tate Law. We save a logarithm for all elliptic
curves, and obtain a power saving for the one-parameter families.

Recently Conrey, Farmer, and Zirnbauer developed the L-functions Ratios
conjecture, which gives a recipe that predicts a wealth of statistics, from
moments to spacings between adjacent zeros and values of L-functions. The
problem with this method is that several of its steps involve ignoring error
terms of size comparable to the main term; amazingly, the errors seem to cancel
and the resulting prediction is expected to be accurate up to square-root
cancellation.

In this article, we discuss the remarkable connection between two very
different fields, number theory and nuclear physics. We describe the essential
aspects of these fields, the quantities studied, and how insights in one have
been fruitfully applied in the other. The exciting branch of modern
mathematics, random matrix theory, provides the connection between the two
fields.