i will discuss four families of random matrices. the first two are
classical: a gaussian measure on the space of $n\times n$ hermitian matrices
(\textquotedblleft gaussian unitary ensemble\textquotedblright) and a gaussian
measure on the space of all $n\times n$ complex matrices (\textquotedblleft
ginibre ensemble\textquotedblright). as $n\rightarrow\infty,$ the eigenvalues
of the gaussian unitary ensemble concentrate onto an interval with a
semicircular density, while the eigenvalues of the ginibre ensemble become
uniformly distributed in a disk in the complex plane.

now, the space of $n\times n$ hermitian matrices can be identified with the
lie algebra $u(n)$ of the unitary group $u(n),$ and the gaussian unitary
ensemble is the distribution of brownian motion in $u(n).$ similarly, the
space of all $n\times n$ matrices is the lie algebra $gl(n;\mathbb{c})$ of the
general linear group $gl(n;\mathbb{c})$ and the ginibre ensemble is the
distribution of brownian motion in $gl(n;\mathbb{c}).$ it is then natural to
consider also brownian motions in the groups $u(n)$ and $gl(n;\mathbb{c})$
themselves.

the eigenvalues for brownian motion in $u(n)$ have a known limiting
distribution in the unit circle. the eigenvalues for brownian motion in
$gl(n;\mathbb{c})$ have received little attention up to now. assuming that the
eigenvalues have a limiting distribution, recent results of mine with kemp
show that the limiting distribution is supported in a certain domain
$\sigma_{t}$ in the complex plane. the figure shows the domain for $t=3.85$,
along with a plot of the eigenvalues for $n=2,000.$ one notably feature of the
domains is that they change topology from simply connected to doubly connected
at $t=4.$ i will give background on all four families of random matrices,
describe our new results, and mention some ideas in the proof.

the theory of congruences of modular forms is a central topic in
contemporary number theory.
congruences between modular forms play a crucial role in understanding
links between geometry and arithmetic: cornerstone example of this is
the proof of serre's modularity conjecture by khare and wintenberger.
congruences of galois representations govern many kinds of
representations of the absolute galois group of number fields. even
though our understanding is improving, many aspects remain very
mysterious, some are theoretically approachable, many are not; and
amongst the latter, some allow numerical studies to reveal first insights.
in this talk i will introduce congruence graphs, which are graphs
encoding congruence relations between classical newforms. then i will
explain first how to construct analogous graphs for congruences of
galois representations, and then how to use these graphs to study
questions regarding hecke algebras and atkin-lehner operators.