The goal is to understand the connections between the geometric structure of sparse (random) matrices and graphs and their spectra. Specifically, I would like to deepen the connections between three distinct research areas, each having their own set of difficult problems and open questions.The first is the study of random matrices with independent entries, started in the statistics community in the 1920s, and further developed by Wigner, Dyson and others in the 1950s and 60s; many of the results have been extended to more sparse matrices recently. A related question, not yet accessible through the random matrix machinery, is what does the top eigenvalue of a random regular graph of bounded degree look like?The second area of group theory related to the so-called Atiyah question/conjecture. What can the atoms in the spectrum in a vertex-transitive graph look like? How does this depend on the local structure of the graph and its group of automorphisms?The third is the study of random Schroedinger operators, originated with Anderson in the 1980s. Given a vertex-transitive graph, such as Z^d or a regular tree, how does the spectrum change when random perturbations are added? Most interesting and difficult is the case when these perturbations are discrete, e.g. adding a loop at each vertex independently at random. Most questions about these models are still open, including localization in higher dimensions and local eigenvalue statistics in any dimension.The interplay between these areas has already been fruitful, and gave rise to new ideas and concepts. Specifically, techniques from random Schroedinger operators have been useful in understanding spectra of lamplighter groups, the Novikov-Shubin invariant and the limiting spectra of random Toeplitz matrices. The limiting operator formalism used in understanding the local eigenvalue statistics of random matrices also helped with critical 1-dimensional random Schroedinger operators.