If a group action is given, then we can naturally define a graph called a Schreier graph. We define self-similar groups and consider the spectra of Schreier graphs associated with them. This talk is based on the material that I studied recently for a conference talk, and is accessible to every math grad student.

http://www.math.uci.edu/~mgsc/talk.php?year=2016&number=5

We consider a drift-diffusion process on a smooth potential landscape

with small noise. We give a new proof of the Eyring-Kramers formula

which asymptotically characterizes the spectral gap of the generator of

the diffusion. The proof is based on a refinement of the two-scale

approach introduced by Grunewald, Otto, Villani, and Westdickenberg and

of the mean-difference estimate introduced by Chafai and Malrieu. The

new proof exploits the idea that the process has two natural

time-scales: a fast time-scale resulting from the fast convergence to a

metastable state, and a slow time-scale resulting from exponentially

long waiting times of jumps between metastable states. A nice feature

of the argument is that it can be used to deduce an asymptotic formula

for the log-Sobolev constant, which was previously unknown.

After defining the spectral theory of orthogonal polynomials on the unit circle (OPUC) and real line (OPRL), I'll describe Verblunsky's version of Szego's theorem as a sum rule for OPUC and the Killip--Simon sum rule for OPRL and their spectral consequences. Next I'll explain the original proof of Killip--Simon using representation theorems for meromorphic Herglotz functions. Finally I'll focus on recent work of Gambo, Nagel and Rouault who obtain the sum rules using large deviations for random matrices.

We consider the finite element solution of the vector Laplace equation on a
domain in two dimensions.  For some choices of boundary conditions, there is a
theory, making use of finite element differential complexes and bounded
cochain projections, that shows that a mixed finite element method using
appropriate choices of finite element spaces, and in which the rotation
of the solution is introduced as a second unknown, leads to a stable,
optimally convergent discretization. However, the theory that leads to these
conclusions does not apply to the case of Dirichlet boundary conditions, in
which both components of the solution vanish on the boundary.  We present
computational examples that demonstrate that such a mixed finite element method
does not perform optimally in this case, and an analysis which theoretically
confirms the suboptimal convergence that does occur and indicates the source
of the problem.  These results also have implications for the solution of the
biharmonic equation and of the Stokes equations using a mixed formulation
involving the vorticity.