Methods of enumeration of spanning trees in a finite graph and relations to
various areas of mathematics and physics have been investigated for more
than 150 years. We will review the history and applications. Then we will
give new formulas for the asymptotics of the number of spanning trees of a
graph. A special case answers a question of McKay (1983) for regular
graphs. The general answer involves a quantity for infinite graphs that we
call ``tree entropy", which we show is a logarithm of a normalized
[Fuglede-Kadison] determinant of the graph Laplacian for infinite graphs.
Proofs involve new traces and the theory of random walks.

Holomorphic functions on a domain are the solutions to a set of homogeneous partial differential
equations called the Cauchy-Riemann equation, and CR functions are the solutions to the analogous
equations on the boundary. Many problems in complex analysis can be reduced to finding appropriate
solutions to the inhomogeneous versions of these equations. These solutions have been successfully
constructed when the geometry of the domain is sufficiently simple. I hope to show how these constructions
follow a pattern based on the singular integral operators of Calder\'on and Zygmund. I then plan to
discuss some more recent examples where this well-understood paradigm breaks down.

I will consider the random walk on Z^d driven by a field of random i.i.d.
conductances.
The law of the conductances is bounded from above; no restriction is posed on the
lower tail (at zero) except that the bonds with positive conductances percolate.
The presence of very weak bonds allows the random walk in a finite box mix pretty
much arbitrarily slowly. However, when we focus attention on the return probability
to the starting point -- i.e., the heat-kernel -- it turns out that in dimensions
d=2,3 the
decay is as for the simple random walk. On the other hand, in d>4 the heat- kernel
at time 2n may decay as slowly as o(1/n^2) and in d=4 as slowly as O(n^{-2}log n).
These upper bounds can be matched arbitrarily closely by lower bounds in particular
examples. Despite this, the random walk scales to Brownian motion under the usual
diffusive scaling of space and time. Based on joint works with N. Berger, C. Hoffman,
G. Kozma and T. Prescott.