Let x and y be points chosen uniformly at random
from the four-dimensional discrete torus with side length n.
We show that the length of the loop-erased random walk from
x to y is of order n^2 (log n)^{1/6}, resolving a conjecture
of Benjamini and Kozma. We also show that the scaling limit
of the uniform spanning tree on the four-dimensional discrete
torus is the Brownian continuum random tree of Aldous. Our
proofs use the techniques developed by Peres and Revelle,
who studied the scaling limits of the uniform spanning tree
on a large class of finite graphs that includes the
d-dimensional discrete torus for d >= 5, in combination with
results of Lawler concerning intersections of
four-dimensional random walks.

There is a rich physical literature on polymer dynamics which presents a number of fascinating challenges for mathematicians. We model thermal fluctuation of a polymer in solvent as a curve or loop obeying a stochastic partial differential equation (SPDE). The simplest instance is the so-called Rouse model which is an infinite dimensional Ornstein-Uhlenbeck process satisfying a linear SPDE. We'll review the Rouse model and then describe recent results (a) of Seung Lee on an SPDE for the Rouse model in a half-space with reflecting boundary conditions; and (b) of Scott McKinley on an SPDE model of the hydrodynamic interaction.