We distinguish a class of random point processes which we call
Giambelli compatible point processes. Our definition was partly
inspired by determinantal identities for averages of products and
ratios of characteristic polynomials for random matrices.
It is closely related to the classical Giambelli formula for Schur symmetric functions.

We show that orthogonal polynomial ensembles, z-measures on
partitions, and spectral measures of characters of generalized
regular representations of the infinite symmetric group generate
Giambelli compatible point processes. In particular, we prove
determinantal identities for averages of analogs of characteristic
polynomials for partitions.

Our approach provides a direct derivation of determinantal
formulas for correlation functions

$p$-Harmonic morphisms are maps between
Riemannain manifolds that preserve solutions of $p$-
Laplace's equation. They are characterized as horizontally
weakly conformal $p$-harmonic maps so, locally, they are
solutions of an over-determined system of PDEs. I will talk
about some background of $p$-harmonic morphisms, some
calssifications and constructions of such maps, and some
applications related to minimal surfaces and biharmnonic
maps.

A gradient Gibbs measure is the projection to the gradient variables $\eta_b=\phi_y-\phi_x$
of the Gibbs measure of the form
$$
P(\textd\phi)=Z^{-1}\exp\Bigl\{-\beta\sum_{\langle x,y\rangle}V(\phi_y-\phi_x)\Bigr\}\textd\phi,
$$
where $V$ is a potential, $\beta$ is the inverse temperature and $\textd\phi$ is the product
Lebesgue measure. The simplest example is the (lattice) Gaussian free field
$V(\eta)=\frac12\kappa\eta^2$. A well known result of Funaki and Spohn (and Sheffield)
asserts that, for any uniformly-convex $V$, the possible infinite-volume measures of this type are
characterized by the \emph{tilt}, which is a vector $u\in\R^d$ such that
$E(\eta_b)=u\cdot b$ for any (oriented) edge $b$. I will discuss a simple example
for which this result fails once $V$ is sufficiently non-convex thus showing that
the conditions of Funaki-Spohn's theory are generally optimal. The underlying
mechanism is an order-disorder phase transition known, e.g., from the context
of the $q$-state Potts model with sufficiently large $q$. Based on joint work
with Roman Koteck\'y.

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.