Alexander Glatzman:
Title: Phase diagram of the loop O(n) model
Phase transitions are natural phenomena in which a small change in an external
parameter, like temperature or pressure, causes a dramatic change in the qualitative
structure of the object. To study this, many scientists (such as Nobel laureates
Pauling and Flory) proposed the abstract framework of lattice models. One of the
most famous lattice models is Bermoulli percolation: given an infinite planar graph,
each vertex is open with some probability p independently of the others. We show
that, for a general class of graphs, the percolation process at p=1/2 exhibits no
(unique) infinite connected component. We use this, together with a novel graphical
representation, to show that, in a large two-dimensional region of parameters when n
is between 1 and 2, the loop O(n) model exhibits macroscopic loops. This behaviour
is reminiscent of scale invariance and, thus, is in agreement with a conformal
invariance conjecture. Previously, existence of macroscopic loops was known in
several sparse regions of parameters, and these are recent works. The difficulty
comes from the lack of monotonicity with respect to parameters: each point of the
phase diagram should be treated separately. This is joint work with Matan Harel and
Nathan Zelesko (Northeastern).

Yvan Velenik:
Title: Interfacial phenomena in the planar Ising model: an overview of
nonperturbative results
Abstract: I'll review a selection of results about the behavior of
interfaces in the planar Ising model obtained during the last three decades. These will include
scaling limits of the interface, the phenomena of wetting and critical
prewetting, equilibrium crystal shapes and the effect of a gravitational field. The
focus will be on nonperturbative results valid in the whole phase coexistence regime.