##### Robert Adler:

#### Title: Phase Transitions and Topology

Abstract: Phase transitions are phenomena occurring in large random systems, as water cools and turns to ice, is heated and turns to vapor, or as iron loses or gains magnetism. Although not usually recognized as such, there is topology in these transitions, generally studied only at the topologically simple level of connectivity.

Recently, there has been growing interest in far more sophisticated phase transitions, studying large random topological structures undergoing transitions in their homological structure. The results in this area, to a large extent initially motivated by questions from topological data analysis, are building the foundations of a new area of research, that one might call Random Topology.

The aim of the lecture will be to describe these developments, mainly via a number of examples, ranging from the asymptotics of the Betti numbers of large simplicial complexes to a version of a random Morse theory. While the details of the examples are not simple, my aim will be to present everything in an audience friendly, widely accessible fashion.

##### Idan Perl:

#### Title: Harmonic functions and polynomials on discrete groups

Abstract: Alexopoulos proved that on a finitely generated nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial. We give a simpler, more direct proof for this result using a more elementary definition for polynomials. We also show that the Laplacian operator is onto, and calculate the precise dimension of the space of harmonic functions of given polynomial growth, refining previous results.