Take a polynomial of degree n with coefficients +/ - 1 taken randomly, independently. What is the probability that it is irreducible over the rationals? This question is still open, but a lot of progress happened around it in recent years. We will survey some of this progress, including joint work with Lior Bary-Soroker, Dimitris Koukoulopoulos and David Hokken.

We consider a model for a growing subset of a euclidean lattice (an "aggregate") where at each step one choose a random point from the existing aggregate, starts a random walk from that point, and adds the point of exit to the aggregate. We show that the limiting shape is a ball. Joint work with Itai Benjamini, Hugo Duminil-Copin and Cyril Lucas.

Lace expansion is a method for solving self-interacting models in high dimension. In the core of the technique is the problem of solving convolution equations. Initially highly dependent upon Fourier transform, in the last decade it has evolved to the degree that it can be applied in some non-commutative settings where Fourier transform is not available.
Joint work with Remco van der Hofstad and Erwin Bolthausen.