In the first half of the talk, I will survey results and open problems on transience of self-interacting martingales. In particular, I will describe joint works with S. Popov, P. Sousi, R. Eldan and F. Nazarov on the tradeoff between the ambient dimension and the number of different step distributions needed to obtain a recurrent process. In the second, unrelated, half of the talk, I will present joint work with Tom Hutchcroft, showing that the component structure of the uniform spanning forest in $\mathbb{Z}^d$ changes every dimension for $d > 8$. This sharpens an earlier result of Benjamini, Kesten, Schramm and the speaker (Annals Math 2004), where we established a phase transition every four dimensions. The proofs are based on a the connection to loop-erased random walks.[-]

The Hopf algebra of Lie group integrators has been introduced by H. Munthe-Kaas and W. Wright as a tool to handle Runge-Kutta numerical methods on homogeneous spaces. It is spanned by planar rooted forests, possibly decorated. We will describe a canonical surjective Hopf algebra morphism onto the shuffle Hopf algebra which deserves to be called planar arborification. The space of primitive elements is a free post-Lie algebra, which in turn will permit us to describe the corresponding co-arborification process.
Joint work with Charles Curry (NTNU Trondheim), Kurusch Ebrahimi-Fard (NTNU) and Hans Z. Munthe-Kaas (Univ. Bergen).
The two triangles appearing at 24'04" and 25'19'' respectively should be understood as a #.[-]

A phylogenetic tree that has been reconstructed from a given gene can describe a different evolutionary history from its underlying species tree. The reasons for this include: error in inferring the gene tree, incomplete lineage sorting, lateral gene transfer, and the absence of the gene in certain species. In this talk, I discuss probabilistic models and mathematical results that help address basic questions concerning the consistency and efficiency of different methods for inferring a species phylogeny from gene trees.[-]

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé–Galton–Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé–Galton–Watson trees.
Section 2 defines Bienaymé–Galton–Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.[-]

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé–Galton–Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé–Galton–Watson trees.
Section 2 defines Bienaymé–Galton–Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.[-]

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé–Galton–Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé–Galton–Watson trees.
Section 2 defines Bienaymé–Galton–Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.[-]