Related papers: From quantum electrodynamics to posets of planar b…
We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables…
We equip the graded polynomial algebra generated by nonplanar rooted binary trees with a Hopf algebra structure by defining a coproduct which disallows cutting both children of any given vertex, refining Connes-Kreimer's notion of…
In the first purpose, we concentrate on the theory of quantum integrable systems underlying the Connes-Kreimer approach. We introduce a new family of Hamiltonian systems depended on the perturbative renormalization process in renormalizable…
We initiate the study of affine actions of groups on $\Lambda$-trees for a general ordered abelian group $\Lambda$; these are actions by dilations rather than isometries. This gives a common generalisation of isometric action on a…
We introduce a correspondence between phylogenetic trees and Brauer diagrams, inspired by links between binary trees and matchings described by Diaconis and Holmes (1998). This correspondence gives rise to a range of semigroup structures on…
We extend the Euler's totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Mobius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero then there is a…
In this chapter, we trace the path from the Tamari lattice, via lattice congruences of the weak order, to the definition of Cambrian lattices in the context of finite Coxeter groups, and onward to the construction of Cambrian fans. We then…
This paper gives a review of Connes-Kreimer formulation of perturbative renormalization in Quantum Field Theory. We begin with the derivation of the Feynman calculus, the Hopf algebra structure on Feynman diagrams and we show the natural…
The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite…
We give a framework for growth models on posets which simultaneously generalizes the Classical Sequential Growth models for posets from causal set theory and the tree growth models of natural growth and simple tree classes, the latter of…
We give a novel combinatorial interpretation to the perturbative series solutions for a class of Dyson-Schwinger equations. We show how binary tubings of rooted trees with labels from an alphabet on the tubes, and where the labels satisfy…
Tree-level Feynman diagrams in a cubic scalar theory can be given a metric such that each edge has a length. The space of metric trees is made out of orthants joined where a tree degenerates. Here we restrict to planar trees since each…
Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems.…
This thesis provides an introduction to the various category theory ideas employed in topological quantum field theory. These theories are viewed as symmetric monoidal functors from topological cobordism categories into the category of…
We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets (which we rephrase as…
This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection $\varphi$ between binary trees and plane trees answers an open problem posed by Bai and Chen.…
A formal expansion for the Green's functions of an interacting quantum field theory in a parameter that somehow encodes its "distance" from the corresponding non-interacting one was introduced more than thirty years ago, and has been…
Non-Abelian physics, originating from noncommutative sequences of operations, unveils novel topological degrees of freedom for advancing band theory and quantum computation. In photonics, significant efforts have been devoted to developing…
The purpose of this paper is twofold. First we answer to a question asked by Steingrimsson and Williams about certain permutation tableaux: we construct a bijection between binary trees and the so-called Catalan tableaux. These tableaux are…
Tonks' projection from the permutohedron to the associahedron and the Loday--Ronco map both send permutations to planar binary trees. We give a syntactic account of these maps in the equational calculus of the free non-symmetric, non-unital…