The Combinatorics of Iterated Loop Spaces

It is well known since Stasheff's work that 1-fold loop spaces can be described in terms of the existence of higher homotopies for associativity (coherence conditions) or equivalently as algebras of contractible non-symmetric operads. The combinatorics of these higher homotopies is well understood and is extremely useful. For $n \ge 2$ the theory of symmetric operads encapsulated the corresponding higher homotopies, yet hid the combinatorics and it has remain a mystery for almost 40 years. However, the recent developments in many fields ranging from algebraic topology and algebraic geometry to mathematical physics and category theory show that this combinatorics in higher dimensions will be even more important than the one dimensional case. In this paper we are going to show that there exists a conceptual way to make these combinatorics explicit using the so called higher nonsymmetric $n$-operads.Comment: 23 page

Integrable Combinatorics

We review various combinatorial problems with underlying classical or quantum integrable structures. (Plenary talk given at the International Congress of Mathematical Physics, Aalborg, Denmark, August 10, 2012.)Comment: 21 pages, 16 figures, proceedings of ICMP1

Rational Combinatorics

We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.Comment: Adv. in Appl. Math. (2007), doi:10.1016/j.aam.2006.12.00

Alexander Duality and Rational Associahedra

A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong, Williams, and the author initiated the systematic study of {\em rational Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics (which is in turn a generalization of classical Catalan combinatorics). The latter paper gave two possible models for a rational analog of the associahedron which attach simplicial complexes to any pair of coprime positive integers a < b. These complexes coincide up to the Fuss-Catalan level of generality, but in general one may be a strict subcomplex of the other. Verifying a conjecture of Armstrong, Williams, and the author, we prove that these complexes agree up to homotopy and, in fact, that one complex collapses onto the other. This reconciles the two competing models for rational associahedra. As a corollary, we get that the involution (a < b) \longleftrightarrow (b-a < b) on pairs of coprime positive integers manifests itself topologically as Alexander duality of rational associahedra. This collapsing and Alexander duality are new features of rational Catalan combinatorics which are invisible at the Fuss-Catalan level of generality.Comment: 23 page

Hopf algebras and the combinatorics of connected graphs in quantum field theory

In this talk, we are concerned with the formulation and understanding of the combinatorics of time-ordered n-point functions in terms of the Hopf algebra of field operators. Mathematically, this problem can be formulated as one in combinatorics or graph theory. It consists in finding a recursive algorithm that generates all connected graphs in their Hopf algebraic representation. This representation can be used directly and efficiently in evaluating Feynman graphs as contributions to the n-point functions.Comment: 10 pages, 2 figures, LaTeX + AMS + eepic; to appear in the proceedings of the Conference on Combinatorics and Physics, MPIM Bonn, March 19-23, 200