123,492 research outputs found
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 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
-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
Recent developments in algebraic combinatorics
A survey of three recent developments in algebraic combinatorics: (1) the
Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and
(3) toric h-vectors and intersection cohomology. This paper is a continuation
of "Recent progress in algebraic combinatorics" (math.CO/0010218), which dealt
with three other topics.Comment: 30 page
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
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