14 research outputs found
Linear lambda terms as invariants of rooted trivalent maps
The main aim of the article is to give a simple and conceptual account for
the correspondence (originally described by Bodini, Gardy, and Jacquot) between
-equivalence classes of closed linear lambda terms and isomorphism
classes of rooted trivalent maps on compact oriented surfaces without boundary,
as an instance of a more general correspondence between linear lambda terms
with a context of free variables and rooted trivalent maps with a boundary of
free edges. We begin by recalling a familiar diagrammatic representation for
linear lambda terms, while at the same time explaining how such diagrams may be
read formally as a notation for endomorphisms of a reflexive object in a
symmetric monoidal closed (bi)category. From there, the "easy" direction of the
correspondence is a simple forgetful operation which erases annotations on the
diagram of a linear lambda term to produce a rooted trivalent map. The other
direction views linear lambda terms as complete invariants of their underlying
rooted trivalent maps, reconstructing the missing information through a
Tutte-style topological recurrence on maps with free edges. As an application
in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent
maps as linear lambda terms containing no closed proper subterms, and conclude
by giving a natural reformulation of the Four Color Theorem as a statement
about typing in lambda calculus.Comment: accepted author manuscript, posted six months after publicatio
A Sequent Calculus for a Semi-Associative Law
We introduce a sequent calculus with a simple restriction of Lambek\u27s product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, tree rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. One combinatorial application of this coherence theorem is a new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice Y_n. Elsewhere, we have also used the sequent calculus and the coherence theorem to build a surprising bijection between intervals of the Tamari order and a natural fragment of lambda calculus, consisting of the beta-normal planar lambda terms with no closed proper subterms
The Internal Operads of Combinatory Algebras
We argue that operads provide a general framework for dealing with
polynomials and combinatory completeness of combinatory algebras, including the
classical -algebras, linear -algebras, planar
-algebras as well as the braided -algebras. We show that every extensional combinatory algebra gives rise to
a canonical closed operad, which we shall call the internal operad of the
combinatory algebra. The internal operad construction gives a left adjoint to
the forgetful functor from closed operads to extensional combinatory algebras.
As a by-product, we derive extensionality axioms for the classes of combinatory
algebras mentioned above
Tropical M\"obius strips and ruled surfaces
We consider the enumeration of tropical curves in M\"obius strips for two
different lattice structures and relate them to the enumeration of curves in
two rational ruled surfaces over a complex elliptic curve. Using this
correspondence, we prove regularity results such as the piecewise
quasi-polynomiality of relative invariants and the quasi-modularity of their
generating series.Comment: 39 pages, comments welcom
Analyticity results for the cumulants in a random matrix model
The generating function of the cumulants in random matrix models, as well as
the cumulants themselves, can be expanded as asymptotic (divergent) series
indexed by maps. While at fixed genus the sums over maps converge, the sums
over genera do not. In this paper we obtain alternative expansions both for the
generating function and for the cumulants that cure this problem. We provide
explicit and convergent expansions for the cumulants, for the remainders of
their perturbative expansion (in the size of the maps) and for the remainders
of their topological expansion (in the genus of the maps). We show that any
cumulant is an analytic function inside a cardioid domain in the complex plane
and we prove that any cumulant is Borel summable at the origin
Recommended from our members
Teichmüller Space (Classical and Quantum)
This is a short report on the conference “Teichmüller Space (Classical and Quantum) ” held in Oberwolfach from May 28th to June 3rd, 2006