A 3–manifold is often described combinatorially, for example by a knot diagram, or by gluing a collection of polyhedra, or by attaching handles to a given manifold. Figure 1 shows a few examples. Just over a decade ago, Perelman posted the outlines of a proof of the Geometrization Theorem on the ArXiv [56, 57], and with his work and that of others it is now known that all 3–manifolds decompose ...

We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Zk of order k. We then demonstrate how such a chain map induces a “Zk-combinatorial Stokes theorem”, which in turn implies “Dold’s theorem” that there is no equivariant map from an n-connected to an n-dimensional free Zk-complex. Thus we build a combinatorial access road to probl...

Combinatorial design theory traces its origins to statistical theory of experimental design but also to recreational mathematics of the 19th century and to geometry. In the past forty years combinatorial design theory has developed into a vibrant branch of combinatorics with its own aims, methods and problems. It has found substantial applications in other branches of combinatorics, in graph th...

This paper is motivated by questions of complexity and com-binatorics of proofs in the sequent calculus. We shall pay particular attention to the role that symmetry plays in these questions. We want to have combinatorial models for proofs, models that will reeect complexity phenomena as well as accommodate other kinds of mathematical structures or relationships. For interpolation in proofs such...

Let Σ be a nonempty set of prime numbers. In the present paper, we continue the study, initiated in a previous paper by the second author, of the combinatorial anabelian geometry of semi-graphs of anabelioids of pro-Σ PSC-type, i.e., roughly speaking, semi-graphs of anabelioids associated to pointed stable curves. Our first main result is a partial generalization of one of the main combinatoria...

For a labeled tree on the vertex set [n] := {1, 2, . . . , n}, define the direction of each edge ij as i → j if i < j. The indegree sequence λ = 1122 . . . is then a partition of n−1. Let aλ be the number of trees on [n] with indegree sequence λ. In a recent paper (arXiv:0706.2049v2) Cotterill stumbled across the following two remarkable formulas aλ = (n− 1)! (n− k)!e1!(1!)1e2!(2!)2 . . . and

We study several canonical decision problems that arise from the most famous theorems from combinatorial geometry. We show that these are W[1]-hard (and NP-hard) if the dimension is part of the input by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision vers...

In spite of Matiyasevic's solution to Hubert's 10th problem some fifteen years ago it is still unknown whether there exists an algorithm to decide the solvability of diophantine equations within the field of rational numbers. In this note we show the equivalence of this problem with a conjecture of B. Grünbaum [6] on rational coordinatizability in combinatorial geometry. Such an algorithm exist...

We describe the application of a physics-inspired renormalization technique to combinatorial games. Although this approach is not rigorous, it allows one to calculate detailed, probabilistic properties of the geometry of the P-positions in a game. The resulting geometric insights provide explanations for a number of numerical and theoretical observations about various games that have appeared i...