Pattern-equivariant cohomology with integer coefficients

We relate Kellendonk and Putnam’s pattern-equivariant (PE) cohomology to the inverse-limit structure of a tiling space. This gives a version of PE cohomology with integer coefficients, or with values in any Abelian group. It also provides an easy proof of Kellendonk and Putnam’s original theorem relating PE cohomology to the Čech cohomology of the tiling space. The inverse-limit structure also allows for the construction of a new non-Abelian invariant, the PE representation variety.

ON THE LIMIT CYCLE STRUCTURE OF THRESHOLD BOOLEAN NETWORKS OVER COMPLETE GRAPHS

In previous work, the limit structure of positive and negative finite threshold boolean networks without inputs (TBNs) over the complete digraph Kn was analyzed and an algorithm was presented for computing this structure in polynomial time. Those results are generalized in this paper to cover the case of arbitrary TBNs over Kn. Although the limit structure is now more complicated, containing, not only fixed-points and cycles of length 2, but possibly also cycles of arbitrary length, a simple algorithm is still available for its determination in polynomial time. Finally, the algorithm is generalized to cover the case of symmetric finite boolean networks over Kn.

MODEL THEORY OF DIFFERENCE FIELDS, II: PERIODIC IDEALS AND THE TRICHOTOMY IN ALL CHARACTERISTICS

We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)