On injective endomorphisms of symbolic schemes

Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties [10], we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular automata in [7]. We extend known results to this more general setting. We also establish several new ones regarding the closed image property, surjunctivity, reversibility, and invertibility for cellular automata over algebraic varieties with coefficients in an algebraically closed field. As a byproduct, we obtain a negative answer to a question raised in [7] on the existence of a bijective complex affine algebraic cellular automaton $\tau \colon A^{\mathbb Z} \to A^{\mathbb Z}$ whose inverse is not algebraic

Solutions of Word Equations over Partially Commutative Structures

We give NSPACE(n log n) algorithms solving the following decision problems. Satisfiability: Is the given equation over a free partially commutative monoid with involution (resp. a free partially commutative group) solvable? Finiteness: Are there only finitely many solutions of such an equation? PSPACE algorithms with worse complexities for the first problem are known, but so far, a PSPACE algorithm for the second problem was out of reach. Our results are much stronger: Given such an equation, its solutions form an EDT0L language effectively representable in NSPACE(n log n). In particular, we give an effective description of the set of all solutions for equations with constraints in free partially commutative monoids and groups

The mathematical research of William Parry FRS

In this article we survey the mathematical research of the late William (Bill) Parry, FRS