We survey currently available algorithms for computing with matrix groups over infinite domains. We discuss open problems in the area, and avenues for further development.

We endow the collection of ω-bifinite domains with the structure of a probability space, and we will show that in this space the collection of all universal domains has measure 1. For this, we present a probabilistic way to extend a finite partial order by one element. Applying this procedure iteratively, we obtain an infinite partial order. We show that, with probability 1, the cpo-completion ...

We give a complete characterization of the class of upward monotone generalized quantifiers Q1 and Q2 over countable domains that satisfy the scheme Q1x Q2y ! Q2y Q1x . This generalizes the characterization of such quantifiers over finite domains, according to which the scheme holds iff Q1 is 9 or Q2 is 8 (excluding trivial cases). Our result shows that in infinite domains, there are more gener...

We study semigroup C*-algebras of ax + b-semigroups over integral domains. The goal is to generalize several results about C*-algebras of ax + bsemigroups over rings of algebraic integers. We prove results concerning K-theory and structural properties like the ideal structure or pure infiniteness. Our methods allow us to treat ax+b-semigroups over a large class of integral domains containing al...

We give a complete characterization of the class of upward monotone generalized quantifiers Q1 and Q2 over countable domains that satisfy the scheme Q1x Q2y ! Q2y Q1x . This generalizes the characterization of such quantifiers over finite domains, according to which the scheme holds iff either Q1 is 9 or Q2 is 8 (excluding trivial cases). Our result shows that in infinite domains, there are mor...

Classes of functions of several variables on arbitrary non-empty domains that are closed under permutation of variables and addition of dummy variables are characterized in terms of generalized constraints, and hereby Hellerstein’s Galois theory of functions and generalized constraints is extended to infinite domains. Furthermore, classes of operations on arbitrary non-empty domains that are cl...

Various issues with regard to chaos and recurrence in infinite dimensions are discussed. The doctrine we are trying to derive is that Sobolev spaces over bounded spatial domains do host chaos and recurrence, while Sobolev spaces over unbounded spatial domains are lack of chaos and recurrence. Local Sobolev spaces over unbounded spatial domains can host chaos and are natural phase spaces e.g. fo...

We present a method that combines offline trajectory optimization and online Model Predictive Control (MPC), generating robust controllers for complex periodic behavior in domains with unilateral constraints (e.g., contact with the environment). MPC offers robust and adaptive control even in high-dimensional domains; however, the online optimization gets stuck in local minima when the domains h...