## 1 Introduction
In gene expression modeling, it is often necessary to include delays to take into account the nontrivial amount of time required to transcribe and translate a gene [13; 14; 15; 16; 17; 18]. Delay-differential equations (DDEs), which are often used to model such systems, are instances of infinite-dimensional dynamical systems, best conceptualized as inducing a time evolution in a function space [6; 19]. A DDE model would nevertheless be formulated in terms of a finite number of variables defining a state space, typically the concentrations of various chemical species in a gene expression model. It is often observed that the behavior of a DDE model, after decay of transients, lies in a low-dimensional manifold in the state space. In some situations, such as a system with an Andronov-Hopf bifurcation, this is in fact guaranteed, at least in the neighborhood of the bifurcation [20]. A differentiable state-space manifold to which the long-term behavior of DDE solutions are confined is called an inertial manifold [21]. The condition for the existence of an inertial manifold in an infinite-dimensional dynamical system is essentially the same as the condition for the existence of a slow manifold in a finite-dimensional system [22]: there must be a separation of time scales between fast relaxation and the modes governing the long-term evolution [23]. It will be briefly noted that the literature is not entirely consistent with respect to the use of the term \"inertial manifold\". Many authors use \"invariant manifold\" when describing inertial manifolds of DDEs. For state-space manifolds of DDEs, \"invariance\" is a problematic concept because the time-evolution operator acts on a function space [6; 19]. The latter is therefore the natural setting for a concept of invariance. \"Inertial manifold\" is thus strongly to be preferred when describing manifolds in the state space of a DDE. On a related note, when there is no opportunity for confusion, the adjectives \"invariant\" or \"inertial\" may be dropped. Thus, in the rest of the paper, \"slow manifold\" may be read as shorthand for either \"slow invariant manifold\" or \"slow inertial manifold\", as appropriate, and similarly for center and unstable manifolds. Constructing an inertial manifold for a system of DDEs is not a trivial undertaking, largely because the phase space, which is a function space, and state space, typically \(\mathbb{R}^{n}\), are very different in nature. Much of the past work in this area has focused on center manifolds, where the existence of purely imaginary eigenvalues of the linearization can be exploited to bootstrap a series solution [24; 25], [PERSON] and coworkers [26] and [PERSON] [27] have described the automation of the relevant tasks using the symbolic algebra system Maple. [PERSON] has proven that, in the case of a DDE with an unstable manifold, numerical integration eventually reaches this manifold [28], which opens up methods for constructing unstable manifolds based on integration, for instance the numerical projection method of [PERSON] and coworkers [29]. [PERSON] and [PERSON] developed a method for computing one-dimensional unstable manifolds of delay-differential equations [30], which was subsequently added to the DDE bifurcation analysis package DDE-BIFTOOL [31]. [PERSON] and [PERSON] have described methods based on integration to compute the two-dimensional unstable inertial manifolds of DDEs [32]. [PERSON] has constructed inertial manifolds as singular perturbation series, using the delay as the singular perturbation parameter [33]. These are just a handful of techniques and applications. With some exceptions, they tend to be highly specialized to one type of manifold, and sometimes to manifolds of a specific dimension. This does not negate their usefulness, but it leaves open the question of whether a more general approach might be available. The problem of computing inertial manifolds of DDEs would be considerably facilitated if an approximating ODE could be constructed.