The relation between Lie symmetries of differential equations and the groups appearing in discretized versions of corresponding dynamical systems (such as Petri nets, Boolean networks, and automata representations) have never been investigated mathematically before in a systematic manner. Such study holds the potential for completely novel relationships to be discovered that allow methods to be transferred between these domains, and also to create an integrative mathematical framework to successfully address both continuous and discrete aspects of dynamics, something the has largely been lacking. In particular, the investigation in BIOMICS of simple Lie algebras interpreted in finite fields and the simple non-abelian groups (SNAGs) in the finite automata of corresponding discrete dynamical systems in relation to the Chevalley correspondence is a fundamental and completely novel direction for dynamical systems theory.
Recently we have sharpened the understanding of symmetries in this context [Attila Egri-Nagy, Chrystopher L. Nehaniv, "Symmetries of Automata" 2011]: (1) any group of (state-)automorphisms of a reachable finite automaton must in fact be realizable by a subgroup of the semigroup of that automaton, and (2) this leads to Noether-type Krohn-Rhodes decomposition, using these symmetries and resultant invariants. In particular, all computations needed for the new coordinate system are present in (as divisors of) the automaton. This strongly suggests the Lie symmetries (and also groups in Lie groupoids) of the corresponding continuous system are indeed likely to occur in some form in the discretized version. BIOMICS will study examples of just such correspondences in order to shed new light on the nature of relation between discrete and continuous dynamical models.
Moreover, BIOMICS will cast these results in a rigorous category-theoretic framework formalizing the particular objects and morphisms in the discrete and continuous categories in such a way as to create adjunct functorial relationships between them to allow the transfer of structure and knowledge between the discrete and continuous mathematical descriptions in a coherent integrative framework. The formulation and extension of appropriate behaviour-realization adjunctions in this context will also connect this work, not only to models of biological systems, but to logics of computational and coalgebras in the specification of dynamic system stability that will enable the development of interaction machines as a mathematical category that can serve as a common ground for biological and computational systems. Such a mathematical framework would have immediate applications and widespread impact in the understanding and manipulation of biological and novel computational systems.
After BIOMICS, the resulting framework will also facilitate extension of dynamical systems theory to what has been required of mathematics for a long time by biology and computation but which it has sorely lacked – namely, an adequate and effective constructive dynamical systems theory of organizations in which the new entities are created and destroyed in ways whose parameters can not be fixed in advance, but in the course of self-production and self-maintenance interact with their environments, at various scales, which they both shaped and are shaped by.