Nils A. Baas

Ansatz for dynamical hierarchies

Complex, robust functionalities can be generated naturally in at least two ways: by the assembly of structures and by the evolution of structures. This work is concerned with spontaneous formation of structures. We define the notion of dynamical hierarchies in natural systems and show the importance of this particular kind of organization for living systems. We then define a framework that enables us to formulate, investigate, and manipulate such dynamical hierarchies. This framework allows us to simultaneously investigate different levels of description together with their interrelationship, which is necessary to understand the nature of dynamical hierarchies. Our framework is then applied to a concrete and very simple formal, physicochemical, dynamical hierarchy involving water and monomers at level one, polymers and water at level two, and micelles (polymer aggregates) and water at level three. Formulating this system as a simple two-dimensional molecular dynamics (MD) lattice gas allows us within one dynamical system to demonstrate the successive emergence of two higher levels (three levels all together) of robust structures with associated properties. Second, we demonstrate how the framework for dynamical hierarchies can be used for realistic (predictive) physicochemical simulation of molecular self-assembly and self-organization processes. We discuss the detailed process of micellation using the three-dimensional MD lattice gas. Finally, from these examples we can infer principles about formal dynamical hierarchies. We present an ansatz for how to generate robust, higher-order emergent properties in formal dynamical systems that is based on a conjecture of a necessary minimal complexity within the fundamental interacting structures once a particular simulation framework is chosen.

Topology, Geometry and Quantum Field Theory: Two-vector bundles and forms of elliptic cohomology

In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology.

Stable bundles over rig categories

The point of this paper is to prove the conjecture that virtual 2-vector bundles are classified by K(ku), the algebraic K-theory of topological K-theory. Hence, by the work of Ausoni and the fourth author, virtual 2-vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology. The main technical step is showing that for well-behaved small rig categories R (also known as bimonoidal categories), the algebraic K-theory space, K(HR), of the ring spectrum HR associated to R is equivalent to K(R) � Z ×| BGL(R)| + ,w here GL(R) is the monoidal category of weakly invertible matrices over R. The title refers to the sharper result that BGL(R) is equivalent to BGL(HR). If π0R is a ring, this is almost formal, and our approach is to replace R by ar ing completed version,¯ R, provided

Hyperstructures and memory evolutive systems

The aim of the paper is to compare two different approaches to the modeling of complex natural systems, in particular of their hierarchical organization with higher-order structures and their emergence processes. These approaches are, respectively, the hyperstructures (HS) of Baas and the memory evolutive systems (MES) of Ehresmann and Vanbremeersch. The HS are “structural” while MES, based on category theory, take dynamics more into account. It is shown how the dynamical organization and mechanisms developed for MES rely on simple ideas of a philosophical nature, that might be disengaged from the categorical setting and extended to the general frame of HS.

On the chemical synthesis of new topological structures

The construction of chemical species with topological properties is an area of increasing interest. Numerous such structures have been synthesized in recent years, particularly from DNA, which is a natural synthon for these molecules. Recently, higher-order topological structures have been introduced. These hyper-structures consist of combinations of simple structures, such as a Brunnian Link of Brunnian Links or a Hopf Ring of Hopf Rings, or combinations thereof. In this article, we discuss the possibilities of constructing these hyper-structures with real molecules, particularly emphasizing the tools that result from the double helical structure of DNA.

Higher-order Brunnian structures and possible physical realizations

We consider few-body bound state systems and provide precise definitions of Borromean and Brunnian systems. The initial concepts are more than a hundred years old and originated in mathematical knot-theory as purely geometric considerations. About thirty years ago they were generalized and applied to the binding of systems in nature. It now appears that recent generalization to higher-order Brunnian structures may potentially be realized as laboratory-made or naturally occurring systems. With the binding energy as measure, we discuss possibilities of physical realization in nuclei, cold atoms, and condensedmatter systems. Appearance is not excluded. However, both the form and the strengths of the interactions must be rather special. The most promising subfields for present searches would be in cold atoms because of external control of effective interactions, or perhaps in condensed-matter systems with nonlocal interactions. In nuclei, it would only be by sheer luck due to a lack of tunability.

On structure and organization: an organizing principle

We discuss the nature of structure and organization and the process of making new things. Hyperstructures are introduced as binding and organizing principles, and we show how they can transfer from one situation to another. A guiding example is the hyperstructure of higher order Brunnian rings and similarly structured many-body systems.

Ring completion of rig categories

Abstract We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category ℛ we construct a natural additive group completion ℛ̅ that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category ℛ (also known as a symmetric bimonoidal category), the additive group completion ℛ̅ will be a commutative ring category. In an accompanying paper we show how to use this construction to prove the conjecture that the algebraic K-theory of the connective topological K-theory ring spectrum ku is equivalent to the algebraic K-theory of the rig category 𝒱 of complex vector spaces.

Hyperstructures as Abstract Matter

We give a new introduction to hyperstructures and higher order structures in general. A specific implementation of a hyperstructure we call Abstract Matter, and we argue that this is a suitable model for studying and simulating various kinds of both organic and inorganic matter — even cognitive matter.

New states of matter suggested by new topological structures

We extend the well-known Borromean and Brunnian rings to new higher order versions. Then we suggest an extension of the connection between Efimov states in cold gases and Borromean and Brunnian rings to these new higher order links. This gives rise to a whole new hierarchy of possible states with Efimov states at the bottom.