We discuss some of the origins and applications of the notion of higher dimensional algebra, with the hope that this will encourage new insights into the development of the mathematics of complex hierarchical systems.

A general notion of operad is given, which includes: (1) the operads that arose in algebraic topology in the 1970’s to characterise loop spaces. (2) the higher operads of Michael Batanin [Bat98]. (3) braided and symmetric analogues of Batanin’s operads which are likely to be important in the study of weakly symmetric higher dimensional monoidal categories. The framework of this paper, links tog...

The quantum cosmology of a higher-derivative gravity theory arising from the heterotic string effective action is reviewed. A new type of Wheeler-DeWitt equation is obtained when the dilaton is coupled to the quadratic curvature terms. Techniques for solving the Wheeler-DeWitt equation with appropriate boundary conditions shall be described, and implications for semiclassical theories of inflat...

A gravity-driven inflation is shown to arise from a simple higher dimensional universe. In vacuum, the shear of n > 1 contracting dimensions is able to inflate the remaining three spatial dimensions. Said another way, the expansion of the 3-volume is accelerated by the contraction of the n-volume. Upon dimensional reduction, the theory is equivalent to a four dimensional cosmology with a dynami...

In this work, which follows a series of studies on the higher-dimensional steady state universe idea and prepared for Professor Tekin Dereli's Festschrift, we show influence dynamical internal (unobservable) space evolution possible anisotropy external (observable) space. We obtain mathematically exactly same Friedmann equation standard $\Lambda$CDM model space, but with some remarkable physica...