The Mathematics of a Series of Manifolds: A Toolkit for a Field Theory of Living Form

Source abstract

Algebraic Biology proposes that a living form is specified by a multiset of roots of the exceptional group E and deployed in space-time as a single four-dimensional manifold, which ₈ an observer reads as a series of three-dimensional shapes. This article gathers, in one place and with uniform notation, the mathematics that this proposal uses. It covers six areas: (1) the root system and Weyl group of E , their orbits, stabilizers and weak order; (2) Morse theory, handle ₈ decompositions, cobordisms and foliations, which give the series of manifolds its structure; (3) fibre bundles with connection, in particular bundles with structure group the normalizer of a maximal torus, which carry a continuous and a discrete gauge at once; (4) prescribed metrics, incompatible elasticity and the homotopy classification of topological defects; (5) combinatorial and computational topology-order complexes, the Möbius function, a theorem of Björner on spherical intervals, persistent homology and its stability; and (6) skew-product dynamics and overdamped stochastic dynamics, which formalize the absence of feedback. For each tool we state the definition, the result used, the role it plays in the theory, and its status: established mathematics, a result proved or computed for this programme, or a modelling choice. We include new computations for E-the orbit sizes and maximal lengths of all eight fundamental ₈ weights and a catalogue of the 154 standard parabolic subgroups of rank 2 to 4-and two short propositions with proofs. The aim is to separate cleanly what is mathematics from what is biology, so that each part can be checked on its own terms.