In this thesis, we apply cochain complexes as an algebraic model of space in a diverse range of mathematical and scientific settings. We begin with an algebraic-discrete Morse theory model of auto-encoding cochain data, connecting the homotopy theory of d ...
There is a classical "duality" between homotopy and homology groups in that homotopy groups are compatible with homotopy pullbacks (every homotopy pullback gives rise to a long exact sequence in homotopy), while homology groups are compatible with homotopy ...
K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other fields of mathematics, like spaces and (not necessarily c ...
Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class (Z) over tilde of 2-local finite spectra and showed that all spectra Z is an element of (Z) over tilde admit a v(2)-self-map of periodicity 1. The aim here is to compute the K(2)-loc ...
As plant fibres are increasingly used in technical textiles and their composites, underlying principles of wound healing in living plant fibres are relevant to product quality, and provide inspiration for biomimetic healing in synthetic materials. In this ...
Consider a fibration sequence of topological spaces which is preserved as such by some functor , so that is again a fibration sequence. Pull the fibration back along an arbitrary map into the base space. Does the pullback fibration enjoy the same property? ...
We describe the bifurcation diagrams of almost toric integrable Hamiltonian systems on a four dimensional symplectic manifold M, not necessarily compact. We prove that, under a weak assumption, the connectivity of the fibers of the induced singular Lagrang ...
Modelling the shoulder's musculature is challenging given its mechanical and geometric complexity. The use of the ideal fibre model to represent a muscle's line of action cannot always faithfully represent the mechanical effect of each muscle, leading to c ...
For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed k >= 2, there is a polynomial-time algorithm that, for a 1-connected topological space X given as a f ...
Hopf-Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group G are the Hopf-Galois extensions with respect to the dual of the group algebra of ...
