Several results obtained by the topological investigations of quantum field theories are reconsidered in the correspondence to macroscopic quantum effects realized in the condensed matter physics. In order to deepen our understanding of anomaly and topological quantization, we introduce the concept of cohomology and give a unified interpretation to anomaly and topological quantization.
The helicity is a topological invariant of an ideal fluid in three function of vorticity as topological invariants. This is extended to axisymmetric flows. We show that these are variants of the cross helicity. Noether's theorem associated with the particle relabeling symmetry underpins this unified view. A comment is given to the bearing of Kelvin's circulation theorem with Noether's second theorem.
Saprotrophic woodland fungi form self-organised transport networks as they forage for resources across the forest floor. These networks adapt during development by selective reinforcement of major transport routes and recycling of the intervening redundant mycelium to support further extension. The predicted transport performance of the resulting weighted networks show improved efficiency in comparison to evenly weighted networks with the same topology, or standard reference networks. Experimental measurement of nutrient movement using radiotracers and scintillation imaging show that fluxes are more dynamic, with synchronised oscillations and switching between different pre-existing routes. The same structures that confer good transport efficiency also show good resilience to both simulated damage and experimental attack by grazing insects, with persistence of a centrally connected core. We argue that fungi grow as self-organised planar spatial networks, honed by evolution, which may exemplify potential solutions to real-world compromises between search strategy, transport efficiency, resilience and cost.