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Thursday, 26 January 2023

Beyond Topology?

 Thinking about classification of phases of matter along with the pragmatist standard of never reaching the answer the easy natural question occurred to me `what next?' Is it possible to have distinct phases of matter with the same topology?

One --not the only-- way to generalize this is to ask is it possible to get even `more global' than topology?

One way to answer this is to apply the axiom of infinite divisibility to the topological invariants like genus, meaning that we can/should/must consider the possibility of topological invariants being real numbers.

This way we are talking about a genuinely differential topology...

First we make topological invariants real and hyperreal (differentials), then we will have a differential topology proper. Then if we recall Differential Geometry=Topology, (Gauss-Bonnet theorem) We might similarly say Differential Topology=`Globaller than Topology'


https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.133903

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