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The modern triathlon “heat-electricity-mechanics” has an indisputable champion, graphene, as a recordman, among all materials in normal conditions, in all three specialties: thermal conductivity, electrical mobility and mechanical strength. On the other hand graphene, being perfectly planar, is the simplest of all possible sp2 pure carbon structures. The graphene family includes curved forms like fullerenes, having gaussian curvature G >0, nanotubes, with G=0 like graphene, and schwarzites with G <0 and vanishing mean curvature. The conjugation of carbon-carbon sp2 bonds makes several global electronic and vibrational properties of graphenes to primarily depend upon the structure topology. Global properties which can be estimated on topological grounds are the growth process, the isomer hierarchy, the vibrational spectrum, the elastic constants, the porosity as a function of the deposition energy, etc. The dynamics of free electrons in graphene is well described by the Dirac quantum-relativistic equation, and some of its consequences like the Zitterbewegung and Klein’s paradox have been proved in graphene. Thus graphene allows for the simulation and validation of fundamental theories in fields hardy accessible to experiments like high-energy physics and cosmology. With some surprising prediction! It is a fact that since the late XIX century topology has become a reference paradigm in many branches of fundamental physics, from Hermann Weyl’s topological theory of electricity and cosmological wormholes, to string theory and present topological field theories in high-energy physics.
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