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Explore the mathematical foundations and physical implications of quasicrystals extended to Lorentzian spacetimes in this advanced physics lecture. Learn about the cut-and-project method for constructing traditional Euclidean quasicrystals, which creates aperiodic tilings that lack translational symmetry by taking irrational slices of higher-dimensional lattices. Discover how this mathematical framework can be extended to spacetime through the development of Spacetime Cut-and-Project (CNP), enabling the construction of the first-ever examples of spacetime quasicrystals in both (1+1)- and (1+3)-dimensional spacetime. Examine the potential relevance of quasi-crystalline spacetime structures to fundamental physics, including their application as a novel string-compactification scheme that could naturally address the hierarchy problem and explain the smallness of the cosmological constant. Investigate the connections between these structures and quantum gravity through their conformal Lorentzian properties and their relationship to quantum error-correcting codes, offering new perspectives on the geometric nature of spacetime itself.
