9531430 Mermin This grant continues on the well recognized work of the PI on the generalization of symmetry classification (of periodic crystals) to include aperiodic crystals. Some specific items include the case of tensorial order parameter, symmetry changes at commensurate-incommensurate phase transition and the newly discovered tetrahedral quasicrystals. In addition, there are questions of the relationship between the recent work in the mathematical literature by Novikov, Piunikhin and Conway and the PI's classification scheme. %%% Until the discovery of quasicrystals in the early eighties, the scientific view of a periodic crystal had been well established for more than 100 years and had been described in the textbooks. The aperiodic or quasiperiodic structures challenged that framework and, in a seminal work by the PI, the symmetry classification has been generalized to include the new developments. The present grant further explores ways of enriching the new classification scheme. There are also plans to study and establish relationships with the work of others. ***
