{"date":"2022-06-13","starts":{"hour":11,"minute":0,"second":0,"second_of_day":39600},"ends":{"hour":12,"minute":0,"second":0,"second_of_day":43200},"room":"seminar room, minus 101","speaker":"N. Christopher Phillips","from":"University of Oregon","title":"Gap labelling for electron motion in quasicrystals and C*-algebra of minimal actions of Z^d on the Cantor set","abstract":"This talk will be a survey of the mathematics of the gap labelling\r\nproblem for quasicrystals, but will assume no knowledge of physics.\r\n\r\nIn one standard approximation,\r\nthe possible energy levels of an electron moving in a crystal\r\nform a collection of bands.\r\nThese energy levels constitute the spectrum of a suitable\r\nSchr$$\\\"{o}$$dinger operator, and the gaps between the bands are\r\ngaps in the spectrum.\r\n\r\nQuasicrystals are not periodic, but exhibit long range order.\r\nThe structure of the spectrum of the Schr$$\\\"{o}$$dinger operator\r\nfor quasicrystals is addressed by the ``Gap Labelling Conjecture''.\r\nThis conjecture was made in 1989, and some results are known.\r\n\r\nAn infinite quasicrystal has an associated action of $${\\mathbb{Z}}^d$$\r\non the Cantor set $$X$$,\r\nand thus a transformation group C*-algebra $A$.\r\nThe physics is supposed to give an invariant measure on $X$,\r\nand hence a tracial state on $$A$$.\r\nThe gaps in the spectrum of Schr\\\"{o}dinger operator\r\ncorrespond to the values of this tracial state on projections in $$A$$,\r\nand the Gap Labelling Theorem states that these values all already occur\r\nas values of the measure on compact open subsets of $$X$$.\r\n\r\nIn this talk, I will give a more careful description of the situation,\r\nincluding sketches of how the objects above\r\nare constructed and how they are related to each other.\r\nThen I will say something about the results that have been proved,\r\nand outline what goes into their proofs.","speaker_web":null,"announcement":null,"descfiles_signed_ids":"[]","online":false,"slug":"2022-06-13","shift":{"beginning":{"hour":11,"minute":0,"second":0,"second_of_day":39600},"ending":{"hour":12,"minute":0,"second":0,"second_of_day":43200},"exclude_end":false,"range":"[39600,43200]"},"announcement_i18n":{"en":"","he":""},"duration":"[2022-06-13 08:00:00 UTC,2022-06-13 09:00:00 UTC]","id":895,"created_at":"2022-06-09T16:25:59.670+03:00","updated_at":"2022-06-13T10:23:06.518+03:00","admin_users":[],"summary":"N. Christopher Phillips: Gap labelling for electron motion in quasicrystals and C*-algebra of minimal actions of Z^d on the Cantor set","full_speaker_text":"N. Christopher Phillips (*University of Oregon*)","unusual":"place","unusual_place":true,"unusual_time":false,"unusual_day":false}