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By Nyssen L. (ed.)

There's a wealthy and historic dating among theoretical physics and quantity concept. This quantity provides a range of difficulties that are presently in complete improvement and encourage loads of examine. all of the seven contributions begins with an introductory survey which makes it attainable even for non-specialists to appreciate the implications and to realize an idea of the good number of topics and methods used. subject matters lined are: part locking in oscillating platforms, crystallography, Hopf algebras and renormalisation concept, Zeta-function and random matrices, Kloosterman sums and the neighborhood Langlands correspondence. meant for examine mathematicians and theoretical physicists in addition to graduate scholars, this quantity provides an summary of modern advancements in a thrilling topic crossing numerous disciplines. A booklet of the eu Mathematical Society. dispensed in the Americas via the yankee Mathematical Society.

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40 43 56 64 65 67 40 Jean-Louis Verger-Gaugry 1 Introduction The mathematics of uniformly discrete point sets and Delone sets developed recently has at least four different origins: (i) the experimental evidence of nonperiodic states of matter in condensed matter physics, so-called aperiodic crystals, like quasicrystals [4], [55], [62], [68], [103] incommensurate modulated crystals phases [67], [69] and their geometric modelization (cf. Appendix), (ii) works of Delone [36], [37], [42], [97] on geometric crystallography (comparatively, see [58], [83], [90], [101] for a classical mathematical approach of periodic crystals), (iii) works of Meyer on now called cut-and-project sets and Meyer sets [80], [81], [82], [92] (for a modern language of Meyer sets in locally compact abelian groups: [84]), (iv) the theory of self-similar tilings [10], [75], [109] and the use of ergodic theory to understand diffractivity [5], [98], [109].

One considers the prime number decomposition d = ri=1 piei , takes its ˜ + 1 MUBs from the tensorial product smallest factor m ˜ = mini (piei ), and gets m (k) r (k) ˜ B = ⊗i=1 Bi , (k = 0, . . , m). At this point it is instructive to enlighten the above-described construction of MUBs by confining ourselves to the Galois ring in d = 6. Let us take the latter as the quotient GR(62 ) = Z6 [x]/(x 2 + 3x + 1) of polynomials over Z6 by a polynomial irreducible over both Z2 and Z3 . GR(62 ) has 36 elements.

Problems pertinent to quantum information theory are touching more and more branches of pure mathematics, such as number theory, abstract algebra and projective geometry. This paper focuses on one of the most prominent issues in this respect, namely the construction of sets of mutually unbiased bases (MUBs) in a Hilbert space of finite dimension. An updated list of open problems related to the development of quantum technologies can be, for example, found in [53]. To begin with, one recalls that two different orthonormal bases A and B of a dd dimensional Hilbert space √H with metrics .

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