Title    name Lastest   |   Most hits   년도선택 최근 50건20102009200820072006200520042003200220012000     The Mathematics of Aperiodic Order On Fibonacci electrons and the stability of soft quasicrystals Ron Lifshitz (Tel Aviv University) 2010-10-01   I would like to take advantage of the informal nature of this workshop to describe three (not necessarily related) results of ongoing research on quasicrystals in my group, which may be of interest to mathematicians. The rst is the observation of log-periodic oscillations in quantities characterizing the quantum dynamics of Fibonacci electrons, such as the probability of an electron to remain in its initial position as a function of time. The second is a conjecture suggesting that the addition of weak disorder, to an otherwise perfect quasiperiodic potential, may provide the necessary conditions for the existence of a generalized Bloch theorem for quasicrystals. The third result is our successful explanation of the thermodynamic stability of quasicrystals, composed of soft isotropic particles. We have demonstrated that, with some help from entropy, very simple (even purely-repulsive) isotropic pair-potentials can lead to the formation of stable dodecagonal quasicrystals.   The Mathematics of Aperiodic Order Aperiodic order and quantum mechanics David Damanik (Rice University) 2010-10-01    We review recent developments in the spectral theory of Schrodinger operators with potentials exhibiting aperiodic order. We will discuss in detail the shape of the spectrum, especially the presence of gaps and their structure.   The Mathematics of Aperiodic Order Rauzy fractals associated with a sequence of substitutions Pierre Arnoux (University d' Aix-Marseilles) 2010-09-30   In several contexts, it would be interesting to describe a Rauzy fractal associated, not with a substitution, but with an innite sequence of substitutions. We explain how this can be realized in a specic case, when the substitutions have the same matrix of the irreducible Pisot type. The generalization of this contruction poses dicult questions.   The Mathematics of Aperiodic Order Toward S-adic Rauzy fractals Valerie Berthe (CNRS) 2010-09-30    Rauzy fractals are compact sets that are associated with substitutive dynamical systems with Pisot type in order to produce a geometric and spectral representation of the underlying symbolic dynamical system. In particular, discrete spectrum property can be stated in terms of tiling properties for the Rauzy fractals. The aim of this lecture is to discuss a possible generalization of this situation to more general symbolic dynamical systems, namely symbolic systems generated by iterating a nite number of substitutions according to a given sequence. Such systems (called S-adic) arise for instance when applying a multidimensional continued fraction algorithm.   The Mathematics of Aperiodic Order Tilings associated with shift radix systems Wolfgang Steiner (CNRS) 2010-09-30    Shift radix systems form a collection of dynamical systems depending on a parameter which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. In this talk, we discuss collections of fractal tiles dened by shift radix systems. For certain classes of parameters, they provide natural families of tiles for betaexpansions with (non-unit) Pisot numbers as well as for canonical number systems with (non-monic) expanding polynomials. In the unit and monic case respectively, they coincide with the well-known Rauzy fractals or fundamental domains of canonical number systems. The case of non-monic expanding polynomials is intimately related to the self-ane tiles with p-adic factors discussed in Jorg Thuswaldner's talk. Under some algebraic conditions on the parameter, the tiles provide multiple (weak) tilings and in many cases, e.g. for the dense family of all rational parameters, even (weak) tilings of the d-dimensional real vector space. Here, we mean by weak tiling that the tiles are compact, cover the space, and the interiors of distinct tiles do not intersect. The obtained weak tilings can have a complicated structure: They may consist of tiles having innitely many dierent shapes, the tiles need not be self-ane (or graph directed self-ane), and we do not know whether the boundary of the tiles has zero measure.   The Mathematics of Aperiodic Order Number systems and tilings with p-adic factors Joerg Thuswaldner (University of Leoben) 2010-09-30    Many results are known about tilings induced by integral self-similar   The Mathematics of Aperiodic Order Discrete planes, two dimensional placement rules and connectivity of Rauzy fractals Anne Siegel (CNRS) 2010-09-30    Rauzy fractals - or central tiles - can be considered as geometrical representations of selfsimilar processes. Their topological properties are related to several questions in number theory, ergodic theory or quasi-crystals. Most of methods allowing studying the topological properties of fractals are connected to an ad-hoc description of the boundary of the tiles. Unfortunately, this description of the graph boundary is not uniform and prevents one to study the properties of families of fractals. In this talk, we investigate the connections between Rauzy fractals and discrete geometry, especially polygonal tilings induced by discrete plane. We will explain how these polygonal tilings can be generated thanks to placement rules inspired by a duality process. As an example of application, we will prove Rauzy fractal connectivity for the full family of Arnoux-Rauzy substitutions. This is a common work with Valerie Berthe and Timo Jollivet.   The Mathematics of Aperiodic Order Relative Cohomology for Tiling Spaces Lorenzo Sadun (University of Texas) 2010-09-30      The Mathematics of Aperiodic Order Words, decomposition rules and invariants Jun Morita (University of Tsukuba) 2010-09-29    For bi-innite words, we introduce some algebraic structure using Kellendonk product, which allows us to discuss their standard modules and decomposition rules. We also obtain several deformations of the dimension map. Using this, we dene certain formal power series as word invariants.   The Mathematics of Aperiodic Order Some comments on a hexagonal monotile Uwe Grimm (The Open University) 2010-09-28      The Mathematics of Aperiodic Order Fusion tilings Natalie Priebe Frank (Vassar College) 2010-09-28   We introduce a class of tilings of Euclidean space that generalizes self-similar tilings. Larger and larger patches of tiles are constructed according to fusion rules that play a similar role to the substitution rules of symbolic dynamics. The construction is extremely general, but conditions can be chosen that allow us to investigate the measure theory and topology. We will discuss which standard results for self-similar tilings hold and if time permits, sketch the proofs.   The Mathematics of Aperiodic Order Topological invariants of aperiodic tilings Franz Gahler (University of Bielefeld) 2010-09-28   An overview is given on dierent ways to construct a space of aperiodic tilings as an inverse limit of a sequence of simpler spaces, and how this can be used to compute topological invariants like Cech cohomology. For two-dimensional canonical projection tilings it is then analysed to what extent the dierent MLD classes of such tiling spaces can be distinguished by their cohomology.   The Mathematics of Aperiodic Order Arithmetics of self-similarities of Finitely Generated Uniformly Discrete Jean-Louis Verger-Gaugry (CNRS) 2010-09-28      The Mathematics of Aperiodic Order Aperiodic structures and notions of order and disorder Shelomo I. Ben-Abraham (Ben-Gurion University) 2010-09-27   Articial aperiodic structures have recently been the subject of extensive and intensive research, resulting in layered quasiregular heterostructures, as well as photonic and phononic metamaterials with possible applications such as optical and acoustic bandpasslters or photonic waveguides. Our main interests are centered on rather fundamental questions concerning determinism, or the old problem of order vs. disorder. We seek to establish possible quantications of such concepts, to develop basic notions of complexity and entropy, and beyond. Our specic objects of study are multidimensional generalizations of the standard substitution sequences. Here we present and discuss some two-dimensional instances of the Prouhet-Thue-Morse and paperfolding systems. We compute their rectangle complexities and nd them to be at most polynomial, which means that the corresponding entropy basically vanishes. We also point out that the perfectly deterministic Champernowne sequence has entropy log 2. Thus the notion of entropy cannot serve as an unqualied measure of disorder, and there still remain many unanswered questions.   The Mathematics of Aperiodic Order Wannier transform for aperiodic solids: preliminary results Jean Bellissard (Georgia Institute of Technology) 2010-09-27   The talk will present some preliminary results on a work in progress concerning the possible extension of the Bloch theory to aperiodic Schrodinger operators on $L^2$($R^d$). The operator giving the potential energy in this specic case will be seen as a sum of short range atomic potentials with atoms located on a Delone set Ll ⊂ $R^d$, which is aperiodic, repetitive with nite local complexity. The main properties of such sets will be summarized, examples will be given, and their associated tilings will be described. The Penrose, the octagonal tilings and the icosahedral phase of several quasicrystals are important examples. The Anderson-Putnam complex and the Lagarias group will be dened. This will lead to a denition of the Wannier transform that is shown to be a unitary map onto a Hilbert space analog to the space of Bloch functions used in Solid State Physics. The Schrodinger operator will be shown to have a Bloch like representation and its domain will be characterized. Comments will be made about the potential applications of such method of analysis depending upon how far this work has progressed.

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