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Continued fractions, modular symbols, and noncommutative geometry
 Selecta Mathematica (New Series) Vol.8 N.3
, 2002
"... Abstract. Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to th ..."
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Cited by 44 (15 self)
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Abstract. Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient PGL(2,Z) \ P 1 (R) should be considered as non–commutative modular curve, and show that the modular complex can be seen as a sequence of K0–groups of the related crossed–product C ∗ –algebras. §0. Introduction and summary In this paper we study the interrelation between several topics: a generalization of the classical Gauss problem on the distribution of continued fractions, certain averages of modular symbols, the properties of geodesics on modular curves, the Mixmaster Universe model in general relativity, and the non–commutative geometry
Holography principle and arithmetic of algebraic curves
"... Abstract. According to the holography principle (due to G. ‘t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating thi ..."
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Cited by 29 (7 self)
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Abstract. According to the holography principle (due to G. ‘t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS3 holography of black holes. Moreover, in the case of Euclidean AdS2 holography, we present some results on bulk/boundary correspondence where the boundary is a non–commutative space. 0.1. Holography principle. Consider a manifold M d+1 (“bulk space”) with boundary N d. The holography principle postulates the existence of strong ties between certain field theories on M and N respectively. For example, in the actively discussed Maldacena’s conjecture ([Mal], [Wi]), M d+1 is the anti de Sitter space
NONCOMMUTATIVE GEOMETRY ON TREES AND BUILDINGS
"... The notion of a spectral triple, introduced by Connes (cf. [9], [7], [10]), provides a powerful generalization of Riemannian geometry to noncommutative spaces. It originates from the observation that, on a smooth compact spin manifold, the infinitesimal line element ds can be expressed in terms of t ..."
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Cited by 7 (3 self)
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The notion of a spectral triple, introduced by Connes (cf. [9], [7], [10]), provides a powerful generalization of Riemannian geometry to noncommutative spaces. It originates from the observation that, on a smooth compact spin manifold, the infinitesimal line element ds can be expressed in terms of the inverse of the classical Dirac operator D, so that the Riemannian
A WALK IN THE NONCOMMUTATIVE GARDEN
"... 2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9 ..."
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Cited by 7 (0 self)
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2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9
New perspectives in Arakelov geometry
 in “Number theory”, 81–102, CRM Proc. Lecture Notes
, 2004
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Buildings and their applications in geometry and topology
 ASIAN J. MATH
"... Buildings were first introduced by J.Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (BruhatTits buildings), topological buildings, Rbuildings, in particular Rtrees. They are useful for many differ ..."
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Cited by 4 (2 self)
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Buildings were first introduced by J.Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (BruhatTits buildings), topological buildings, Rbuildings, in particular Rtrees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic varieties, algebraic Ktheories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and Sarithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparametric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurston’s geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications: 1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric
intersection indices of linear cycles and the geometry of buildings and symmetric spaces
 Duke Math. J
"... This paper generalizes Yu. Manin’s approach toward a geometrical interpretation of Arakelov theory at infinity to linear cycles in projective spaces. We show how to interpret certain nonArchimedean Arakelov intersection numbers of linear cycles on P n−1 with the combinatorial geometry of the Bruhat ..."
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Cited by 3 (0 self)
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This paper generalizes Yu. Manin’s approach toward a geometrical interpretation of Arakelov theory at infinity to linear cycles in projective spaces. We show how to interpret certain nonArchimedean Arakelov intersection numbers of linear cycles on P n−1 with the combinatorial geometry of the BruhatTits building associated to PGL(n). This geometric setting has an Archimedean analogue, namely, the Riemannian symmetric space associated to SL(n, C), which we use to interpret analogous Archimedean intersection numbers of linear cycles in a similar way. 1.
Stability of lattices and the partition of arithmetic quotients
 Asian J. Math
"... 1. Introduction. Elements of the group G = SL2(R) act on the upper half plane H = {z = x + iy  y> 0} by linear fractional transformations ..."
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Cited by 2 (0 self)
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1. Introduction. Elements of the group G = SL2(R) act on the upper half plane H = {z = x + iy  y> 0} by linear fractional transformations
MODULAR INDEX INVARIANTS OF MUMFORD CURVES
, 905
"... Abstract. We continue an investigation initiated by Consani–Marcolli of the relation between the algebraic geometry of padic Mumford curves and the noncommutative geometry of graph C ∗algebras associated to the action of the uniformizing padic Schottky group on the Bruhat–Tits tree. We reconstruc ..."
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Cited by 1 (1 self)
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Abstract. We continue an investigation initiated by Consani–Marcolli of the relation between the algebraic geometry of padic Mumford curves and the noncommutative geometry of graph C ∗algebras associated to the action of the uniformizing padic Schottky group on the Bruhat–Tits tree. We reconstruct invariants of Mumford curves related to valuations of generators of the associated Schottky group, by developing a graphical theory for KMS weights on the associated graph C ∗algebra, and using modular index theory for KMS weights. We give explicit examples of the construction of graph weights for low genus Mumford curves. We then show that the theta functions of Mumford curves, and the induced currents on the Bruhat–Tits tree, define functions that generalize the graph weights. We show that such inhomogeneous graph weights can be constructed from spectral flows, and that one can reconstruct theta functions from such graphical data. 1.
Hawkin’s Theory of Progress: Progress does not consist
"... of replacing a theory that is wrong with one that is right. It consists of replacing a theory that is wrong with one that is more subtly wrong. To A.M. Vershik This text aims to present and discuss a number of situations in analysis, geometry, number theory and mathematical physics which can profit ..."
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of replacing a theory that is wrong with one that is right. It consists of replacing a theory that is wrong with one that is more subtly wrong. To A.M. Vershik This text aims to present and discuss a number of situations in analysis, geometry, number theory and mathematical physics which can profit from developing their nonstandard description or interpretation (i.e. the study of the saturated theory) and then using it to prove standard results and/or establish standard theories. Entirely nonstandard proofs are quite rare of much use in standard mathematics. One of the main obstructions is that it is hard to find a satisfactory standard shape of nonstandard (=hyper) structures. One of the great values of nonstandard mathematics is in serving as a "guiding star" and often offering a conceptually simple and elegant interpretation and generalization of classical structures. This provides a broader view of the standard theory and sometimes leads to new concrete standard results. The text can be divided into five parts. The first short part comprises sections 1 and 2. Section 1