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29
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
Quantum invariants of 3manifolds: integrality, splitting, and perturbative expansion
 In Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of ThreeManifolds
, 1999
"... Abstract. We consider quantum invariants of 3manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that t ..."
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Cited by 36 (9 self)
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Abstract. We consider quantum invariants of 3manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that the projective quantum invariant is always an algebraic integer, if the quantum parameter is a prime root of unity. We also show that the projective quantum invariant of rational homology 3spheres has a perturbative expansion a la Ohtsuki. The presentation of the theory of quantum 3manifold is selfcontained. 0.1. For a simple Lie algebra g over C with Cartan matrix (aij) let d = maxi̸=j aij. Thus d = 1 for the ADE series, d = 2 for BCF and d = 3 for G2. The quantum group associated with g is a Hopf algebra over Q(q 1/2), where q 1/2 is the quantum parameter. To fix the order let us point out that our q is q 2 in [Ka, Ki, Tu] or v 2 in the book [Lu2]. For example, the quantum
Ramanujan’s mock theta functions and their applications, Séminaire Bourbaki
"... One of the most romantic stories in the history of mathematics is that of the friendship between Hardy and Ramanujan. It began and ended with two famous letters. The first, sent by Ramanujan to Hardy in 1913, presents its author as a penniless clerk in a Madras shipping office who has made some disc ..."
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Cited by 26 (0 self)
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One of the most romantic stories in the history of mathematics is that of the friendship between Hardy and Ramanujan. It began and ended with two famous letters. The first, sent by Ramanujan to Hardy in 1913, presents its author as a penniless clerk in a Madras shipping office who has made some discoveries that “are termed by the
ϑfunctions and real analytic modular forms, qSeries with Applications to Combinatorics, Number Theory
 School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
"... Abstract. In this paper we examine three examples of Ramanujan’s third order mock ϑfunctions and relate them to Rogers ’ false ϑseries and to a realanalytic modular form of weight 1/2. 1. ..."
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Cited by 18 (4 self)
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Abstract. In this paper we examine three examples of Ramanujan’s third order mock ϑfunctions and relate them to Rogers ’ false ϑseries and to a realanalytic modular form of weight 1/2. 1.
A unified WittenReshetikhinTuraev invariant for integral homology spheres
, 2006
"... We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTurae ..."
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Cited by 14 (2 self)
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We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTuraev invariants of M. As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) as a function on ζ behaves like an “analytic function ” defined on the set of roots of unity. That is, the τζ(M) for all roots of unity are determined by a “Taylor expansion ” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τζ(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.
The YangMills measure in the Kauffman bracket skein module
 Comment. Math. Helv
"... Abstract. For each closed, orientable surface Σg, we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module Kt(Σg ×I). The trace is defined when t  is neither 0 nor 1, and at certain roots of unity. At t = −1, the trace is integration against the symplectic measure ..."
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Cited by 7 (2 self)
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Abstract. For each closed, orientable surface Σg, we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module Kt(Σg ×I). The trace is defined when t  is neither 0 nor 1, and at certain roots of unity. At t = −1, the trace is integration against the symplectic measure on the SU(2) character variety of the fundamental group of Σg. 1.
Cyclotomy and analytic geometry over F1
"... Abstract. Geometry over non–existent “field with one element ” F1 conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the no ..."
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Cited by 7 (0 self)
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Abstract. Geometry over non–existent “field with one element ” F1 conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the notion of “analytic functions” over F1. The paper combines a focused survey with some new constructions. To Alain Connes, for his sixtieth anniversary 0. Introduction: many faces of cyclotomy 0.1. Roots of unity and field with one element. The basics of algebraic geometry over an elusive “field with one element F1 ” were laid down recently in [So], [De1], [De2], [TV], fifty years after a seminal remark by J. Tits [Ti]. There are many motivations to look for F1; a hope to imitate Weil’s proof for Riemann’s
Analytic asymptotic expansions of the Reshetikhin–Turaev invariants of Seifert 3–manifolds for sl2(C
"... Abstract. We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or nonorientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asy ..."
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Cited by 5 (3 self)
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Abstract. We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or nonorientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asymptotic formula thereby proving the socalled asymptotic expansion conjecture (AEC) due to J. E. Andersen [An1], [An2] for these manifolds. For the case of Seifert manifolds with base S 2 we actually prove a little weaker result, namely that the asymptotic formula has a form as predicted by the AEC but contains some extra terms which should be zero according to the AEC. We prove that these ‘extra ’ terms are indeed zero if the number of exceptional fibers n is less than 4 and conjecture that this is also the case if n≥4. For the case of Seifert fibered rational homology spheres we identify the Casson–Walker invariant in the asymptotic formula. Our calculations demonstrate a general method for calculating the large r asymptotics of a finite sum Σ r k=1f(k), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to Rozansky [Ro1], [Ro3], is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis. Contents
qseries and Lfunctions related to halfderivatives of the Andrews
 Gordon identity, Ramanujan J. (2004
"... Abstract. Studied is a generalization of Zagier’s qseries identity. We introduce a generating function of Lfunctions at nonpositive integers, which is regarded as a halfdifferential of the Andrews– Gordon qseries. When q is a root of unity, the generating function coincides with the quantum inv ..."
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Cited by 5 (5 self)
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Abstract. Studied is a generalization of Zagier’s qseries identity. We introduce a generating function of Lfunctions at nonpositive integers, which is regarded as a halfdifferential of the Andrews– Gordon qseries. When q is a root of unity, the generating function coincides with the quantum invariant for the torus knot. In Ref. [13], Zagier studied the qseries, (1) X(q)= n=0
Modularity of the concave composition generating function, Algebra and Number Th., accepted for publication
"... Abstract. A composition of an integer constrained to have decreasing then increasing parts is called concave. We prove that the generating function for the number of concave compositions, denoted v(q), is a mixed mock modular form in a more general sense than is typically used. We relate v(q) to gen ..."
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Cited by 3 (0 self)
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Abstract. A composition of an integer constrained to have decreasing then increasing parts is called concave. We prove that the generating function for the number of concave compositions, denoted v(q), is a mixed mock modular form in a more general sense than is typically used. We relate v(q) to generating functions studied in connection with “Moonshine of the Mathieu group ” and the smallest parts of a partition. We demonstrate this connection in four different ways. We use the elliptic and modular properties of Appell sums as well as qseries manipulations and holomorphic projection. As an application of the modularity results we give an asymptotic expansion for the number of concave compositions of n. For comparison, we give an asymptotic expansion for the number of concave compositions of n with strictly decreasing and increasing parts. The generating function of which is related to a false theta function rather than a mock theta function. 1.