Noncommutative differential geometry on the quantum two sphere of podle?. I: An algebraic viewpoint

We compute the cyclic homology of the coordinate ringA(S _q ² (c, d)) of the quantum two sphereS _q ² (c, d) of Podle. The results are then interpreted from the point of view of Connes' noncommutative index theorem.     

格点上的非交换微分运算及其应用

By introducing the noncommutative differential calculus on the function space of the infinite/finite set and construct a homotopy operator, one prove the analogue of the Poincare lemma for the difference complex. As an application of the differential calculus, a two dimensional integral model can be derived from the noncommutative differential calculus.     

Noncommutative Geometry (An Introduction to Selected Topics)

We give an introduction to noncommutative geometry and to some of its applications. Emphasis will be on noncommutative manifolds, notably noncommutative tori and spheres.     

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.     

Cartan calculus for quantum differentials on bicrossproducts

We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups k(M) k G associated to finite group factorizations X = GM and a field k. The irreducible calculi are associated to certain conjugacy classes in X and representations of isotropy groups. We find the full exterior algebras and show that they are inner by a bi-invariant 1-form which is a generator in the noncommutative de Rham cohomology H ¹. The special cases where one subgroup is normal are analysed. As an application, we study the noncommutative cohomology on the quantum codouble D ^*(S ₃)k(S ₃) k₆ and the quantum double D(S ₃) \triangleleft $$ " align="middle" border="0"> k S ₃, finding respectively a natural calculus and a unique calculus with H ⁰ = k.1.     

Equivariant Spectral Triples on the Quantum SU(2) Group

We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L ₂-space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SU_q(2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU(2), and we prove that for p < 4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dimension p acting on the L ₂-space.The first author would like to acknowledge support from the National Board of Higher Mathematics, India.     

On the algebraic geometry ofS-duality

This paper deals with algebro-geometric questions arising in the verification of theS-duality conjecture for supersymmetric Yang-Mills quantum field theories in the four-dimensional case. We describe all the cases for the gauge groups of rank 1 and 2, where the Gell-Man-Law beta-function is either zero or negative, and point out some series of such cases for gauge groups of arbitrary rank. Realization of one of these series on the complex projective plane demonstrates a relationship with exceptional bundles. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 163–178, February, 1997. Translated by S. K. Lando     

Notes on 2-parameter quantum groups I

A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form. Their positive parts turn out to be 2-cocycle deformations of each other under some conditions. An operator realization of the positive part is given. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10431040, 10728102), the TRAPOYT, the FUDP and the Priority Academic Discipline from the MOE of China, the SRSTP from the STCSM, the Shanghai Priority Academic Discipline from the SMEC     

Twisted Homology of Quantum SL(2)

We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of the coordinate algebra of the quantum SL(2) group relative to twisting automorphisms acting by rescaling the standard generators a,b,c,d. We discover a family of automorphisms for which the “twisted” Hochschild dimension coincides with the classical dimension of , thus avoiding the “dimension drop” in Hochschild homology seen for many quantum deformations. Strikingly, the simplest such automorphism is the canonical modular automorphism arising from the Haar functional. In addition, we identify the twisted cyclic cohomology classes corresponding to the three covariant differential calculi over quantum SU(2) discovered by Woronowicz.     

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds.     

Classical differential geometry solution of the brachistochrone tunnel problem

The Frenet-Serret equations of classical differential geometry are used to describe the quickest descent tunneling path problem. The optimal tunnel is shown to have a constant turn rate with zero torsion and is equivalent to Edelbaum's hypocycloid solution. The solutions are obtained using the maximum principle and singular arc conditions. The optimal curvature is a first-order singular arc and the optimal torsion is a second-order singular arc. Our treatment includes both the normal and the abnormal optimal control problems. Our problem is abnormal for the case where the final speed is zero. Analytical solutions for the optimal time histories are derived for all states and all adjoint states. One of Leitmann's sufficiency field theorems is used to establish optimality of the solutions.     

Revival of a classical topic in differential geometry: the exploration of envelopes in a computerized environment

Learning mathematics in a technology-rich environment enables us to revive classical topics which have been removed from the curriculum a long time ago. Both theoretical issues and applications can be studied with an experimental process. We present how envelopes of 1-parameter families of plane curves and some of their applications can be presented early in the curriculum either for pre-service teachers or for in-service teachers. This approach may be useful for students in an engineering curriculum. Working with technology yields important effects, such as reviving classical topics, broadening perspectives on already known topics, and enhancing the learner's experimental skills, where conversion between various registers of representation is an important issue.     

Body‐based activities in secondary geometry: An analysis of learning and viewpoint

Body‐based activities have the potential to support mathematics learning, but we know little about the impact they have in the classroom. This study compares high school geometry students learning through either body‐based or analogous non‐body‐based activities over the course of a two‐week unit on similarity. Pre‐ and post‐tests revealed that while students in both conditions showed gains in content area comprehension over the course of the study, the body‐based condition showed significantly greater gains. Further, there were differences in the language students used to describe the learning activities at the end of the unit that may have contributed to the differences in learning gains. The students in the body‐based condition included more mathematical and nonmathematical details in their recollections. Additionally, students in the body‐based condition were more likely to recall their experiences from a first‐person perspective, while students in the control condition were more likely to use a third‐person perspective.     

Contributions to differential geometry of isotropic curves in the complex space

This work deals with classical differential geometry of isotropic curves in the complex space C⁴. First, we study spherical isotropic curves and pseudo helices. Besides, in this section we introduce some special isotropic helices (type-1, type-2 and type-3 isotropic slant helices) and express some characterizations of them in terms of É. Cartan equations. Thereafter, we prove that position vector of an isotropic curve satisfies a vector differential equation of fourth order. Finally, we investigate position vector of an arbitrary curve with respect to É. Cartan frame by a system of complex differential equations whose solution gives components of the position vector. Solutions of the mentioned system and vector differential equation have not yet been found. Therefore, in terms of special cases, we present some special characterizations.     

Gheorghe Ţiţeica and the origins of affine differential geometry

We describe the historical and ideological context that brought to the fore the study of a centro-affine invariant that subsequently received much attention. The invariant was introduced by ?i?eica in 1907, and this discovery has been viewed by many as a consequence of Klein's Erlangen program. We thus present the starting point of affine differential geometry, as it was discovered by ?i?eica after his years in the Ph.D. program in Paris (1896–1899) under the guidance of Gaston Darboux.     

Dirac operators on the quantum group SU(2) and the quantum sphere

The problem of constructing the Dirac operators on the quantum groupSU(2) and the quantum sphere S_qμ ² are discussed. In both cases, the constructions presented have the sameSU _q(2)-invariant form and are directly connected with the corresponding Laplace operators. Bibliography:16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 49–65. Translated by P. N. Bibikov.     

An algorithmic construction of group automorphisms and the quantum Yang-Baxter equation

The structure group G of a non-degenerate symmetric set (X,S) is a Bieberbach and a Garside group. We describe a combinatorial method to compute explicitly a group of automorphisms of G and show this group admits a subgroup that preserves the Garside structure. In some special cases, we could also prove the group of automorphisms found is an outer automorphism group.     

The Local Index Formula for SU q (2)

We discuss the local index formula of Connes–Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula. (Received: January 2005)