Related papers: On tea, donuts and non-commutative geometry
Classical theory of Complex Multiplication (CM) shows that all abelian extensions of a complex quadratic field $K$ are generated by the values of appropriate modular functions at the points of finite order of elliptic curves whose…
The Serre-Swan theorem provides the link between projective modules of finite rank and vector bundles over compact manifolds, and plays a prominent role in non-commutative geometry. Its extension to non-compact manifolds is discussed.
An introduction is given to some selected aspects of noncommutative geometry. Simple examples in this context are provided by finite sets and lattices. As an application, it is explained how the nonlinear Toda lattice and a discrete time…
Noncommutative geometry is based on an idea that an associative algebra can be regarded as "an algebra of functions on a noncommutative space". The major contribution to noncommutative geometry was made by A. Connes, who, in particular,…
This PhD thesis aims at describing the applications of noncommutative geometry to particle physics and quantum field theory. It includes a brief survey of the basic principles and definitions of noncommutative geometry such as spectral…
The book covers basics of noncommutative geometry and its applications in topology, algebraic geometry and number theory. A brief survey of main parts of noncommutative geometry with historical remarks, bibliography and a list of exercises…
A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played…
We suggest to compactify the universal covering of the moduli space of complex structures by non-commutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of…
In this article, following an insight of Kontsevich, we extend the famous Weil conjecture (as well as the strong form of the Tate conjecture) from the realm of algebraic geometry to the broad noncommutative setting of dg categories. As a…
We introduce a differential extension of algebraic K-theory of an algebra using Karoubi's Chern character. In doing so, we develop a necessary theory of secondary transgression forms as well as a differential refinement of the smooth…
This informal introduction is an extended version of a three hours lecture given at the 6th Peyresq meeting ``Integrable systems and quantum field theory''. In this lecture, we make an overview of some of the mathematical results which…
There are theories of coverings of $C^*$-algebras which can be included into a following list: coverings of commutative $C^*$-algebras, coverings of $C^*$-algebras of groupoids and foliations, coverings of noncommutative tori, the double…
In this paper, a Riemannian geometry of noncommutative super surfaces is developed which generalizes [4] to the super case. The notions of metric and connections on such noncommutative super surfaces are introduced and it is shown that the…
This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…
We outline a brief description of non commutative geometry and present some applications in string theory. We use the fuzzy torus as our guiding example.
We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum-Connes…
Using a multidimensional large sieve inequality, we obtain a bound for the mean square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an…
According to Taubes, the Gromov invariants of a symplectic four-manifold X with b_+ > 1 satisfy the duality Gr(A) = +/- Gr(K-A), where K is Poincare dual to the canonical class. Extending joint work with Simon Donaldson in math.SG/0012067,…
The purpose of this note is to present a construction of an infinite family of symplectic tori T_{p} representing an arbitrary multiple of the homology class of the fiber of an elliptic surface E(n), for n > 2, such that, for i \neq j,…
We recall a construction of non-commutative algebras related to a one-parameter family of (deformed) spheres and tori, and show that in the case of tori, the *-algebras can be completed into C*-algebras isomorphic to the standard…