Related papers: Noncommutative geometry of algebraic curves
For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a…
There exists a covariant non-injective functor from the space of generic Riemann surfaces to the so-called toric AF-algebras; such a functor maps isomorphic Riemann surfaces to the stably isomorphic toric AF-algebras. We use the functor to…
A covariant functor from the category of the complex tori to the category of the Effros-Shen algebras is constructed. The functor maps isomorphic complex tori to the stably isomorphic Effros-Shen algebras. Our construction is based on the…
A covariant functor from the category of mapping tori to a category of AF-algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding AF-algebras. We use this functor…
We construct a functor which maps conjugate pseudo-Anosov automorphisms of a surface to the so-called stably isomorphic stationary AF-algebras; the functor gives new topological invariants of three dimensional manifolds coming from the…
We construct a functor from the category of elliptic curves to a category of noncommutative tori. Our proof is based on an isomorphism between the Sklyanin algebras and dense sub-algebras of the noncommutative tori.
We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence associate to every such curve a noncommutative deformation algebra. This new invariant…
We associate to each unital $C^*$-algebra $A$ a geometric object---a diagram of topological spaces representing quotient spaces of the noncommutative space underlying $A$---meant to serve the role of a generalized Gel'fand spectrum. After…
The "noncommutative geometry" of complex algebraic curves is studied. As first step, we clarify a morphism between elliptic curves, or complex tori, and C*-algebras T_t={u,v | vu=exp(2\pi it)uv}, or noncommutative tori. The main result says…
After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois…
Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…
We provide bar and cobar constructions as functors between some categories of curved algebras and curved augmented coalgebras over a graded commutative ring. These functors are adjoint to each other.
We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some…
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The…
After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois…
Tannaka duality and its extensions by Lurie, Sch\"appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete…
Following the guidelines of classical differential geometry the `building material' for the tensor calculus in non-commutative geometry is suggested. The algebraic account of moduli of vectors and covectors is carried out.
It is shown that the differential geometry of space-time, can be expressed in terms of the algebra of operators on a bundle of Hilbert spaces. The price for this is that the algebra of smooth functions on space-time has to be made…
We study the interplay between noncommutative tori and noncommutative elliptic curves through a category of equivariant differential modules on $\mathbb{C}^*$. We functorially relate this category to the category of holomorphic vector…
We develop a formulation for non-commutative derived analytic geometry built from differential graded (dg) algebras equipped with free entire functional calculus (FEFC), relating them to simplicial FEFC algebras and to locally…