Igor Nikolaev

Noncommutative geometry of algebraic curves

A covariant functor from the category of generic complex algebraic curves to a category of the AF-algebras is constructed. The construction is based on a representation of the Teichmuller space of a curve by the measured foliations due to Douady, Hubbard, Masur and Thurston. The functor maps isomorphic algebraic curves to the stably isomorphic AF-algebras.

Noncommutative Geometry: A Functorial Approach

The text consists of an introduction, table of contents and Chapters 1, 4, 5 and 6 of a 300 pages bookThe 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 is attached. Our notes are intended for the graduate students and faculty with interests in noncommutative geometry; they can be read by non-experts in the field.

Langlands reciprocity for the even-dimensional noncommutative tori

We conjecture an explicit formula for the higher dimensional Dirichlet character; the formula is based on the K-theory of the so-called noncommutative tori. It is proved, that our conjecture is true for the two-dimensional and one-dimensional (degenerate) noncommutative tori; in the second case, one gets a noncommutative analog of the Artin reciprocity law.

On the AF-algebra of a Hecke eigenform

An AF-algebra is assigned to each cusp form f of weight two; we study properties of this operator algebra, when f is a Hecke eigenform.

Tate curves and UHF-algebras

Abstract It is proved that (a stabilization of) the norm-closure of a self-adjoint representation of the twisted homogeneous coordinate ring of a Tate curve contains a copy of the UHF-algebra.

Geodesic Laminations Revisited

The Bratteli diagram is an infinite graph which reflects the structure of projections in an AF-algebra. We prove that every strictly ergodic unimodular Bratteli diagram of rank 2g+m−1 gives rise to a minimal geodesic lamination with the m principal regions on a hyperbolic surface of genus g≥1. The proof is based on a Morse theory of the recurrent geodesics on the hyperbolic surfaces.

On Embedding of the Bratteli Diagram into a Surface

We study C*-algebras O λ which arise in dynamics of the interval exchange transformations and measured foliations on compact surfaces. Using Koebe-Morse coding of geodesic lines, we establish a bijection between Bratteli diagrams of such algebras and measured foliations. This approach allows us to apply K-theory of operator algebras to prove a strict ergodicity criterion and Keane’s conjecture for the interval exchange transformations.