By Matilde Marcolli

ISBN-10: 0821838334

ISBN-13: 9780821838334

Marcolli works from her invited lectures brought at a number of universities to handle questions and reinterpret effects and buildings from quantity concept and arithmetric algebraic geometry, often is that they are utilized to the geometry and mathematics of modular curves and to the fibers of archimedean locations of mathematics surfaces and types. one of many effects is to refine the boundary constitution of definite sessions of areas, equivalent to moduli areas (like modular curves) or arithmetric types accomplished by way of compatible fibers at infinity by way of including barriers that aren't seen inside algebraic geometry. Marcolli defines the noncommutative areas and spectral triples, then describes noncommutable modular curves, quantum statistical mechanics and Galois conception, and noncommutative geometry at arithmetric infinity.

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These alternating phases separate cycles within each era. 42). 44) admits SO(3) symmetry on the space-like hypersurfaces, and that it dt presents a singularity at t → 0. In terms of logarithmic time dΩ = − abc , the mixmaster universe model of Belinsky, Khalatnikov, and Lifshitz admits a discretization with the following properties: 1. The time evolution is divided in Kasner eras [Ωn , Ωn+1 ], for n ∈ Z. 43). 2. Each era, where the parameter u decreases with growing Ω, can be subdivided in cycles corresponding to the discrete steps un , un − 1, un − 2, etc.

The Lyapunov spectrum measures how the Hausdorff dimension varies, as a function h(c) = dimH (Lc ). 3. Limiting modular symbols and iterated shifts. 28) ϕ(s) = {g(0), g(i∞)}G, where g ∈ PGL(2, Z) (or PSL(2, Z)) is a representative of the coset s ∈ P. 23) in the following way. 2. 26). 24) and t0 ∈ P. Without loss of generality, one can consider the geodesic γβ in H×P with one end at (i∞, t0 ) and the other at (β, t0 ), for ϕ(t0 ) = {0, i∞}G . 3 of [70] is based on the fact that one can replace the homology class defined by the vertical geodesic with one obtained by connecting the successive rational approximations to β in the continued fraction expansion (Figure 5).

The first case for which there is not yet a complete solution to the explicit √ class field theory problem is the case of real quadratic fields, K = Q( d), for some positive integer d. It is natural to ask whether the approach outlined above, based on noncommutative geometry, may provide any new information on this case. Recent work of Manin [62] [63] suggests a close relation between the real quadratic case and noncommutative geometry. We will discuss the possible relation between his approach and the GL2 -system.

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