By Matilde Marcolli

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, more often than not is that they are utilized to the geometry and mathematics of modular curves and to the fibers of archimedean areas of mathematics surfaces and types. one of many effects is to refine the boundary constitution of sure sessions of areas, akin to moduli areas (like modular curves) or arithmetric types accomplished through appropriate fibers at infinity through including limitations that aren't obvious inside algebraic geometry. Marcolli defines the noncommutative areas and spectral triples, then describes noncommutable modular curves, quantum statistical mechanics and Galois idea, and noncommutative geometry at arithmetric infinity.

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**Extra resources for Arithmetic Noncommutative Geometry **

**Example text**

In turn is identified with GL+ 1 (Q)\GL1 (Af ). 12) can be thought of geometrically as the Shimura variety associated to the cyclotomic tower (cf. [31]). This is the tower V of arithmetic varieties over V = Spec(Z), with Vn = Spec(Z[ζn ]), where ζn is a primitive n-th root of unity. 15) ˆ AutV (V) = lim AutV (Vn ) = GL1 (Z). 12) and of the cyclotomic tower. 16) Sh(nc) (GL1 , {±1}) := GL1 (Q)\(Af ×{±1}) = GL1 (Q)\A· /R∗+ , where A· := Af × R∗ are the ad`eles of Q with invertible archimedean component.

In such cases, there is a global underlying group G of symmetries of the algebra A, but in certain ranges of temperature the choice of an equilibrium state ϕ breaks the symmetry to a smaller subgroup Gϕ = {g ∈ G : g ∗ ϕ = ϕ}, where g ∗ denotes the induced action on states. Various systems can exhibit one or more phase transitions, or none at all. A typical situation in physical systems sees a unique KMS state for all values of the parameter above a certain critical temperature (β < βc ). This corresponds to a chaotic phase such as randomly distributed spins in a ferromagnet.

Kn , kn+1 ) = kn+1 Qn (k1 , . . , kn ) + Qn−1 (k1 , . . , kn−1 ), with Q−1 Pn (k1 , . . , kn ) = Qn−1 (k2 , . . , kn ), = 0, Q0 = 1. Thus, we obtain [k1 , . . , kn−1 , kn + xn ] Pn−1 (k1 , . . , kn−1 ) xn + Pn (k1 , . . , kn ) Pn−1 Pn = (xn ), Qn−1 Qn Qn−1 (k1 , . . , kn−1 ) xn + Qn (k1 , . . , kn ) with the standard matrix notation for fractional linear transformations, az + b a b = z→ (z). c d cz + d = If α ∈ (0, 1) is an irrational number, there is a unique sequence of integers kn (α) ≥ 1 such that α is the limit of [ k1 (α), .