By Matilde Marcolli

Marcolli works from her invited lectures added at numerous universities to handle questions and reinterpret effects and buildings from quantity concept and arithmetric algebraic geometry, mostly 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 sure periods of areas, similar to moduli areas (like modular curves) or arithmetric kinds accomplished through appropriate fibers at infinity by means of including limitations that aren't obvious inside of 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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Kr , kr+1 , . ] , ω − = [k−1 ; k−2 , . . , k−n , k−n−1 , . ] hence the corresponding geodesics are coded by (ω, s), s ∈ P, where ω is a doubly infinite sequence ω = . . k−(n+1) k−n . . k−1 k0 k1 . . kn . . The equivalence relation of passing to the quotient by the group action is implemented by the invertible (double sided) shift: 1 1 1 −[1/ω + ] 1 , − , ·s . 1 0 ω+ ω + ω − − [1/ω + ] In particular, the closed geodesics in XG correspond to the case where the endpoints ω ± are the attractive and repelling fixed points of a hyperbolic element in the group.

For a group acting on a tree there is a Pimsner six terms exact sequence, computing the K-theory of the crossed product C ∗ -algebra (for A = C(X)) K0 (A) α O K1 (A Γ) o β˜ / K0 (A Γσ ) ⊕ K0 (A Γτ ) K1 (A Γσ ) ⊕ K1 (A Γτ ) o α ˜ / K0 (A Γ) β K1 (A) A direct inspection of the maps in this exact sequence shows that it contains a subsequence, which is canonically isomorphic to the algebraic presentation of the modular complex, compatibly with the covering maps between modular curves for different congruence subgroups.

During each era the mixmaster universe goes through a volume compression. Instead of resulting in a collapse, as with the Kasner metric, high negative curvature develops resulting in a bounce (transition to a new era) which starts again a behavior approximated by a Kasner metric, but with a different value of the parameter u. Within each era, most of the volume compression is due 8. INTERMEZZO: CHAOTIC COSMOLOGY 47 to the scale factors along one of the space axes, while the other scale factors alternate between phases of contraction and expansion.