By Dmitri Burago, Yuri Burago, Sergei Ivanov
"Metric geometry" is an method of geometry in accordance with the proposal of size on a topological area. This procedure skilled a truly quick improvement within the previous couple of a long time and penetrated into many different mathematical disciplines, equivalent to team idea, dynamical platforms, and partial differential equations. the target of this graduate textbook is twofold: to offer a close exposition of easy notions and strategies utilized in the speculation of size areas, and, extra typically, to provide an easy advent right into a large number of geometrical subject matters with regards to the inspiration of distance, together with Riemannian and Carnot-Caratheodory metrics, the hyperbolic airplane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic areas, convergence of metric areas, and Alexandrov areas (non-positively and non-negatively curved spaces). The authors are likely to paintings with "easy-to-touch" mathematical gadgets utilizing "easy-to-visualize" equipment. The authors set a not easy target of creating the middle elements of the publication obtainable to first-year graduate scholars. so much new options and techniques are brought and illustrated utilizing least difficult circumstances and warding off technicalities. The booklet comprises many workouts, which shape an essential component of exposition.
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Extra info for A Course in Metric Geometry (Graduate Studies in Mathematics, Volume 33)
Cf. 4 below for the general definition but, for now, we define overconvergence only for integral weights: Let wk for k E Z denote the k-th tensor power of w, and consider the vector space of rigid analytic sections of wk over the affinoid Z, (N p-) (v). More explicitly, letting K be a complete subfield of Cp, and the affinoid over K induced from Zl (N pm) (v) by base change, define the K-vector space Mk (Npm, v; K) := wk (Zl (N pm) (V) 1K), which has a natural K-Banach space structure, as described in section B2 of [C-BMF].
And [K-2]. 1. The q-expansion E(q) is the q-expansion of a Katz p-adic modular function 6 over A(°) with the identity weight-character. Proof. Fix an integer n > 0. Let Rn = A(°)/([1 + q]Pn - (1 + q)2Pn). Then A = limRn. Let En(q) be the restriction of E(q) to Rn. Let x(a) _ ((a))2V,(a) where 0 is a character of finite order on 1 + qZr. p(l + q)), [l + q]rn (l - + q)2Pn = 0 where the product is over all characters 0 on 1 + qZp trivial on 1 + qpn Zp, En (q) is the q-expansion of a Katz p-adic modular function over the normal- ization of R.
We drop Y from the notation when it is a point and we drop W when it is understood from the context. In particular, the ring of overconvergent families of rigid analytic functions on Z1(Nq) parameterized by V is denoted At(Z1(Nq)v/V). If BK[0,1] and BK(0,1) denote the affinoid and wide open unit disks over K, At(BQp[0,1]BQp(° 1)/Bq (0,1))° may be identified as the ring of power series in the parameter variable T over AM which are of the form E000 A,,Tn Ilan] where Ii is the maxsuch that for some positive real number a, An E imal ideal of AM.