Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.

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We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces embedded in some Euclidean space that are locally bi-Lipschitz equivalent to a ball in the plane.

This work would not have been possible without the aid of many people and institutions. My apologies to those whose contributions I may have forgotten; your assistance is recorded by the existence of this dissertation.

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This review is dedicated to the memory of Oded Schramm, who worked in circle packing before his discovery of stochastic Loewner evolution and its applications to critical phenomena. This extraordinary mathematician’s untimely death on 01 September 2008 in a hiking accident was a

Quasiconformal geometry of fractals

and quasisymmetric structure of hyperspaces

We prove that the conformal dimension of any metric space is at least one unless it is zero. This confirms a conjecture of J. T. Tyson.

1. The context: A personal reminiscence Two important stories in the recent history of mathematics are those of the geometrization