Colloquium De Giorgi 2009 (Publications of the Scuola Normale Superiore Book 3)

Since 2001 the Scuola Normale Superiore di Pisa has organized the "Colloquio De Giorgi", a series of colloquium talks named after Ennio De Giorgi. The Colloquio is addressed to a general mathematical audience, and especially meant to attract graduate students and advanced undergraduate students. The lectures are intended to be not too technical, in fields of wide interest. They must provide an overview of the general topic, possibly in a historical perspective, together with a description of more recent progress. The idea of collecting the materials from these lectures and publishing them in annual volumes came out recently, as a recognition of their intrinsic mathematical interest, and also with the aim of preserving memory of these events. ​ Cover......Page 1 Title Page......Page 4 Copyright Page......Page 5 Table of Contents......Page 6 Preface......Page 7 1.1. Cosets and affine maps......Page 14 1.2. Definition of Ap(G)......Page 16 1.3. The connection between cosets and function algebras......Page 19 2. Coset geometry of groups......Page 20 3. An analysis of cosets and coset-preserving maps......Page 22 4. Coset-preserving maps of connected Lie groups......Page 27 5. Isometries of the algebras Ap(G)......Page 28 References......Page 30 Classical analysis and nilpotent Lie groups......Page 32 1. Classical Fourier series......Page 33 2. The Heisenberg group......Page 34 3. Representations and coadjoint orbits......Page 36 4. Square integrable representations......Page 37 5. Commutative spaces – generalities......Page 41 6. Commutative nilmanifolds......Page 43 References......Page 45 1. Basic discourse......Page 46 2. Historical background......Page 48 3. Leibniz conjecture from a modern point of view......Page 50 4. The period ring of Kontsevich......Page 51 5. Schanuel’s conjecture......Page 52 6. Period ring of 1-motifs......Page 53 References......Page 55 The geometry and curvature of shape spaces......Page 56 References......Page 66 Back matter......Page 67 Contributions by several authors such as Michael G. Cowling.- Joseph A. Wolf.- Gisbert Wustholz and David Mumford Contributions by several authors such as Michael G. Cowling Joseph A. Wolf Gisbert Wüstholz and David Mumford.