Colloquium De Giorgi 2013 and 2014 (Publications of the Scuola Normale Superiore Book 5)

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 1 Title Page 4 Copyright Page 5 Table of Contents 6 Preface 7 The Mathematical Truth 16 New developments in Galois theory 18 1. A return to Galois methods 18 2. Differential Galois theory 18 3. Periods 20 Trace functions over finite fields and their applications 22 1. Motivation 22 1.1. Notation 27 2. Trace functions: definition 27 3. Trace functions: examples 31 3.1. Characters 31 3.2. Point-counting functions 33 3.3. Exponential sums 35 3.4. Operating on trace functions 36 4. Quasi-orthogonality of trace functions 38 5. Distribution of arithmetic functions in arithmetic progressions 44 References 48 Topological methods in algebraic geometry 51 Prologue 52 1. Applications of algebraic topology: non existence and existence of continuous maps 53 2. Projective varieties which are K(π, 1) 58 2.1. Rational K(π, 1)’s: basic examples 61 3. Regularity of classifying maps and fundamental groups of projective varieties 62 3.1. Harmonic maps 62 3.2. Kähler manifolds and some archetypal theorem 64 3.3. Siu’s results on harmonic maps 65 3.4. Hodge theory and existence of maps to curves 67 4. Inoue type varieties 69 5. Moduli spaces of symmetry marked varieties 73 5.1. Moduli marked varieties 73 5.2. Moduli of curves with automorphisms 74 5.3. Numerical and homological invariants of group actions on curves 75 5.4. Classification results for certain concrete groups 79 6. Connected components of moduli spaces and the action of the absolute Galois group 81 6.1. Galois conjugates of projective classifying spaces 81 6.2. Arithmetic of moduli spaces and faithful actions of the absolute Galois group 84 6.3. Change of fundamental group 85 References 85 Grothendieck at Pisa: crystals and Barsotti-Tate groups 92 1. Grothendieck at Pisa 92 2. From formal groups to Barsotti-Tate groups 93 2.1. The Tate module of an abelian variety 93 2.2. Dieudonné theory, p-divisible groups 94 2.3. The theorems of Tate and Serre-Tate 96 3. Grothendieck’s letter to Tate: crystals and crystalline cohomology 98 3.1. Crystals 99 3.2. De Rham cohomology as a crystal 100 4. Grothendieck’s letter to Illusie: deformations of Barsotti-Tate groups 103 4.1. Deformations of flat commutative group schemes. 104 4.2. Deformations of BT’s and BTn’s 106 5. Grothendieck’s letter to Barsotti: Newton and Hodge polygons 111 5.1. The specialization theorem 112 5.2. Specialization of BT’s 112 5.3. Katz’s conjecture 113 5.4. New viewpoints on slopes 115 References 115 Riemann’s hypothesis 121 1. Gauss 121 2. Riemann 122 3. How many zeros are there 124 4. Approaches to RH 125 4.1. Almost periodicity 125 5. A spectral interpretation 126 References 128 Stability results for the Brunn-Minkowski inequality 130 1. Introduction 130 2. Setting and statement of the results 131 3. Conceptual path 133 3.1. Stability on convex sets 133 3.2. Stability when A = B 134 3.3. Stability when A ≠ B 135 4. Concluding remarks 135 References 136 Schanuel’s Conjecture: algebraic independence of transcendental numbers 139 1. The origin of Schanuel’s Conjecture 139 2. Related results 140 3. Algebraic independence of logarithms of algebraic numbers 142 4. Further consequences of Schanuel’s Conjecture 144 5. Roy’s program towards Schanuel’s Conjecture 145 6. Ubiquity of Schanuel’s Conjecture 146 COLLOQUIA 148 Front Matter....Pages i-xv The Mathematical Truth 1 ....Pages 1-2 New developments in Galois theory....Pages 3-6 Trace functions over finite fields and their applications....Pages 7-35 Topological methods in algebraic geometry....Pages 37-77 Grothendieck at Pisa: crystals and Barsotti-Tate groups....Pages 79-107 Riemann’s hypothesis....Pages 109-117 Stability results for the Brunn-Minkowski inequality....Pages 119-127 Schanuel’s Conjecture: algebraic independence of transcendental numbers....Pages 129-137 Back Matter....Pages 139-139