Topology of Singular Spaces and Constructible Sheaves

This volume is based on the lecture notes of six courses delivered at a Cimpa Summer School in Temuco, Chile, in January 2001. Leading experts contribute with introductory articles covering a broad area in probability and its applications, such as mathematical physics and mathematics of finance. Written at graduate level, the lectures touch the latest advances on each subject, ranging from classical probability theory to modern developments. Thus the book will appeal to students, teachers and researchers working in probability theory or related fields.

Contenu

1 Thom-Sebastiani Theorem for constructible sheaves.- 1.1 Milnor fibration.- 1.1.1 Cohomological version of a Milnor fibration.- 1.1.2 Examples.- 1.2 Thom-Sebastiani Theorem.- 1.2.1 Preliminaries and Thom-Sebastiani for additive functions.- 1.2.2 Thom-Sebastiani Theorem for sheaves.- 1.3 The Thom-Sebastiani Isomorphism in the derived category.- 1.4 Appendix: Künneth formula.- 2 Constructible sheaves in geometric categories.- 2.0.1 The basic results.- 2.0.2 Definable spaces.- 2.1 Geometric categories.- 2.2 Constructible sheaves.- 2.3 Constructible functions.- 3 Localization results for equivariant constructible sheaves.- 3.1 Equivariant sheaves.- 3.1.1 Equivariant sheaves and monodromic complexes.- 3.1.2 Equivariant derived categories.- 3.1.3 Examples and stalk formulae.- 3.2 Localization results for additive functions.- 3.3 Localization results for Grothendieck groups and trace formulae.- 3.3.1 Grothendieck groups.- 3.3.2 Trace formulae.- 3.4 Equivariant cohomology.- 4 Stratification theory and constructible sheaves.- 4.1 Stratification theory.- 4.1.1 A cohomological version of the first isotopy lemma.- 4.1.2 Comparison of different regularity conditions.- 4.1.3 Micro-local characterization of constructible sheaves.- 4.2 Constructible sheaves on stratified spaces.- 4.2.1 Cohomologically cone-like stratifications.- 4.2.2 Stability results for constructible sheaves.- 4.3 Base change properties.- 4.3.1 Some constructions for stratifications.- 4.3.2 Base change isomorphisms.- 5 Morse theory for constructible sheaves.- 5.0.1 Real stratified Morse theory.- 5.0.2 Complex stratified Morse theory.- 5.0.3 Introduction to characteristic cycles.- 5.1 Stratified Morse theory, part I.- 5.1.1 Local Morse data.- 5.1.2 Normal Morse data.- 5.1.3 Morse theory for a stratified space with corners.- 5.2 Characteristic cycles and index formulae.- 5.2.1 Index formulae and Euler obstruction.- 5.2.2 A specialization argument.- 5.3 Stratified Morse theory, part II.- 5.3.1 Normal Morse data are independent of choices.- 5.3.2 Splitting of the local Morse data.- 5.3.3 Normal Morse data and micro-localization.- 5.4 Vanishing cycles.- 6 Vanishing theorems for constructible sheaves.- Introduction: Results and examples.- 6.0.1 (Co)stalk properties.- 6.0.2 Intersection (co)homology and perverse sheaves.- 6.0.3 Vanishing results in the complex context.- 6.0.4 Nearby and vanishing cycles.- 6.0.5 Artin-Grothendieck type theorems.- 6.0.6 Applications to constructible functions.- 6.1 Proof of the results.