synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
In microlocal sheaf theory one generalizes the concepts of microlocal analysis, such as the wave front set of a distributions, to sheaf theory, notably to a concept of microsupport? of a sheaf on a smooth manifold, as a subset of the cotangent bundle.
This microlocal point of view was then extended to sheaf theory by Kashiwara-Schapira 82, Kashiwara-Schapira 82) who introduced the notion of microsupport? of sheaves giving rise to microlocal sheaf theory (see Schapira 17). Beware that some authors still use the term “microlocal analysis” for this sheaf theoretic concept.
The generalization of the concepts of microlocal analysisto microlocal sheaf theory via the concept of microsupport? of sheaves was introduced in spring
Masaki Kashiwara, Pierre Schapira, Microlocal study of sheaves, CAstérisque, vol. 128 (1985)
Masaki Kashiwara, Pierre Schapira, Microsupport des faisceaux; applications aux module différentielle, C. R. Acad. Sci. Paris 295, 8 (1982)
Review of this incldues
See also:
V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84, 327–402 (1986), MR87j:32030, doi
Masaki Sato, Restriction, localization and microlocalization, (expository paper) pp. 195–205 in collection “Quadrature domains and their application” (Operator theory: advances and applications 156), doi
M. Kashiwara, P. Schapira, F. Ivorra, I. Waschkies, Microlocalization of ind-sheaves, in “Studies in Lie theory”, Progress in Math. 243, Birkhäuser 2006.
Dmitry Tamarkin, Microlocal condition for non-displaceablility, arxiv/0809.1584 (application of microlocal analysis to symplectic/Lagrangean geometry).
Goro Kato, Daniele Carlo Struppa, Fundamentals of algebraic microlocal analysis, M. Dekker 1999, googB
Masaki Kashiwara, Systems of microdifferential equations, Birkhäuser 1983, 87 pp.
Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of algebraic analysis, Transl. from Japanese by Goro Kato. Princeton Mathematical Series 37, 1986. xii+255 pp. MR87m:58156; Jean-Luc Brylinski, Book Review: Foundations of algebraic analysis. Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 104–108, doi
Masaki Kashiwara, Pierre Schapira, Hochschild homology and microlocal Euler classes, arxiv/1203.4869
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