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Three Lectures on Algebraic Microlocal Analysis

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Algebraic and Analytic Microlocal Analysis (AAMA 2013)

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Abstract

This is a survey talk with some historical comments. I will first explain the notions of Sato’s hyperfunctions and microfunctions, at the origin of the story, and I will describe the Sato’s microlocalization functor which was first motivated by problems of analysis. Then I will briefly recall the main features of the microlocal theory of sheaves with emphasize on the functor \(\mu \)hom which will be an essential tool in the sequel. Then, I will construct the microlocal Euler class associated with trace kernels. This construction applies in particular to constructible sheaves on real manifolds and \(\mathscr {D}\)-modules (or more generally, elliptic pairs) on complex manifolds. Finally, I will first recall the construction of the sheaves of holomorphic functions with temperate growth or with exponential decay. These are not sheaves on the usual topology, but ind-sheaves, or else, sheaves on the subanalytic site. I will explain how these objects appear naturally in the study of irregular holonomic \(\mathscr {D}\)-modules.

Research supported by the ANR-15-CE40-0007 “MICROLOCAL”.

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Notes

  1. 1.

    This difficulty is overcome in [8] by adding a variable.

  2. 2.

    This result is now proved in [33, Th. 2.5.13].

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, Sorbonne Universités, UPMC Univ Paris 6, Paris, France

    Pierre Schapira

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  1. Pierre Schapira
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Correspondence to Pierre Schapira .

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Editors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, USA

    Michael Hitrik

  2. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Dmitry Tamarkin

  3. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Boris Tsygan

  4. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Steve Zelditch

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Schapira, P. (2018). Three Lectures on Algebraic Microlocal Analysis. In: Hitrik, M., Tamarkin, D., Tsygan, B., Zelditch, S. (eds) Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proceedings in Mathematics & Statistics, vol 269. Springer, Cham. https://doi.org/10.1007/978-3-030-01588-6_2

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