Quantitative Tamarkin Theory (CRM Short Courses)

Preface
Contents
1 Introduction
	1.1 A Brief Background of Symplectic Geometry
	1.2 New Methods in Symplectic Geometry
	1.3 Singular Support and Its Geometry
	1.4 Different Appearances of Tamarkin Category
	1.5 Inspirations from Persistence k-Modules
	1.6 Sheaf Quantization and the Hofer Norm
	1.7 U-Projector and Symplectic Homology
	1.8 Further Discussions
2 Preliminaries
	2.1 i-th Derived Functor
		2.1.1 Construction of Ri F
		2.1.2 Computation of RiF
	2.2 Derived Category
	2.3 Upgrade to Functor RF
	2.4 Triangulated Structure
	2.5 Applications to Sheaves
		2.5.1 Base Change Formula
		2.5.2 Adjoint Relation
	2.6 Persistence k-Modules
	2.7 Persistence Interleaving Distance
	2.8 Definition of the Singular Support
	2.9 Properties of the Singular Support
		2.9.1 Geometric and Functorial Properties
		2.9.2 Microlocal Morse Lemma
3 Tamarkin Category Theory
	3.1 Categorical Orthogonal Complement
	3.2 Definitions of Tamarkin Categories
	3.3 Sheaf Convolution and Composition
		3.3.1 Definitions of Operators
		3.3.2 Characterization of Elements in T(M)
	3.4 Geometry of Convolution
	3.5 Lagrangian Tamarkin Categories
		3.5.1 TA(M) when A is a Lagrangian Submanifold
		3.5.2 Convolution and Lagrangians
	3.6 Shift Functor  and Torsion Elements
	3.7 Separation Theorem and the Adjoint Sheaf
		3.7.1 Restatement of the Separation Theorem
		3.7.2 Adjoint Sheaf
	3.8 Proof of the Separation Theorem
	3.9 Sheaf Barcodes  from Generating Functions
	3.10 Interleaving Distance  in a Tamarkin Category
		3.10.1 Definition of Sheaf Interleaving
		3.10.2 Torsion Criterion
	3.11 Examples of Interleaving Based on the Torsion Criterion
4 Applications in Symplectic Geometry
	4.1 Guillermou-Kashiwara-Schapira Sheaf Quantization
		4.1.1 Statement and Corollaries
		4.1.2 Proof of a Simplified Sheaf Quantization
		4.1.3 Constraints Coming from Singular Supports
		4.1.4 Proof of ``Dual Sheaf Identity\'\'
	4.2 Stability with Respect to dT(M)
	4.3 Energy-Capacity Inequality (Following Asano and Ike)
	4.4 Symplectic Ball-Projector (Following Chiu)
		4.4.1 Construction of a Ball-Projector
		4.4.2 Fourier-Sato Transform (Brief)
	4.5 Geometry of PB(r), H (Joint Work with L. Polterovich)
		4.5.1 Singular Support of PB(r), H
		4.5.2 Geometric Interaction with Projectors
	4.6 U-Projector and Its Properties
	4.7 Sheaf Barcode from a U-Projector
	4.8 Comparison with Symplectic Homology
	4.9 A Sheaf Invariant of Domains in R2n
	4.10 Proof of the Gromov Non-squeezing Theorem
A Supplements
	A.1 Persistence k-Modules vs. Sheaves
	A.2 Computations of RHom
	A.3 Dynamics of the GKS Sheaf Quantization
References
Index