Quantitative Tamarkin category

TL;DR

This paper provides a detailed introduction to Tamarkin category theory from a quantitative perspective, illustrating its applications in symplectic topology and its relation to persistent homology, with numerous examples for geometric intuition.

Contribution

It offers a comprehensive exposition of Tamarkin category theory, connecting it to symplectic techniques and generalizing persistent homology, with practical applications and illustrative examples.

Findings

Tamarkin category effectively encodes symplectic invariants.

Sheaf-theoretic methods unify various symplectic techniques.

Generalizations of persistent homology are naturally incorporated.

Abstract

This is a lecture note from a seminar course given at Tel Aviv University in Spring 2018. Part of the preliminary section is built from Kazhdan's seminar organized in the Hebrew University of Jerusalem in Fall 2017. The main topic of this note is a detailed introduction of Tamarkin category theory from a quantitative perspective, followed by a demonstration of various applications in symplectic topology. Many examples are provided in order to obtain certain geometric intuitions of those abstract algebraic constructions in Tamarkin category. In this note, we try to explain how standard symplectic techniques, for instance, generating function, capacities, symplectic homology, etc., are elegantly packaged in the language of sheaves as well as related intriguing sheaf operators. In addition, many concepts developed in Tamarkin category theory are natural generalizations of persistent homology theory, so their relations are emphasized along with the development of the note.