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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2024 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Functoriality in categorical symplectic geometry
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by Mohammed Abouzaid and Nathaniel Bottman;
Bull. Amer. Math. Soc. 61 (2024), 525-608
DOI: https://doi.org/10.1090/bull/1808
Published electronically: August 15, 2024

Abstract:

Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya $A_\infty$-category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim and Woodward’s quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya’s alternate approach to defining functors between Fukaya $A_\infty$-categories, and the second author’s ongoing construction of the symplectic $(A_\infty ,2)$-category. In the last section, we describe a number of direct and indirect applications of this circle of ideas, and propose a conjectural version of the Barr–Beck monadicity criterion in the context of the Fukaya $A_\infty$-category.
References
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Bibliographic Information
  • Mohammed Abouzaid
  • Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
  • MR Author ID: 734175
  • ORCID: 0000-0003-0896-9898
  • Email: abouzaid@math.columbia.edu
  • Nathaniel Bottman
  • Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
  • MR Author ID: 883870
  • ORCID: 0000-0003-3907-9265
  • Email: bottman@mpim-bonn.mpg.de
  • Received by editor(s): October 19, 2022
  • Received by editor(s) in revised form: June 15, 2023
  • Published electronically: August 15, 2024
  • Additional Notes: The first author was supported by an NSF Standard Grant (DMS-2103805), the Simons Collaboration on Homological Mirror Symmetry, a Simons Fellowship award, and the Poincaré visiting professorship at Stanford University.
    The second author was supported by an NSF Standard Grant (DMS-1906220) during the preparation of this article.
  • © Copyright 2024 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 61 (2024), 525-608
  • MSC (2020): Primary 53D40
  • DOI: https://doi.org/10.1090/bull/1808
  • MathSciNet review: 4803602