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Categorification of invariants in gauge theory and symplectic geometry

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Abstract

This is a mixture of survey article and research announcement. We discuss instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds.

During the year 1998–2012, those problems have been studied emphasizing the ideas from analysis such as degeneration and adiabatic limit (instanton Floer homology) and strip shrinking (Lagrangian correspondence). Recently we found that replacing those analytic approach by a combination of cobordism type argument and homological algebra, we can resolve various difficulties in the analytic approach. It thus solves various problems and also simplify many of the proofs.

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Authors and Affiliations

  1. Simons Center for Geometry and Physics, State University of New York, Stony Brook, NY, 11794-3636, U.S.A.

    Kenji Fukaya

  2. Center for Geometry and Physics, Institute for Basic Sciences (IBS), Pohang, Republic of Korea

    Kenji Fukaya

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  1. Kenji Fukaya
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Correspondence to Kenji Fukaya.

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Communicated by: Kaoru Ono

This article is based on the 17th Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on June 18, 2016.

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Fukaya, K. Categorification of invariants in gauge theory and symplectic geometry. Jpn. J. Math. 13, 1–65 (2018). https://doi.org/10.1007/s11537-017-1622-9

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  • DOI: https://doi.org/10.1007/s11537-017-1622-9

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