The legacy of Jean Bourgain in geometric functional analysis. (English) Zbl 1471.46010
This paper is one of the contributions to the volume of the “Bulletin” devoted to the memory of Jean Bourgain (1954–2018). The paper is dedicated to the work in geometric functional analysis with no intention to provide a comprehensive overview of Bourgain’s work in geometric functional analysis because of the enormous breadth and depth of his contributions.
The paper discusses aspects of the following three directions: metric geometry, the geometry of high-dimensional convex bodies, and the study of random operators.
The author provides the necessary background for understanding the following of Bourgain’s contributions to the geometry of high-dimensional convex bodies: the reverse Santaló inequality [J. Bourgain and V. D. Milman, Invent. Math. 88, 319–340 (1987; Zbl 0617.52006)] and Bourgain’s estimate [J. Bourgain, Lect. Notes Math. 1469, 127–137 (1991; Zbl 0773.46013)] related to his slicing conjecture. Subsequent developments are also discussed.
In metric geometry, the author describes Bourgain’s contributions to what Bourgain named the “Ribe program”. The goals of this program can be described in the following way: (1) To find explicit metric characterizations of the isomorphic invariants of Banach spaces determined by finite-dimensional structures; (2) to build a theory of finite metric spaces using an analogy with Banach spaces and results of the first part of the program.
Bourgain achieved the first successes in the Ribe program and thus initiated this very active and fertile direction of research. Bourgain’s main contributions were: (1) sharp estimates for the distortion of embeddings of an arbitrary \(n\)-element metric space into a Euclidean space [J. Bourgain, Isr. J. Math. 52, 46–52 (1985; Zbl 0657.46013)]; (2) metric characterization of superreflexivity [J. Bourgain, Isr. J. Math. 56, 222–230 (1986; Zbl 0643.46013)]; (3) metric characterization of type [J. Bourgain et al., Trans. Am. Math. Soc. 294, 295–317 (1986; Zbl 0617.46024)]; (4) nonlinear Dvoretzky theorem [J. Bourgain et al., Isr. J. Math. 55, 147–152 (1986; Zbl 0634.46008)].
The paper contains descriptions of these contributions and a sketch of the proof of the result (1), and presents some of the important further developments.
As for the study of random operators, the author sketches the proof of the result of [J. Bourgain and L. Tzafriri, Isr. J. Math. 57, 137–224 (1987; Zbl 0631.46017)] on restricted invertibility. This result was used by Marcus, Spielman, and Srivastava in their solution of the Kadison-Singer problem [A. W. Marcus et al., Ann. Math. (2) 182, No. 1, 327–350 (2015; Zbl 1332.46056)].
The paper is a very useful source for becoming acquainted with the mentioned directions of Bourgain’s research.
Reviewer’s remarks: I find it natural to mention (1) the recent breakthrough on the slicing conjecture: [Y.-S. Chen, “An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture”, Preprint (2020), arXiv:2011.13661]; (2) the recent breakthrough in the Ribe program: [P. Ivanisvili et al., Ann. Math. (2) 192, No. 2, 665–678 (2020; Zbl 1458.46012)]; (3) the important recent survey [A. Naor, in: Proceedings of the international congress of mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1–9, 2018. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemťica (SBM). 759–837 (2018; Zbl 1444.46019)].
The paper discusses aspects of the following three directions: metric geometry, the geometry of high-dimensional convex bodies, and the study of random operators.
The author provides the necessary background for understanding the following of Bourgain’s contributions to the geometry of high-dimensional convex bodies: the reverse Santaló inequality [J. Bourgain and V. D. Milman, Invent. Math. 88, 319–340 (1987; Zbl 0617.52006)] and Bourgain’s estimate [J. Bourgain, Lect. Notes Math. 1469, 127–137 (1991; Zbl 0773.46013)] related to his slicing conjecture. Subsequent developments are also discussed.
In metric geometry, the author describes Bourgain’s contributions to what Bourgain named the “Ribe program”. The goals of this program can be described in the following way: (1) To find explicit metric characterizations of the isomorphic invariants of Banach spaces determined by finite-dimensional structures; (2) to build a theory of finite metric spaces using an analogy with Banach spaces and results of the first part of the program.
Bourgain achieved the first successes in the Ribe program and thus initiated this very active and fertile direction of research. Bourgain’s main contributions were: (1) sharp estimates for the distortion of embeddings of an arbitrary \(n\)-element metric space into a Euclidean space [J. Bourgain, Isr. J. Math. 52, 46–52 (1985; Zbl 0657.46013)]; (2) metric characterization of superreflexivity [J. Bourgain, Isr. J. Math. 56, 222–230 (1986; Zbl 0643.46013)]; (3) metric characterization of type [J. Bourgain et al., Trans. Am. Math. Soc. 294, 295–317 (1986; Zbl 0617.46024)]; (4) nonlinear Dvoretzky theorem [J. Bourgain et al., Isr. J. Math. 55, 147–152 (1986; Zbl 0634.46008)].
The paper contains descriptions of these contributions and a sketch of the proof of the result (1), and presents some of the important further developments.
As for the study of random operators, the author sketches the proof of the result of [J. Bourgain and L. Tzafriri, Isr. J. Math. 57, 137–224 (1987; Zbl 0631.46017)] on restricted invertibility. This result was used by Marcus, Spielman, and Srivastava in their solution of the Kadison-Singer problem [A. W. Marcus et al., Ann. Math. (2) 182, No. 1, 327–350 (2015; Zbl 1332.46056)].
The paper is a very useful source for becoming acquainted with the mentioned directions of Bourgain’s research.
Reviewer’s remarks: I find it natural to mention (1) the recent breakthrough on the slicing conjecture: [Y.-S. Chen, “An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture”, Preprint (2020), arXiv:2011.13661]; (2) the recent breakthrough in the Ribe program: [P. Ivanisvili et al., Ann. Math. (2) 192, No. 2, 665–678 (2020; Zbl 1458.46012)]; (3) the important recent survey [A. Naor, in: Proceedings of the international congress of mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1–9, 2018. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemťica (SBM). 759–837 (2018; Zbl 1444.46019)].
Reviewer: Mikhail Ostrovskii (New York)
MSC:
| 46B07 | Local theory of Banach spaces |
| 46B85 | Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science |
| 51F30 | Lipschitz and coarse geometry of metric spaces |
| 52A23 | Asymptotic theory of convex bodies |
| 46-03 | History of functional analysis |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
Biographic References:
Bourgain, JeanCitations:
Zbl 1458.46012; Zbl 1444.46019; Zbl 0617.52006; Zbl 0773.46013; Zbl 0657.46013; Zbl 0643.46013; Zbl 0617.46024; Zbl 0634.46008; Zbl 0631.46017; Zbl 1332.46056References:
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