Geometry and physics in the 20th century.

*(English)*Zbl 0994.01011
Bitar, Khalil (ed.) et al., Proceedings of the international conference on the mathematical sciences after the year 2000, Beirut, Lebanon, January 11-15, 1999. Singapore: World Scientific. 1-9 (2000).

This is a brief survey of the striking and totally unexpected new interaction between geometry and quantum theory during the last 25 years. After a description of the classical era and the quantum era the author gives short summaries of the essential features of quantum cohomology, Jones-Witten invariants, Donaldson invariants, and topological quantum field theories. Then he deals with the question what we should expect in the 21st century. The reader is warned that the biggest break-throughs cannot be predicted and there will inevitably be surprises. Nevertheless one can identify the following three different philosophical strands: (1) the main physics community led by Witten which sees string theory (or its new off-shoot M-theory) as the way forward; (2) the noncommutative geometry of Alain Connes; (3) Penrose’s view that some new physical insight is required which will alter our fundamental approach.

It is possible that these three scenarios will merge in some way to give different aspects of the same reality.

For the entire collection see [Zbl 0966.00022].

It is possible that these three scenarios will merge in some way to give different aspects of the same reality.

For the entire collection see [Zbl 0966.00022].

Reviewer: Manfred Stern (Halle)

##### MSC:

01A67 | Future perspectives in mathematics |

81-03 | History of quantum theory |

53-03 | History of differential geometry |

58-03 | History of global analysis |

01A60 | History of mathematics in the 20th century |

01A61 | History of mathematics in the 21st century |

81T75 | Noncommutative geometry methods in quantum field theory |

58B34 | Noncommutative geometry (à la Connes) |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

81T45 | Topological field theories in quantum mechanics |

57R57 | Applications of global analysis to structures on manifolds |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |