an:01648081
Zbl 0994.01011
Atiyah, Michael
Geometry and physics in the 20th century
EN
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).
2000
a
01A67 81-03 53-03 58-03 01A60 01A61 81T75 58B34 53D45 81T45 57R57 81T30
brief survey; interaction between geometry and quantum theory; quantum cohomology; Jones-Witten invariants; Donaldson invariants; topological quantum field theories
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].
Manfred Stern (Halle)