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Mathematics in the 20th century. (English) Zbl 1038.01014
Chern, Shiing-Shen (ed.) et al., Contemporary trends in algebraic geometry and algebraic topology. Proceedings of the Wei-Liang Chow and Kuo-Tsai Chen memorial conference on algebraic geometry and algebraic topology, Tianjin, China, October 9–13, 2000. River Edge, NJ: World Scientific (ISBN 981-02-4954-3/hbk). Nankai Tracts Math. 5, 1-21 (2002).
The author declares that he is giving a personal view, leaving out significant parts of the story, listing some themes and talking around there.
The first passage is headed: Local to Global. It is explained that in the 20th century the emphasis has shifted to try and understand the global, large-scale behavior, these topological ideas became very important. Poincaré and Hilbert are compared from this point of view. Poincaré forecasted that topology would be an important ingredient in 20th century mathematics. Incidentally, Hilbert who made his famous list of problems, did not. For Weierstrass the functions were formulas, the explicit things, but, then the functions became defined more by their global properties. The local expansion was only one way of looking at it. A similar sort of story occurs with differential equations, in differential geometry, and even in number theory. In physics you have to study the large-scale behavior of a physical system.
The second theme is increase in dimensions. The idea that you take the higher dimensions seriously is considered as a product of the 20th century. More drastically was going to infinite dimensions, from linear space to Hilbert space.
A third theme is the shift from commutative to noncommutative. The introduction of noncommutative multiplication is said to be “the bread and butter of 20th century algebraic machinery”. The next theme is the passage from linear to nonlinear. In differential equations this has thrown up a whole range of new phenomena, e.g. soliton and chaos, in physics from Maxwell equations to Yang-Mills equations; here we can see an interesting link between nonlinearity and noncommutativity.
Then the dichotomy between geometry and algebra is referred to. Arnold is named as the inheritor of the Poincaré-Newton tradition, and Bourbaki as the most famous disciple of Hilbert. It is said that algebra is concerned with manipulation in time, and geometry is concerned with space. In algebra a sequence of operations is performed one after the other and this means you have to have time. Putting things into geometrical form enables us to use our spatial intuition.
Among the techniques and common methods the following are described: homology theory, \(K\)-theory, Lie groups, finite groups (here the role of the Monster is emphasized). Impact of physics is especially described: general relativity, quantum mechanics. There has been a tremendous incursion of new ideas from physics into mathematics: Donaldson’s work on 4-dimensional manifolds, Vaughan-Jones’ work on knot invariants, mirror symmetry, quantum groups, string theory, M-theory.
About the 21st century it is said that this might be the era of quantum mathematics, as to get quite rigorous proofs of all the beautiful things the physicists have been speculating about.
For the entire collection see [Zbl 1021.00012].
01A60 History of mathematics in the 20th century