several key themes that have characterised mathematics in the 20th century. The impact of physics is also discussed, and some speculations are made about possible developments in the 21st century. - pdf / HN

Key Themes Highlighted by Atiyah

1. From the Local to the Global

Atiyah emphasizes a fundamental shift during the 20th century—from focusing on local, explicit investigations to embracing global, qualitative structures. Examples include:

• Complex analysis: moving beyond power-series expressions to studying functions via their global aspects, like singularities and domains.

• Differential equations: from explicit local solutions to understanding global behavior in terms of singularities.

• Differential geometry: evolving from local curvature computations (à la Gauss) to exploring the large-scale topology of manifolds.

• Number theory: the interplay between local (individual primes) and global perspectives.

• Physics: inherently concerns scaling behavior from local differential laws to global phenomena.

1. Rise of Higher Dimensions

There was a dramatic move from the familiar lower-dimensional settings to much higher—even infinite—dimensional contexts:

• Complex variables: expansion from one variable to multi-variable theory, introducing entirely new phenomena.

• Geometry: from tangible curves and surfaces to abstract, high-dimensional manifolds.

• Analysis: development of infinite-dimensional spaces (e.g., Hilbert spaces) and functionals such as those in the calculus of variations.

1. From Commutative to Non-commutative Structures

Atiyah notes the growing prominence of non-commutative algebra:

• Rooted in Hamilton’s quaternions and Grassmann’s exterior algebra.

• Now central in areas like operator algebras and quantum theory.