The 'Useless' Perspective That Transformed Mathematics

Abstraction: Representation theory maps abstract groups to concrete matrices

Key points:

• Representation theory assigns a matrix to each element of a group such that the group multiplication structure is preserved; converts abstract groups into tractable linear algebra
• Character table summarizes all representations of a group: rows are distinct representations, columns correspond to generators, entries are traces (diagonal sums)
• William Burnside dismissed representation theory as useless in 1897; it later became central to algebra, topology, geometry, number theory, and the Langlands program
• Modular representations (matrices over finite/clock-arithmetic number systems) can distinguish groups that have identical real-valued character tables
• Andrew Wiles' 1994 proof of Fermat's Last Theorem critically depended on modular representations of an elliptic curve to show certain objects cannot exist
• Geordie Williamson (University of Sydney) described it as having "gobbled vast tracts of mathematics in the second half of the 20th century"

Connections: Representation Theory · Group Theory · Linear Algebra

Source: https://www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609/