In the seventeenth century, the algebraic approach to geometry proved to be enormously fruitful. When René Descartes (1596-1650) developed coordinate geometry, the study of equations (algebra) and shapes (geometry) became inextricably interlinked. The move towards greater abstraction can make mathematics appear more abstruse and impenetrable, but it brings greater clarity and power, and can lead to surprising unifications.
From the mid-nineteenth century, the focus of mathematical study moved from numbers and the solution of individual equations to consideration of more general structures such as permutations and transformations. In studying whether equations higher than the fourth degree could be solved, Evariste Galois had shown the connection between groups of transformations and the roots of polynomials. Attention moved towards transformation groups and then towards more abstract groups, and a variety of other structures, like rings and fields.
The link between discrete (finite) groups and polynomial equations made by Galois inspired Sophus Lie to seek a similar link between infinite (continuous) groups and differential equations. The theory of Lie groups and Lie algebras that emerged from this has had a profound impact on mathematics. A Lie group is an algebraic group and also a topological manifold (a space that, locally, looks like Euclidean space). Moreover, the two aspects are entwined: the algebraic group operations are continuous in the topology.
Through the twentieth century, the trend towards greater abstraction continued. One of the greatest contributors to this movement was the outstanding German mathematician Emmy Noether. Following Noether’s abstraction of algebra, the nature of algebraic geometry was profoundly changed.
Emmy Noether has been called The Mother of Modern Algebra. The renowned algebraist Saunders MacLane wrote that abstract algebra started with Noether’s 1921 paper “Ideal Theory in Rings” and Hermann Weyl said that she “changed the face of algebra by her work” (Kleiner, 2007). Noether’s algebraic work was truly ground-breaking and hugely influential.
Continuous transformation groups are intimately related to symmetry, which is a powerful organizing principle. Noether used the theory of Lie groups to derive her theorems relating symmetries of the Lagrangian to conserved quantities of a dynamical system [see last week’s Thats Maths post]. Noether’s theorems play an important role in dynamics and quantum mechanics. Although Noether’s theorems are well known in physics, her most important work was in abstract algebra, especially in the area known as ring theory.
Whole numbers can be added and multiplied. Systems with these two operations came to be known as rings. Many other examples of rings were found, comprising objects other than numbers. For example, the set of even numbers is a ring and the set of all polynomials in the variable x with integer coefficients is a ring.
But many rings do not have all the properties of whole numbers: square matrices of a given size can be added and multiplied and form a ring. In general they do not commute: AB ≠ BA, and division may be impossible, since a matrix may not have an inverse.
Noetherian rings are of fundamental importance in both commutative and noncommutative ring theory, because they have simplified ideal structures. Ideals are to rings what normal subgroups are to groups. A Noetherian ring is a ring that satisfies an ascending chain condition on ideals. The ring of integers and the polynomial ring over a field are both Noetherian rings.
Algebraic Geometry gets more Abstract
In the mid 20th century, Oskar Zariski studied shapes called varieties, which are associated with rings. Every variety was associated with a ring, but the converse connection was not clear: not every ring corresponded to a variety. The extraordinary and eccentric genius Alexander Grothendieck found the connection by introducing new abstract structures called schemes, and revolutionised the subject of algebraic geometry.
Algebraic geometry is now intertwined with number theory, coding theory and even theoretical physics. Many connections have emerged within the past few decades and more may be expected. Varieties and schemes are central to modern research in algebraic geometry. They are the abstract offspring of the geometric shapes and algebraic equations unified by Descartes so long ago.
Kleiner, Israel (2007): “Emmy Noether and the Advent of Abstract Algebra”. Chapter 6 in A History of Abstract Algebra. ISBN: 978-0-8176-4684-4.