Of all the realms of mathematics, there are few where more people feel more at home than in the safe harbors of algebra. Since the age of eight, we have been made familiar with all of the classic moves involved in Solving For X, how a(b+c) = ab + ac, how if a = c then a-c = 0, and a fleet of other manipulations besides that have become so second nature to us that we rarely stop and think about just What we are doing. What sort of a thing is it we are engaging in when we employ the laws of algebra, and might the moves that we have gotten used to be nothing more than very restricted cases of a much bigger world?

These questions began intriguing mathematicians in the late nineteenth century, leading to the explosion of the field of Universal Algebra in the 1930s and 1940s, which sought to answer the question of what makes algebra algebra, and how we might construct things that follow the rules we most associate with the algebra we’ve learned since youth, but which present us with often strange new mathematical vistas. Some of the most interesting results of universal algebraic thinking came when mathematicians began playing with the properties possessed by different categories of algebras, including the field of “Varieties of Groups” pioneered by the mathematical power couple of Hanna and Bernhard Neumann.

Hanna’s mathematical journey would take her all over the world, first in flight and later in triumph. Born Hanna von Caemmerer on Febraury 12, 1914 to a family of long-standing Prussian military background, she lost her tradition-flouting historian father early in World War I, and her Huguenot-descended mother had to make do on a slim war pension that required she and her two siblings to contribute to the household as best they could. Hanna’s academic gifts were such that, at the age of thirteen, she was able to earn money coaching other children. From 1922 to 1932 she attended the Augusta-Victoria Schule, where she took fifteen subjects and, on her final examinations for college, scored highly on all areas save for music.

Hanna began her first year at the University of Berlin in 1932, where the foundational topologist Georg Feigl and function theorist Ludwig Bieberbach were among the mathematical luminaries offering courses to undergraduates. Hanna allowed herself to experience the full breadth of the university’s offerings, taking courses from Gestalt theory psychological pioneer Wolfgang Kohler, Nobel prize winning physicist Walther Nernst, and Germany’s most famous academic lawyer Martin Wolff. She threw herself into the coffee break academic discussion culture of the university, and soon became good friends with doctoral student Bernhard H. Neumann.

Their romance, for soon it became such, would determine the course of their lives, for with the ascension of the Nazis to power in 1933, Bernhard, of Jewish descent, knew his days in the German university system were numbered, and emigrated to England to continue his studies at Cambridge. He and Hanna became engaged secretly to avoid reprisals against the members of Bernhard’s family still in Germany, though Hanna soon caught the official attention of the Nazis anyway by her work with a group of fellow students who sought to keep non-enrolled Nazi rabble rousers out of classes taught by Jewish professors. Hanna continued her mathematical studies, but in her fourth year she was faced with the decision of applying for her doctorate, which is what she wanted to do, or taking the Staatsexamen, which was a broader degree for those desiring to go into public service and the teaching profession. Unfortunately, she knew that, if she opted for the doctorate, a pro-Nazi member of the administration would be sitting on her examination panel, and would question her about her “political knowledge,” which would serve as a thinly veiled evaluation of her Nazi political purity.

Hanna elected to take the Staatsexamen instead, which she completed in 1936, and then accepted a position as a research student to Helmut Hasse at the University of Gottingen, which was by 1937 a shadow of its pre-Nazi self. But with the Nazi territorial expansions of 1938, Hanna realized that it would no longer be possible to simply wait out the regime, and she broke off her course of study early to go to England. Here, she and Bernhard could be together, though lingering fear for Bernhard’s family compelled them to marry secretly in 1938. Hanna continued her work (mostly, thanks to the housing problems created by the Second World War, on a card table in a caravan parked near a haystack on a market gardener’s farm) under the light supervision of matrix legend Olga Taussky-Todd about the “sub-group structure of free products of groups with an amalgamated subgroup.”

So. It’s probably time to talk some math.

The area of math that Hanna Neumann worked in is one that, taken piece by piece, is something anybody could probably understand, but that is so enwrapped in specialized vocabulary that few make the attempt. Neumann’s most enduring work was in the area of “Varieties of Groups”, about which she wrote a book in 1967 that is still a widely referenced classic. To understand the field she was working in, you need to know about mathematical varieties, and to understand *those*, we need to go back to our opening question- just what is an algebra, anyway?

Basically, when we talk in the most general possible terms about an algebra, what we’re talking about is a set of Numbers or elements, with a set of operations and a set of defining equations. The operations are mathematical actions that combine elements from the number set and produce an element from that same set. They are classified on the basis of how many elements from the number set they need to function. Addition is a “binary” operation because, to add two numbers together you need… *two* numbers! Inversion is a “unary” operation because all I need in order to give you the inverse of a number is the *one* number you want the inverse of. There are even “nullary” operations that you don’t need any input for because the output number never changes – for example, if you want to know what number, added to any number, gives that number back again, I can tell you the answer is “0” without you providing me with an input value. That’s just how zero works, regardless of what it’s being combined with.

So much for the operations, what about the defining equations (or axioms as they’re sometimes known)? Well, these are just the special properties that we want this number set, with whatever operations we’ve loaded on top of it, to possess. The most common universal algebra is the Group, which has three operations – a binary one (like addition or multiplication), a unary one (inverse finding), and a nullary one (the existence of an identity element that always results in the return of the initial input) – and three axioms, which you’ve used since elementary school:

Associativity : Whatever your operation is (I’m going to call it *), it has to be true that (a*b)*c = a*(b*c). You’re allowed to do the operations in whatever order you want.

The Identity Axiom: If “e” is your identity element in the set, then x*e=x=e*x.

The Inverse Axiom: If ~x is the inverse of element x, then x*~x = e = ~x*x.

Most of us have lived our lives within the thin slice of the algebraic universe given by the group defined by the integers with the binary operation of addition, and another algebraic structure called a Ring, which allows you to have two binary operations, like the real numbers under multiplication and addition. But there are so many more out there waiting to be played around with, and once you realize that it’s natural to ask the question, if there are multiple possible different things out there called algebras, how do we start grouping them together, and what properties do those collections of algebras possess?

Enter the “variety.” A variety is just a set of algebraic structures, all of which share a given “signature” (set of binary, unary, and nullary operations), and which have the same rules for determining when two things are “equal” to each other, or alternately, what combinations of elements can be counted on to equal 1.

Hanna Neumann’s work explored the properties of Varieties, and in addition to proving some important theorems (such as the proof that all finite free products of finitely generated Hopf-type groups are themselves Hopf-type) she proposed a number of stimulating questions and challenges that motivated the field throughout the late-mid 20th century. She received her PhD in 1944, and was given a D.Sc. by Oxford in 1955 in recognition of her group theoretic publications, and somewhere in the interim from 1939 to 1951 found the time to have five children.

From 1946 to 1958 she lectured at the University College of Hull, where she was instrumental in injecting the joys and challenges of pure mathematics to a curriculum that focused on rote applied problem solving. She brought that same spirit of horizon-opening enthusiasm to her next position, overseeing the pure mathematics curriculum at the University of Manchester, and introducing thereby a generation of young British mathematicians to the most advanced methods and exciting prospects from the European abstract tradition, while she herself expanded her studies of the properties of Varieties that are created by groups which are themselves built from the combination of a finite set of elements.

Finally, in 1963, she and Bernhard were offered positions at the Australian National University, which they accepted, and where they would remain until Hanna’s death in 1971. In Australia, she devoted herself not only to her research and department, but to the wider, and perhaps more vital, question of secondary school curriculum restructuring, and the perpetual problem of preparing a nation’s teachers, all of vastly different ages and experiences, how to cope with the new standards. She was elected in 1966 as Vice-President of the Australian Association of Mathematics Teachers upon the founding of that group, and used her intercontinental experience of different mathematical teaching systems to help guide the creation of a series of pamphlets to help teachers grasp the new mathematical topics being introduced to the Australian curriculum. In recognition of both her contributions to pure mathematics and to Australian education, she was elected in 1969 as a fellow of the Australian Academy of Science.

The last years of her life were spent traveling the world, introducing her ideas (including her recent Hopf proof) to audiences in Germany, Canada, the United States, and France, where her natural ear for languages stood her in good stead. In demand as a speaker, educational reformer, department overseer, and fundamental researcher, with a growing and successful family, she had every reason to expect several more decades of full life when, on tour in Ottawa, she checked into a hospital the evening of November 12, 1971, reporting that she was feeling slightly ill. That night, she slipped into a coma, and died on November 14.

**FURTHER READING:** Neumann’s classic *Varieties of Groups *(Springer Verlag 1967) is not too hard to find, but is not a particularly inviting introductory text to the topic. To get your foot in the door, I’d recommend, as I probably have multiple times in the past, Dummit & Foote’s classic *Abstract Algebra *text, which introduces what you need to know about groups, rings, fields, and various morphisms, from essentially the ground up. Neumann has a few pages in the indispensable *Women of Mathematics: A Biobibliographic Sourcebook *edited by Grinstein and Campbell, but the Australian Academy of Science memorial to her is both more thorough and online!

*Photo credit: Photo of the mathematician Hanna Neumann By Konrad Jacobs, Erlangen – The Oberwolfach Photo Collection, CC BY-SA 2.0 de, via Wikimedia Commons*