# Norman Larrabee Alling

### Born: 8 February 1930 in Rochester, New York, USA

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**Norman Alling**was the son of Harold Lattimore Alling and Merle Kolb. He entered Bard College, Annandale-on-Hudson, New York, in 1948. Bard College was founded in 1860 as St Stephen's, an Episcopal college for men, but in 1928 it became the undergraduate college for Columbia University. However it ended its relationship with Columbia in 1944 so by the time Alling entered it was a private, coeducational institution of higher learning. He studied there for four years obtaining a B.A. in 1952.

After the award of his Bachelor's Degree, Alling entered Columbia University in New York and was awarded his Master's Degree in 1954. Then he undertook research under Robert Taylor but in 1955 he began lecturing in mathematics which he continued at Columbia until 1957. It was on 20 August 1957 that he married Katharine McPherson Page; they had two children Elizabeth Larrabee Alling and Margaret Tilden Alling. In 1958 he was awarded his Ph.D. for his thesis

*On Ordered Divisible Groups*. A paper containing the main results of his doctoral thesis was published in the

*Transactions of the American Mathematical Society*in 1960. Alling did not remain at Columbia University to finish off writing his thesis, however, but rather took up an appointment as an assistant professor at Purdue University in West Lafayette, Indiana, in 1957 after he stopped lecturing at Columbia. Awarded a National Sciences Foundation postdoctoral fellowship, he spent the year 1961-62 at Harvard University, being promoted to associate professor at Purdue in 1962. He had, by this time, begun working in new areas of algebra publishing papers such as

*On exponentially closed fields*(1962),

*An application of valuation theory to rings of continuous real and complex-valued functions*(1963), and

*The valuation theory of meromorphic function fields over open Riemann surfaces*(1963).

It was around this time that Ribenboim and Alling first met. Ribenboim writes in the Preface of [1]:-

Alling lectured at the Massachusetts Institute of Technology from 1962 to 1964 when he was awarded a National Sciences Foundation Senior Postdoctoral Fellowship which enabled him to spend 1964-65 undertaking research at the Massachusetts Institute of Technology. He was appointed to the University of Rochester, in Rochester, New York, in 1965 as an associate professor. Now back in the town of his birth, he was promoted to full professor in 1970. He remained at Rochester for the rest of his career, retiring in 1992 when he was made professor emeritus.Both Norman and I shared a strong interest in the theory of valuations and this topic continues to link us.

David Singerman, reviewing [1], writes that the topics that Alling studied:-

C Earle, reviewing... cover a wide range of mathematics, from the theory of ordered groups to Riemann surfaces. Perhaps the breadth of Alling's interests can be deduced from the titles of his books. One such book, written jointly with N Greenleaf, is 'Foundations of the theory of Klein surfaces'. Klein surfaces are surfaces with a dianalytic structure, and those that are compact represent real algebraic curves in the same way as compact Riemann surfaces represent complex algebraic curves. This led to the publication of another book 'Real elliptic curves' in1981. This is a very different type of book which started with181987^{th}-century work on elliptic functions. The third book 'Foundations of analysis over surreal number fields' appeared in, and includes an account of Conway's theory of surreal numbers. Many of[his]papers are concerned with related topics.

*Foundations of the theory of Klein surfaces*, writes:-

Manfred Knebusch described Alling's approach to his topic in... the authors develop basic function theory on Klein surfaces and explore the relation between compact Klein surfaces and real algebraic function fields. They show that the compact Klein surfaces are related to real algebraic function fields in the same way that the closed Riemann surfaces are to complex algebraic function fields. They also show that every Klein surface can be represented as the quotient of a(perhaps not connected)Riemann surface by a conjugate analytic involution. Klein surfaces are thus natural objects of study, and this book provides a useful introduction to their properties.

*Real elliptic curves*. Here is a short extract from his description:-

The book consists of an introduction and three parts with the headings "Elliptic integrals", "Elliptic functions" and "Real elliptic curves". The first two parts offer a detailed and scrupulous presentation of the historical development of the theory of elliptic integrals and functions in the1819^{th}and^{th}centuries, from Giulio Fagnano and Euler through Legendre, Gauss, Abel and Jacobi to Riemann and Weierstrass. Although these two parts amount to roughly two-thirds of the book and form an effective self-contained whole, they only supply the background for the last part, in which the author deals with the actual theme of the book and also with his own contribution to the theory of real elliptic curves.

*Foundations of analysis over surreal number fields*is described by L Márki:-

A student who was taught by Alling at Rochester in the 1970s wrote:-As indicated by the title, the book aims at laying the foundations of analysis over these fields. This is done in such a way that all the preliminaries are developed in full detail, which makes the book accessible at the graduate student level. Its expository nature is also revealed by the wealth of examples and by the comments comparing various approaches when tackling certain problems. On the other hand, the book contains many new results - in fact, analysis over surreal number fields seems to make its first appearance here.

After he retired, Alling continued to undertake research. For example, in 1995 at a conference on 'Ordered algebraic structure' in Curaçao he presented a paperDuring our four semesters together, we students formed strong friendships with each other that were to continue well past graduation. Just as important, our prolonged contact with the same professor - Norman Alling in my year - gave us a deeper appreciation of the mindset of a professional mathematician.

*Continuous ultraproducts*which appeared in the conference proceedings in 1997.

Finally we note Alling's love for skiing on the slopes in Colorado.

**Article by:** *J J O'Connor* and *E F Robertson*

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