Died: March 1973 in Germany

**Margaret Hilary Ashworth**'s father was the Assistant Head Postmaster in Halifax in the north of England. Her primary school education was at Haugh Shaw Junior & Infants' School, one of the earliest schools built by the Halifax School Board, which she entered in 1949 when she was five years old and left at the age of eleven. Then she began her secondary education at the Crossley and Porter School for Girls. The Crossley Halifax carpet manufacturers founded the school in 1857 as an orphanage school and later the name of Thomas Porter was added when he made a large financial donation. By the time that Ashworth studied there it no longer served as an orphanage school but simply as a good grammar school. She completed her studies in 1962 having shown an extraordinary talent for mathematics. She then entered St Mary's College of Durham University and embarked on an amazing undergraduate career [1]:-

Her undergraduate career was quite outstanding and she must surely be placed amongst the most brilliant mathematics undergraduates at Durham. She obtained more than twice the marks required for a first class honours degree and, together with an equally successful undergraduate contemporary, Sheila Trelease(now Sheila Greaves), also of St Mary's, set a record in that respect which is unlikely to be surpassed.

Clearly Ashworth's performance meant that a research career beckoned. At Durham she had known Oliver Atkin who had left for a position at Oxford University and the Atlas Computer Laboratory. She now applied to work for a doctorate at Oxford with Atkin as her thesis advisor but before beginning research she took the Diploma in her first year at Oxford [1]:-

Atkin recalls how favourably she impressed her examiners at the first year Diploma oral examinations, with her poise, her speed of response and her sound knowledge of mathematics.

In 1966-67 her supervisor spent a year in the United States with visiting research fellowships at the University of Maryland and the University of Wisconsin. Ashworth went with Atkin spending the year at these two universities working on various problems related to modular forms. Returning to Oxford, she submitted her thesis in 1968 and was awarded a D.Phil. The year in the United States had been highly significant for Atkin who emigrated to the United States in 1970 to take up a position in the University of Arizona. In 1968, the year in which she completed her doctorate, Ashworth married Lieutenant A H Millington of the Royal Electrical and Mechanical Engineers. She was awarded a two-year Science Research Council Fellowship which meant that she could undertake research at any university of her choice. This was very suitable for her since she moved around the country with her husband as he was posted to various different places. She worked at the nearest university to the various locations at which they lived. Then Lieutenant Millington was given a two-year posting to Germany and Margaret Millington went with him teaching mathematics at an Army Education Centre. However, she began to suffer health problems and a brain tumour was diagnosed. By a tragic coincidence, Oliver Atkin's wife Raynor had died a couple of years earlier in the first year after they emigrated.

Margaret Millington had little opportunity to make a significant mathematical contribution. However, ten years after her death in 1983 the London Mathematical Society organised a Durham Symposium on Modular Forms and [1]:-

... the importance of her contribution to the subject was made manifest at the Symposium.

This contribution was made in her doctoral thesis in which she studied three different topics on modular forms. The first of these is contained in a section entitled *Vanishing of coefficient of certain modular forms*. The second topic involved an investigation of subgroups of the modular group and resulted in two papers both published in 1969: *Subgroups of the classical modular group* (in the Journal of the London Mathematical Society) and *On cycloidal subgroups of the modular group* (in the Proceedings of the London Mathematical Society). In *Subgroups of the classical modular group* Millington gives a classification of subgroups in terms of their cusp split and shows that that certain arithmetically possible combinations of type and cusp split do not occur for any subgroup. In *On cycloidal subgroups* she extends results of Hans Petersson (1953) and also investigates inclusion relations among cycloidal groups, especially for a class of such groups constructed by Petersson. The third topic in her thesis is described in [1]:-

Her third thesis topic was an attempt to synthesise and clarify the numerous congruences of Ramanujan, Wilson, Wilton, Kolberg, Lahiri, et al., between the Fourier coefficients of cuspforms and sigma functions(sums of powers of divisors), extending to noncuspforms with poles and to more recondite functions such as sums over divisors in complex quadratic fields.

Millington continued her research while holding the Science Research Council Fellowship but her illness prevented her from classifying the new forms introduced by Atkin and Lehner. The work that she began at this time was completed by other mathematicians in the following few years. Atkin ends his tribute to Millington with these words [1]:-

Margaret was somewhat conservative both as a mathematician and as a person, inclined to speak when spoken to, but then to say her piece clearly and at reasonable length. ... I have no doubt that, had she lived, she would have made exciting original contributions to a field which has at last come into its own again, after nearly a quarter century in the doldrums, and where there are now at least twenty first rate people of her generation working actively.

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

**February 2010**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Millington.html]