Ayoub was the daughter of
William Lloyd Garrison Williams, also a Canadian and American mathematician, and his wife, pianist Anne Sykes. She was born on February 7, 1922, in
Cincinnati. Although her father was working at
Cornell University in
Ithaca, New York at the time, her mother, originally from Cincinnati, went to her family home in Cincinnati for the births of both Ayoub and her older sister, Hester. In 1924, her father moved to
McGill University in
Montreal, Canada, and she grew up in Montreal. Her first school, in 1928, was "an Italian school in Rome", where her mother was wintering; the family trip to Italy also included the 1928
International Congress of Mathematicians in
Bologna. Later, she attended both English-language and French-language schools in Montreal, including the
Trafalgar School for Girls, from which she graduated young, in 1938. Intent on studying mathematics, but avoiding her father's department at McGill despite earning top admission scores there, she entered
Bryn Mawr College in 1939, where (after the 1935 death of
Emmy Noether) the mathematics department was headed by
Anna Johnson Pell Wheeler. After graduating "at the top of her class", she did a master's degree program at
Radcliffe College. There, she took courses with
Saunders Mac Lane and
Hassler Whitney and, inspired by Mac Lane, decided to focus on algebra, despite Wheeler's preference for
mathematical analysis. Because her Radcliffe master's degree did not have a thesis, she returned to McGill University for a second master's degree, in 1945, the year that her father was starting the
Canadian Mathematical Congress. She completed a Ph.D. in 1947, at
Yale University, with the dissertation
A Theory of Normal Chains. The
Mathematics Genealogy Project lists this work as jointly supervised by
Reinhold Baer and
Nathan Jacobson. In a 2014 interview, she stated that it would have been directed by
Øystein Ore had he not been on leave that year, that instead it was directed by Baer, whom she had visited at the
University of Illinois, and that Jacobson, newly arrived at Yale, served as her outside examiner. ==Career and later life==