Shreeram Shankar Abhyankar

Born: 22 July 1930 in Ujjain, India
Died: 2 November 2012 in West Lafayette, Indiana, USA

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Shreeram Shankar Abhyankar's parents were Shankar Keshav Abhyankar and Uma Tamhankar. Although Shreeram (known to his friends as Ram) was born in Ujjain, he only spent the first two years of his life there. His father taught mathematics in Ujjain from 1928 to 1932 but then moved to Gwalior where he was a mathematics teacher at a college. Shreeram was brought up in Gwalior where his father Shankar later became principal of the college. Shreeram was the second oldest of his parents' six surviving children having two sisters and three brothers. Yvonne Abhyankar writes [5]:-
Mathematics was the household business, and Ram was surrounded by it since he was a child. Ram fell in love with mathematics as soon as he learned to count and would spend hours on end in the pursuit of further knowledge. Ram became so obsessed with mathematics at a young age that it worried his father; once Ram's father locked up his maths books. This disturbed young Ram greatly; with the help of his uncle he would wake at the crack of dawn before anyone else and sneak away and unlock the hidden treasure.
Later in life he told Avinash Sathay that [8]:-
... as a young child he was weak and suffered from many ailments. This continued until he discovered mathematics. Once he started reading mathematics, he did not get sick again. Even when sick, he could push aside the pain and get well by immersing in mathematics.
Shreeram attended school in Gwalior and, after graduating, went to the Royal Institute of Science of Bombay University intending to major in physics. It may seem strange that he wanted to major in physics when he was passionate about mathematics but, it appears, he did this because his father was a mathematician. However, while studying at the Royal Institute of Science, he also attended mathematics lectures at the Tata Institute of Fundamental Research. In particular, he attended lectures by Marshall Stone, who was visiting from Chicago, and this experience persuaded him that he must study outside India. He was also influenced by Damodar Dharmananda Kosambi (1907-1966), who had studied under George David Birkhoff, and by Pesi Masani (1919-1999), who had studied for his Ph.D. advised by Garrett Birkhoff. Kosambi strongly advised Abhyankar to make a career in mathematics while Masani, who had connections in Harvard, managed to arrange for Abhyankar to be admitted there to study for his Ph.D.

He could not go to Harvard without financial support, so he applied for funds from various Indian sources until he believed he had enough to live on in the United States. He was awarded a B.Sc. in Mathematics from Bombay University in 1951 and prepared to travel. He arranged a free passage on a ship sailing to the USA but took ill on the voyage and was put ashore in England where he recovered in hospital over a period of two months. Once better, he continued to the United States but the semester was more than half finished before he reached Harvard. He arrived in the mathematics department on Saturday morning and few people were about. The only mathematician who was there that morning was Oscar Zariski, so Abhyankar had his first meeting with Zariski who immediately began asking Abhyankar what courses he would take. Abhyankar had already talked to Masani before leaving India and been advised by him on suitable courses but Zariski, quickly realising Abhyankar's potential, suggested the most advanced courses.

Abhyankar was awarded a Master's Degree by Harvard in 1952 and continued to study there for his doctorate advised by Zariski. He received a Ph.D. from Harvard in 1955 for his thesis Local Uniformization on Algebraic Surfaces over Modular Ground Fields. Steven Dale Cutkosky explains in [5] the background to Abhyankar's thesis:-

Abhyankar's interest in resolution of singularities began when he was a graduate student at Harvard in the mid-1950s. This was a subject that his advisor and mentor Zariski was particularly interested in. Zariski had discovered the general definition of nonsingularity over all fields and in mixed characteristic: the local ring of a nonsingular point is a regular local ring. Zariski had also proven resolution of singularities of three-dimensional varieties over an algebraically closed field of characteristic zero and proven local uniformization in any dimension over an arbitrary field of characteristic zero, introducing general valuation theory into the subject. Zariski was very interested in the question of resolution of singularities of positive characteristic surfaces and mentioned this to Abhyankar as an important problem which was probably too difficult for a Ph.D. problem. Abhyankar became fascinated with this problem, and after a tremendous effort solved it as his Ph.D. thesis.
Although he produced remarkable results while a graduate student at Harvard, nevertheless these were difficult times financially for the young man. One of the grants that he had been promised when he applied in India did not materialise and this put him in considerable difficulty. However, after one year he was awarded financial support by Harvard and his financial problems were eased. Around the time that Abhyankar was a research student working in algebraic geometry, Alexander Grothendieck was taking a totally new approach to the topic. David Mumford explains how Abhyankar was unconvinced by the direction Grothendieck was taking the topic:-
I first met Ram when I was a graduate student several years behind him. We were both studying under Oscar Zariski. ... We were good friends, but our paths diverged. Grothendieck came to visit, and I found his new setting for algebraic geometry very congenial. Ram did not. Ram had an independent streak that always led him to look at problems from his unique point of view, often different from that of the crowd. A small example concerns the words 'analytic geometry', which stood at that time for the high school study of conic sections and their equations. Ram said nonsense, algebraic geometry is the study of the geometry of algebraic equations, so analytic geometry must be the study of the geometry of all real or complex analytic equations. His opinion won the day, and this has become common usage. Not so in the case of the value of Grothendieck's schemes, on which he fought the consensus all his life. Our tastes in mathematics were always different, but I admired (and admire) his traditional algebraic approach, which he pursued with such great skill and insight. Few could match him in polynomial calculations, for example, in the construction of extensions of curves and number fields with given Galois group.
After completing his Ph.D. in 1955 he was appointed as an Instructor in Mathematics at Columbia University. One of the students he taught soon after arriving at Columbia was Yvonne Kraft. She explains in [5] how their relationship developed:-
After some weeks he suggested we go out, and then for a time we went out quite often. Of course, as my friends tell me, this was a highly improper thing to do. Eventually we drifted apart. After some time I heard that Ram had been in a car accident in Maine, so I wrote him a short letter hoping that he would soon be well. Somehow we continued corresponding. He had, by that time, begun teaching at Cornell. I was by then a graduate student at Boston University. Since Ram had studied at Harvard, getting his Ph.D. under Oscar Zariski, he came frequently to Boston to see Zariski, and since I was also there, he visited me. In this way our courtship and lifelong friendship began.
On 5 June 1958, he married Yvonne Kraft in New York City. By this time, following his appointments at Columbia and Cornell, he was an associate professor at Johns Hopkins. Shreeram and Yvonne Abhyankar had two children: a son, Hari, born in 1970, and a daughter, Kashi, born in 1973. Hari went on to obtain a Ph.D. in Operations Management from the Massachusetts Institute of Technology in 1999 and Kashi obtained a Ph.D. in Mathematics from Berkeley in 2001. In 1963 Abhyankar was appointed to Perdue University as a full professor and he was named Marshall Distinguished Professor of Mathematics at Perdue in 1967.

Many of the references quoted give details of his mathematical contributions. Here is Balwant Singh's overview in [10]:-

Abhyankar's style of work, according to my observation, was to work thoroughly in one area of algebraic geometry for a few years, making substantial and deep contributions to it, and then move on to a different area or return to one of his earlier favourites. Some specific areas encompassed in his vast research work are resolution of singularities, tame coverings and algebraic fundamental groups, affine geometry, enumerative combinatorics of Young tableaux and Galois groups and equations.
Let us quote from several of the people who knew Abhyankar and his approach to mathematics. Avinash Sathaye writes [3]:-
Abhyankar had a unique perspective of mathematics. He often rebelled against "fancy mathematics", demanding that all theorems should have detailed concrete proofs. He also believed that papers should spell out all the necessary details and he practiced this rigorously. As a result, several of his papers are difficult to read, because you have to keep your concentration on every little detail that he has laid down. He would often say that the proofs should be so logical that a computer should be able to verify them! He also had a sense of poetry and rhythm in his papers. He would create multiple subsections with matching words and equal number of subitems, so that the paper had a natural symmetry. Sometimes, he would spend enormous amount of time to create such intricate structures.
Steven Dale Cutkosky writes [1]:-
Professor Abhyankar was a charismatic man, and an excellent speaker, who could mesmerise an audience. He had a special way of talking with people about mathematics. He would insist that people explain what they were doing in elementary terms. Often when you began talking with him you realised how poor your understanding was, but at the end of the conversation you had a much deeper knowledge. Throughout his mathematical life, he took polynomials and power series, and related concepts such as determinants and discriminants as the focus of his interest, only considering the most fundamental and important problems. He liked to see things in the simplest way possible, without affectation. I have always admired his strength, being willing to stand alone if necessary, following what he believed in.
Rajendra V Gurjar writes [4]:-
The total body of his work (research papers, books, a very large number of conference lectures around the world, . . . ) is truly staggering. It shows his tremendous hard work, technical abilities and total passion for mathematics. This in itself is greatly inspiring, but he will always be remembered for his indomitable spirit, a strong patriotic feeling towards his Indian roots, his confidence in his knowledge and a just pride in knowing the value, and place, of his research work in algebraic geometry. It can be said that his interests lay more in classical mathematics. Perhaps I am wrong, but I think he did not even once use tensor product in his research! One of his motto was never to use a result whose proof he had not read. He broke this rule the first time when he used the classification of finite simple groups.
Not everyone appreciated Abhyankar's style, however, particularly his love of arguing with others. In particular he loved to get into deep arguments with other professors when he was at the University of Pune and this annoyed the vice-chancellor so much that he banned Abhyankar from the premises. Dinesh Thakur writes [5]:-
Provoked by these circumstances and also recognising the need for a new, independent research institute in India, Abhyankar then founded "Bhaskaracharya Pratishthan." In May 1977 I attended a summer school under the auspices of this new institute, and I remember being quite impressed by Abhyankar's ability to penetrate to the heart of the matter directly.
Thakur goes on to describe being with Abhyankar [5]:-
Later, during my graduate years, whenever he came to give a seminar he would take me for dinner and we would talk for hours. He loved to gather an audience, tell anecdotes, and whenever possible, stir up a heated debate. Even for mathematical questions, he seldom gave straightforward answers. Instead, he preferred long rambling discourses, often unfocused and tangential, where he elaborated on related things and mixing mathematics with other things. He would not necessarily follow a logical or linear path, but believed in a "repeat, meditate, and meaning will be revealed to you" philosophy of old Indian masters speaking in sutra/mantra. But these "mathematical ramblings" were full of interesting insights and spontaneous, original viewpoints. In his book, 'Lectures on Algebra', one single lecture is four hundred pages! For him, research developed through discussion; he always preferred to visit, call up, and discuss mathematics in person. He genuinely enjoyed arguments and was proud that he could argue from any stance. Often he would quickly size up the viewpoint of the listener and then take the opposite side. It was a cultivated, competitive sport for him, and he was ready to fight the battle from both sides! Though he often went to extremes, he would often be speaking "deep truths" (it is the hallmark of any deep truth that its negation is also a deep truth: Bohr). Once he started working on something, he became energized and completely devoted himself to his work. I remember the day before his son's wedding he became so engrossed in our mathematical discussions at his house that he totally ignored the wedding party and even got angry when the relatives tried to involve him. Finally, to avoid trouble in his household, I had to escape under some pretence!
Abhyankar has received many honours for his remarkable mathematical achievements. These include the McCoy Prize from Purdue University, the Lester Ford Prize and the Chauvenet Prize from the Mathematical Association of America, a Medal of Honour from the University of Valliadolid, Spain, and a Medal from the University of Brasilia, Brazil. He was given the honorary title of Vidnyan Sanstha Ratna from the Institute of Science, Mumbai. The University of Angers, France, awarded him an honorary doctorate in 1998, and, for this occasion, Heisuke Hironaka wrote:-
Your long and powerful works deserve far more than the honorary doctorate you are receiving. Even so, I am happy to hear the good news. Your originality has been a gold mine for many other algebraic geometers, including myself. Now the mined gold is receiving rays of sunlight, facets after facets ...
Abhyankar was elected a Fellow of the Indian National Science Academy (1987) and a Fellow of the Indian Academy of Sciences (1988). The American Mathematical Society introduced a new honour of 'Fellow of the American Mathematical Society' in 2012 and Abhyankar was among the initial list of those receiving this honour on 1 November 2012. Several conferences have been organised in his honour, such as those at Purdue University in 1990 (to celebrate his 60th birthday), 2000 (to celebrate his 70th birthday), 2010 (to celebrate his 80th birthday), and 2012. A conference was also held in Pune, India, in December 2010 in his honour.

Finally let us mention his interests outside mathematics. Balwant Singh writes [11]:-

Two other passions of Abhyankar were Marathi and Sanskrit languages and Indian mythology. His long stay in India in the 1970s was partly influenced by his desire to have his children learn Marathi language and culture in a genuinely desi environment. His knowledge of Indian mythology was immense and it was not hard to detect his passion for it even during his conversations in mathematics.
Abhyankar suffered a heart attack in his home while sitting at his desk working on mathematics on the morning of Friday 2 November 2012. He was rushed to St Elizabeth's Hospital in Lafayette where he was pronounced dead. His funeral service was held on Thursday 8 November at Soller-Baker West Lafayette Chapel.

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

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