Shreeram S. Abhyankar - Algebraic Geometry

The Indian Professor Shreeram S. Abhyankar obtained his PhD from Harvard University in 1955, under the supervision of Professor Oscar Zariski. At the time of his passing in 2012, he was a Professor at Purdue University (Indiana) with the title of Marshall Distinguished Professor, a title he received in 1967. He has had an outstanding career punctuated by invitations to prestigious universities and international conferences (Princeton, Harvard, Tata Institute. Kyoto, etc). He was a specialist in algebraic geometry, the author of a dozen monographs and over a hundred research articles. 

Early Life and Education

Prof. Abhyankar's journey into mathematics began at a young age. As a child, his father introduced him to both mathematics and Sanskrit poetry through Bhaskaracharya's treatise on algebra (1150). He pursued his undergraduate studies at the Royal Institute of Science in Mumbai, where he obtained a Bachelor of Science degree in 1951. Demonstrating an early aptitude for mathematics, he went on to complete his Master's degree at Harvard University, under the supervision of the eminent Professor Oscar Zariski, a leading figure in algebraic geometry. In his PhD thesis, Prof. Abhyankar tackled the problem of resolving singularities for surfaces in positive characteristic, a domain where the existing methods in zero characteristic (as developed by Prof. Zariski) did not apply. His work provided a comprehensive resolution theorem for these surfaces, establishing new techniques later extended to higher-dimensional varieties. He was part of a tradition that, through Zariski, Noether and many others, goes back to Newton and Bhaskara, an Indian mathematician from the 12th century.

Career and Research

One of Prof. Abhyankar's notable achievements was his extension of his resolution theorem to arithmetic surfaces and three-dimensional varieties in characteristic greater than five. This work, accomplished a decade after his initial findings, marked a significant milestone in algebraic geometry. His results, combined with the contemporary work of Heisuke Hironaka in zero characteristic, represented the most substantial progress in the field for half a century. Despite the challenges, the resolution of singularities in higher dimensions and positive characteristic remains an open question, underscoring the depth and complexity of Prof. Abhyankar's contributions.

Prof. Abhyankar has opened up many other avenues of research. For example, Abhyankar and Moh's theorem topologically and geometrically characterises smooth polynomial applications. The method of approximate roots that he invented in the 1970's was an innovative technique that involved approximating the roots of polynomials to simplify the process of resolving singularities. By providing a more efficient way to handle the roots of polynomials, the method of approximate roots has become a valuable tool in algebraic geometry, facilitating further advancements in the study of singularities and the structure of algebraic varieties.

In 1978, he was awarded the Chauvenet Prize by the Mathematical Association of America for his article "Historical ramblings in algebraic geometry and related algebra" published in the American Mathematical Monthly. In this article, he defended the "high school algebra" point of view, algorithmic methods and the explicit solution of equations. 

Faithful to the ancient but still living tradition of his native country, in 1976 he played an active part in setting up the Bhaskaracharya Pratishthana, a research institute in Poona.

He has received numerous awards and honours and has been a visiting professor at more than twenty universities in the USA and India, as well as in Germany, Japan, France and Australia. His fame was exceptionally confirmed at the international congress held in Purdue on his sixtieth birthday, which brought together most of the great names in algebraic geometry. 

The University of Angers awarded Professor Sheeram S. Abhyankar the Doctor Honoris Causa title on October 29, 1998