Education:

• B.A. Bombay University 1965.
• M.Sc. TIFR-Bombay University 1969.
• Ph.D. TIFR-Bombay University 1970.

Position:

• Joined Tata Institute of Fundamental Research, Bombay, in 1966, and retired from there in 2000.

Post Retirement Positions:

• Visiting Professor at Indian Institute of Technology Bombay from May 2000 to December 2005 and from July 2007 to June 2009.
• Adjunct Professor at Indian Institute of Technology Bombay since July 2009.
• Adjunct Professor at the University of Mumbai-DAE Centre for Excellence in Basic Sciences since January 2009.

Visiting Positions:

• Visiting positions at Mathematisch Instituut (Holland), Purdue University (USA), E.T.H. (Switzer- land), Martin-Luther-University at (Germany), University of Poona (India), Punjab University (India), University of Bonn (Germany), Queen's University (Canada), McGill University (Canada), for varying periods over the last 40 years.

Areas of Specialization:

• Algebraic Geometry and Commutative Algebra

Current Research Interests:

• Subintegrality and seminormality and their relationship with invertible modules and the Picard group.
• The Jacobian problem: If f and g are polynomials in two variables X and Y with coe±cients in a ¯eld of characteristic zero such that the Jacobian of f and g with repect to X & Y is a nonzero constant then are X and Y polynomials in f and g?
• Study of the coe±cients, particulary the coe±¯ent e1 (which is the Chern number), of the Hilbert-Samuel polynomial of a Noetherian local ring with respect to a primary ideal.

Selected Research Papers:

• Unique factorization in power series rings, Inventiones math. 3 (1967) 348-355.
• On the group of automorphisms of a function ¯eld of genus at least two, J. Pure and Applied Algebra 4 (1974) 205-229.
• E®ect of a permissible blowing-up on the local Hilbert functions, Inventiones math. 26 (1974) 201-212.
• A numerical criterion for the permissibility of a blowing-up, Compositio Mathematica 33 (1976) 15-28.
• Embeddings of certain curves in the a±ne plane, Amer. J. Math. 100 (1978) 99-175 (with S.S. Abhyankar).
• Di®erential operators on a hypersurface, Nagoya Math. J. 103 (1986) 67-84.
• Subintegrality, invertible modules and the Picard group, Compositio Mathematica 85 (1993) 249-279 (with L.G. Roberts).
• Picard group and subintegrality in positive characteristic, Compositio Mathematica 95 (1995) 309-321.
• Subintegrality and invertible modules in graded rings, J. Algebra 217 (1999) 73-94.
• On a question of Huneke-Ja®e concerning seminormality, Contemporary Mathematics 390 (2005) 157-164.

Selected Monographs/Lecture Notes/Books:

• Lectures on commutative algebra, Mathematisch Instituut, Amsterdam, 1973.
• S.S. Abhyankar (Notes by B. Singh), Expansion techniques in algebraic geometry, TIFR Lecture Notes, 1977.
• Polynomial variables and the Jacobian problem, Resonance 4 (1999) 20-30.
• Basic Commutative Algebra, book under preparation.

Other Academic Activities:

• Participated in and gave invited lectures at several National and International conferences/seminars; delivered several invited lectures at National and International institutions.