Vijay Kumar Patodi

Vijay Kumar Patodi (born March 12, 1945 in Guna , Madhya Pradesh ; † December 21, 1976 in Bombay , Maharashtra ) was an Indian mathematician who dealt with topology and differential geometry .

Life

Patodi attended school in Guna and studied mathematics at Vikram University in Ujjain up to a bachelor's degree. For his master’s degree, he moved to Banaras Hindu University , where he made his master’s degree in mathematics in 1966. He then spent a year at the Center for Advanced Study at the University of Bombay , then in 1967 he went to the Tata Institute of Fundamental Research in Bombay, where he remained until his death. From 1971 to 1973 Patodi conducted research at the Institute for Advanced Study in Princeton with Michael Francis Atiyah and also worked with Isadore Singer (at MIT ) and Raoul Bott . Back at the Tata Institute, he was appointed Associate Professor in 1973 and Professor in 1976. At that time, his health deteriorated, which was already affecting him during his studies. He died at the age of only 31 from complications from a planned kidney transplant, which was planned as a last resort for a life-threatening illness from which he had suffered for several years.

Scientific work

As a student he had already attended courses at the Tata Institute and in 1969 became aware of a conjecture in a paper by Henry McKean and Isadore Singer (which dealt with an alternative approach to Gauss-Bonnet's theorem ), which he proved in his dissertation . In 1971 he received his PhD from the University of Bombay with the thesis Heat equation and the index of elliptic operators with MS Narasimhan and S. Ramanan , both from the Tata Institute. In it, the heat conduction equation was used for the first time to prove the Atiyah-Singer index theorem , which makes statements about the topology of manifolds from the study of elliptic differential operators like in the Laplace equation , which can be viewed as a stationary limit case of the heat conduction equation . The dissertation resulted in two publications, both published in the Journal of Differential Geometry: Curvature and the eigenforms of the Laplace operator (Vol. 5, 1971, pp. 233–249) and An analytic proof of the Riemann-Roch-Hirzebruch Theorem for Kaehler Manifolds (Vol. 5, 1971, pp. 251-283).

During his years at Princeton, in their joint work with Atiyah and Singer, he developed Spectral Asymmetry and Riemannian Geometry 1-3 (Atiyah, Patodi, Singer Mathematical Proceedings Cambridge Philosophical Society, Vol. 77, 1975, p. 43, Vol. 78, 1975 , P. 403, Vol. 79, 1976, p. 71 and in Bulletin of the London Mathematical Society, Vol. 5, 1973, pp. 229-234) a definition for the -invariant (or Atiyah-Patodi-Singer invariant ). Further work by Patodi concerns the Riemann structure and triangulation of manifolds and a combinatorial formula for the Pontryagin classes . ${\displaystyle \eta }$

In 1974 he was invited speaker at the International Congress of Mathematicians in Vancouver (Riemannian structures and triangulations of manifolds).

literature

• Atiyah, Narasimhan (Ed.) Collected Papers of VKPatodi, World Scientific, 1997
• Atiyah: The heat equation in riemannian geometry. Séminaire Bourbaki 1973
• Atiyah, Bott, Patodi: On the Heat Equation and the Index Theorem. Inventiones Mathematicae , Vol. 19, 1973, pp. 279-330

Web links

• John J. O'Connor, Edmund F. Robertson :  Vijay Kumar Patodi. In: MacTutor History of Mathematics archive .