Vijay Kumar Patodi (12 March 1945 – 21 December 1976) was an Indian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who made fundamental contributions to

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

. He was the first mathematician to apply

heat equation In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...

methods to the proof of the

index theorem Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...

for elliptic operators. He was a

professor Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...

Tata Institute of Fundamental Research Tata Institute of Fundamental Research (TIFR) is a leading research Institute under the Department of Atomic Energy of the Government of India. It is a public deemed university located at Navy Nagar, Colaba in Mumbai. It also has a centres in ...

Mumbai (Bombay) Mumbai ( ; ), also known as Bombay ( ; its official name until 1995), is the capital city of the Indian state of Maharashtra. Mumbai is the financial capital and the most populous city proper of India with an estimated population of 12.5 ...

Education

Patodi was a graduate of Government High School,

Guna Guna may refer to: People * Guna people, Indigenous peoples of Panama and Colombia Philosophy * Guṇa, a Hindu philosophical concept * Guṇa (Jainism), a philosophical concept Places * Guna district, in Madhya Pradesh, India ** Guna, Indi ...

Madhya Pradesh Madhya Pradesh (; ; ) is a state in central India. Its capital is Bhopal and the largest city is Indore, Indore. Other major cities includes Gwalior, Jabalpur, and Sagar, Madhya Pradesh, Sagar. Madhya Pradesh is the List of states and union te ...

. He received his bachelor's degree from Vikram University,

Ujjain Ujjain (, , old name Avantika, ) or Ujjayinī is a city in Ujjain district of the Indian state of Madhya Pradesh. It is the fifth-largest city in Madhya Pradesh by population and is the administrative as well as religious centre of Ujjain ...

, his master's degree from the

Benaras Hindu University Banaras Hindu University (BHU), formerly Benares Hindu University, is a collegiate, central, and research university located in Varanasi, Uttar Pradesh, India, and founded in 1916. The university incorporated the Central Hindu College, ...

, and his Ph.D. from the

University of Bombay University of Mumbai is a public university, public List of largest universities and university networks by enrollment, state university in Mumbai. It is one of the List of largest universities and university networks by enrollment, largest univ ...

under the guidance of M. S. Narasimhan and S. Ramanan at the

. In the two papers based on his Ph.D. thesis, "Curvature and Eigenforms of the Laplace Operator" (''Journal of Differential Geometry''), and "An Analytical Proof of the Riemann-Roch-Hirzebruch Formula for Kaehler Manifolds" (also ''Journal of Differential Geometry''), Patodi made his fundamental breakthroughs.

Research career

He was invited to spend 1971–1973 at the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

Princeton Princeton University is a private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the Unit ...

New Jersey New Jersey is a U.S. state, state located in both the Mid-Atlantic States, Mid-Atlantic and Northeastern United States, Northeastern regions of the United States. Located at the geographic hub of the urban area, heavily urbanized Northeas ...

, where he collaborated with

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...

Isadore Singer Isadore Manuel Singer (May 3, 1924 – February 11, 2021) was an American mathematician. He was an Emeritus Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology and a Professor Emeritus of Mathemat ...

, and

Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott function ...

. The joint work led to a series of papers, "Spectral Asymmetry and Riemannian Geometry" with Atiyah and Singer, in which the

η-invariant In mathematics, the eta invariant of a self-adjoint elliptic differential operator, elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbe ...

was defined. This invariant was to play a major role in subsequent advances in the area in the 1980s. Patodi was promoted to full professor at Tata Institute at age 30, however, he died at age 31, as a result of complications prior to surgery for a kidney transplant.

References

External links

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Concise Biography
20th-century Indian mathematicians Differential geometers Indian topologists Tata Institute of Fundamental Research alumni Institute for Advanced Study visiting scholars Vikram University alumni Banaras Hindu University alumni University of Mumbai alumni 1945 births 1976 deaths Scientists from Madhya Pradesh {{asia-mathematician-stub