Bangalore Srinivasarao Madhava Rao (29 May 1900 – 11 June 1987) was an Indian mathematician and theoretical physicist who served as Professor of Mathematics and later University Professor and Principal at Central College, Bangalore. He held a sustained dual affiliation with the Indian Institute of Science (IISc) from 1935 to 1938, during which he worked with Max Born on Born's nonlinear electromagnetic field theory, producing eight published papers and a doctoral thesis on the subject. Homi J. Bhabha, writing from Bombay in November 1939 having recently returned from Cambridge, wrote to Madhava Rao asking if he, Bhabha, could come to Bangalore to work together on relativistic wave equations. Madhava Rao agreed, and for the next three to four years the two men — along with Harish-Chandra — worked closely on the subject. The mathematical constructs produced by this collaboration — the Bhabha–Madhavarao theory, the Madhavarao algebra and the Madhavarao ring — are named in the physics and mathematics literature. In 1938 he published two papers in the German journal Physikalische Zeitschrift engaging directly with Johannes Stark — Nobel laureate and leading proponent of Deutsche Physik — in defence of Max Born's field theory and modern quantum mechanics. He also served from 1934 as Secretary and Editor of the *Proceedings* of the Indian Academy of Sciences, and held the Lokmanya Tilak Chair in Applied Mathematics at the University of Poona.

Early life and education

Rao was born on 29 May 1900 in Chamrajnagar, in the Kingdom of Mysore. His father, M. Srinivasa Rao, served as a Head Master for over forty years across several middle schools in Mysore State. He attended A.V. School, Chamarajanagar (1905–1912), then Maharaja's College, Mysore (1912–1915), obtaining his Secondary School Leaving Certificate with first class in Mathematics and Science, and then Central College, Bangalore (1915–1919), where he obtained his B.Sc. from the University of Mysore with first class in Mathematics and Science, winning a medal and coming first in the University — the degree certificate bearing the signature of Krishna Raja Wadiyar IV, Maharaja of Mysore, as Chancellor. C.V. Raman, then Palit Professor of Physics at Calcutta, described him as having "a distinct taste for mathematical research" and recommended him strongly for a foreign scholarship to European universities. He was subsequently awarded a scholarship to the University of Calcutta (1919–1921), obtaining his M.Sc. in Pure Mathematics, standing second in the first class and winning a medal.

He returned to Calcutta for advanced research and was awarded the D.Sc. in Mathematics by the University of Calcutta in 1938 (certificate issued 11 March 1939) for his thesis, Contributions to Born's Field Theory.

The D.Sc. was a degree by submitted published works, examined externally. The University of Calcutta appointed Hermann Weyl, then at the Institute for Advanced Study, Princeton, as external examiner. In his report dated 26 January 1938, Weyl described the eight submitted papers as forming "an impressive whole dealing with all aspects of Born's non-linear, electro-magnetic field theory," showing their author to possess "a versatile and imaginative mind, and the mathematical skill necessary for handling such problems successfully." While offering technical criticisms and expressing broader reservations about Born's field theory programme, Weyl concluded that Madhava Rao "has turned many stones which before had been left unturned" and that his work was "solid enough to impart convincing power" to the impression that Born's theory had fundamental limitations — and approved the thesis, recommending conferment of the D.Sc. The original examination report, signed by Weyl from Princeton, is preserved in the Madhava Rao family archive. A covering letter from Weyl to Dr. B.B. Dutt, Controller of Examinations at the University of Calcutta, reveals that the University had simultaneously approached Jacques Hadamard in Paris and Oswald Veblen at Princeton as co-examiners. Although Weyl noted in January 1938 that he had been unable to reach Hadamard, all three examiners ultimately submitted reports; the University of Calcutta Syndicate recorded on 31 May 1938 that it had read "a joint report from Prof. Jacques Hadamard, Professor Oswald Veblen and Professor Hermann Weyl" and resolved to admit Madhava Rao to the degree.

Career

Central College, Bangalore

Madhava Rao joined the engineering college of the University of Mysore as Assistant Professor in 1922, following a brief period as Research Scholar in 1921. In 1938 he was appointed to a newly created Chair of Professor of Applied Mathematics at Central College, Bangalore, where he had previously been a student (1915–1919). The head of the mathematics department at Central College at that time was Prof. K.S.K. Iyengar, who had initiated a programme of modernising the curriculum to include post-graduate courses in advanced mathematics and mathematical physics; Madhava Rao collaborated fully in teaching this new programme. After Iyengar's death in 1944, Madhava Rao assumed sole charge of the department. He was promoted to Professor Class I in 1948, and to University Professor and Principal in 1953. His earliest known research dates to 1923, when he published three papers on algebraic geometry in the Journal of the Indian Mathematical Society and the Bulletin of the Calcutta Mathematical Society.

Concurrently with his professorship, Madhava Rao served as Secretary and Editor of the *Proceedings* of the Indian Academy of Sciences from the Academy's founding in 1934. Writing in September 1955 upon Madhava Rao's departure for Pune, C.V. Raman expressed regret that the Council had not been able to formally recognise "the very valuable services you have rendered during the many years you have been Secretary to the Academy."

Affiliation with the Indian Institute of Science (1935–1938)

In October 1935, Max Born — who had left Germany following the Nazi rise to power and was briefly based in Cambridge — joined IISc at C.V. Raman's invitation as Reader in Theoretical Physics. During Born's residency, Madhava Rao began working at the IISc Department of Physics, an affiliation he sustained for at least two years after Born's departure. All eight of his physics papers published between 1936 and 1938, indexed in the INSPIRE HEP database, carry the dual institutional affiliation of the University of Mysore and the Indian Institute of Science, Bangalore. The NCBS Archives records that "he had worked with Max Born on non-linear field theory."

Several photographs of Born and Hedi Born at Madhava Rao's home in Bangalore, preserved in the Madhava Rao family archives, attest to the personal as well as professional nature of their connection. The family archive also holds correspondence on the headed notepaper of Born's subsequent department, the Department of Natural Philosophy at the University of Edinburgh, where Born held the chair of Natural Philosophy from 1936 to 1953 and for which he was awarded the Nobel Prize in Physics in 1954.

Collaboration with Raman and Bhabha

Under the influence of Born and in collaboration with C. V. Raman and Homi J. Bhabha, Madhava Rao moved into the mainstream of theoretical physics. A letter from Bhabha dated 10 November 1939, written from his family home in Bombay having recently returned from Cambridge, proposes that the two collaborate on calculating the "Compton effect" of X-rays on charged mesons, and asks whether he, Bhabha, could come to Bangalore to continue this work together on relativistic wave equations. Madhava Rao agreed, and for a period of three to four years, the two — together with Harish-Chandra — worked on the topic, though mostly publishing individual rather than joint papers. Madhava Rao also noted that he was the first to prove the inadequacy of a neutrino theory of light proposed by Raman and Nagendranath, and that Raman was "gracious enough to accept the correctness" of the criticism. The physicist E. C. George Sudarshan, in a 1988 tribute published by the Indian Academy of Sciences, confirmed that Madhava Rao's mastery of matrix algebra was the foundation of the joint work with Bhabha, and noted that Harish-Chandra later left for Princeton where he did significant work on Lie groups, becoming one of the leading mathematicians of the twentieth century. The physicist N. Mukunda similarly noted Madhava Rao's influence on Bhabha in a 2011 appreciation published in Current Science.

The Stark Exchange (1938)

In 1938, Madhava Rao engaged in a direct scientific exchange with Johannes Stark — Nobel laureate in physics and the most prominent advocate of Deutsche Physik, the Nazi-aligned movement that sought to marginalise quantum mechanics and theoretical physics in Germany. The exchange was conducted in the pages of the Physikalische Zeitschrift, a major German physics journal of the period, and unfolded across three consecutive papers in the same issue (Vol. 39, Issue 5):

In his paper "Bemerkungen zur Ringstruktur des Elektrons" (Remarks on the Ring Structure of the Electron, pp. 187–189), Madhava Rao challenged Stark's classical model of the electron as an axial ring structure, arguing that Born's modified electromagnetic field equations were the only mathematically consistent path to resolving the divergence problem in electrodynamics. Stark published a response (pp. 189–193) in the same issue. Madhava Rao replied in "Antwort auf Starks Bemerkungen" (Reply to Stark's Remarks, pp. 193–194), received by the journal on 2 September 1937, explicitly defending Werner Heisenberg's theoretical methods and Arnold Sommerfeld's models against Stark's experimentalist critique, and arguing that the advance of theoretical physics — not experiment alone — was essential to scientific progress. The paper concludes with a quotation from Hermann von Helmholtz in defence of theoretical understanding. Both papers are signed "Bangalore, University of Mysore." The family archive holds the original author offprints (Sonderdruck) of both papers.

Later career

Between 1955 and 1960, Madhava Rao served as Professor of Ballistics at the Institute of Armament Technology, Pune. From 1960 to 1965, he held the Lokmanya Tilak Chair in Applied Mathematics at the University of Poona. Towards the close of his life he became an Associate of the Centre for Theoretical Studies at the Indian Institute of Science, Bangalore.

Research

Early work: algebraic geometry and classical mechanics

Madhava Rao's earliest published research dates to 1923, when he published three papers on algebraic geometry including work on cubic curves. His theorem on apolar cubics — investigating a special type of cubic curve first studied by Newton using elliptic function methods — was cited in the literature as "Madhava Rao's theorem on apolar cubics" by T. Ray Pastor in a paper published by the Accademia dei Lincei, Rome, in 1937. He also worked in analytical dynamics, publishing on the theorems of Liouville and Stäckel for separable Hamiltonian systems in 1935 simultaneously in the German journal Zeitschrift für Physik and in Mathematische Annalen, the latter being cited in several German books on mechanics. He published a further series of dynamics papers in 1940.

Born's nonlinear field theory