Veeravalli S. Varadarajan

1937 - 2019

Veeravalli S. Varadarajan: Architect of Modern Representation Theory

Veeravalli Seshadri Varadarajan (1937–2019) was a titan of 20th-century mathematics whose work bridged the elegant abstractions of Lie groups with the rigorous demands of quantum physics. A central figure in the "Indian School" of mathematics that rose to global prominence in the 1960s, Varadarajan spent over half a century at the University of California, Los Angeles (UCLA), where he became one of the world’s foremost authorities on representation theory and the mathematical foundations of supersymmetry.

1. Biography: From Bangalore to Los Angeles

Early Life and Education

Born on May 18, 1937, in Bangalore (now Bengaluru), India, V.S. Varadarajan displayed early mathematical precocity. He attended Presidency College in Madras (Chennai), earning his B.A. (Honors) in 1957. His intellectual trajectory was set when he joined the Indian Statistical Institute (ISI) in Kolkata for his doctoral studies.

At ISI, he became part of a legendary cohort often referred to as the "Famous Four"—alongside K. R. Parthasarathy, R. Ranga Rao, and S. R. S. Varadhan. Under the mentorship of the great statistician C. R. Rao, this group transformed ISI into a global hub for probability and harmonic analysis. Varadarajan completed his Ph.D. in 1960 at the age of 23.

Academic Career

After a brief period as a researcher in India and visiting positions in the United States (including the Institute for Advanced Study in Princeton), Varadarajan joined the faculty at UCLA in 1965. He remained there for the rest of his career, serving as a Professor of Mathematics and eventually Professor Emeritus until his death on April 27, 2019.

2. Major Contributions: Symmetry and Quantum Reality

Varadarajan’s work is characterized by its breadth, moving from pure probability to the deep intersections of mathematics and theoretical physics.

Probability on Topological Spaces: In his early career, he made fundamental contributions to measure theory, specifically regarding measures on topological spaces. His work provided a rigorous framework for understanding how probability behaves in infinite-dimensional settings.
Representation Theory of Lie Groups: This was his magnum opus. Representation theory is the study of symmetry; it explores how algebraic structures (groups) act on vector spaces. Varadarajan extended the work of Harish-Chandra, focusing on semisimple Lie groups. He was instrumental in developing the "discrete series" representations, which are essential for understanding the spectral analysis of symmetric spaces.
Mathematical Physics and Supersymmetry: Varadarajan was a pioneer in providing a rigorous mathematical basis for quantum mechanics. Later in his career, he focused on Supersymmetry (SUSY), the theory that every fundamental particle has a "superpartner." He developed the theory of supermanifolds and Lie superalgebras, translating the intuitive ideas of physicists into the precise language of mathematics.
The VRP Theorem: Along with Parthasarathy and Ranga Rao, he developed the "Varadarajan-Ranga Rao-Parthasarathy" theorem, which provides a construction of the discrete series representations of semisimple Lie groups.

3. Notable Publications

Varadarajan was a prolific author whose textbooks are considered bibles in their respective fields.

Geometry of Quantum Theory (1968, 1970): A two-volume set that remains a definitive reference for the mathematical foundations of quantum mechanics, particularly the role of Hilbert spaces and symmetry groups.
Lie Groups, Lie Algebras, and Their Representations (1974): This graduate-level text is celebrated for its clarity and depth, serving as the standard introduction for generations of mathematicians.
Harmonic Analysis on Semi-simple Lie Groups (1989): A comprehensive treatment of the field, building on the legacy of Harish-Chandra.
Supersymmetry for Mathematicians: An Introduction (2004): This work bridged the gap between the physics community’s use of supersymmetry and the rigorous requirements of pure mathematics.
Reflections on Quanta, Symmetries, and Supersymmetries (2011): A more reflective work detailing the historical and philosophical evolution of these ideas.

4. Awards and Recognition

While Varadarajan was known for his humility, his contributions were widely recognized by the global scientific community:

Guggenheim Fellowship (1968): Awarded for his early work in representation theory.
The Lars Onsager Medal (2012): Awarded by the Norwegian University of Science and Technology for his outstanding contributions to mathematical physics.
Honorary Doctorate, University of Geneva (1998): Recognized for his profound impact on harmonic analysis.
Fellow of the American Mathematical Society (AMS): Inducted as part of the inaugural class of fellows.
The 80th Birthday Conference (2017): UCLA hosted a major international conference in his honor, titled "Symmetries in Mathematics and Physics," reflecting his dual legacy.

5. Impact and Legacy

Varadarajan’s legacy is twofold: he was both a discoverer and an expositor.

He played a crucial role in the "Langlands Program," a vast web of conjectures connecting number theory and representation theory. By clarifying and extending the work of Harish-Chandra, Varadarajan made complex theories accessible to the broader mathematical community.

His work on the "p-adic" aspects of physics and his exploration of "non-Archimedean" space-time have opened new doors in string theory and quantum gravity. Beyond his research, he was a beloved mentor, known for his "Socratic" teaching style and his ability to see the unity between disparate branches of science.

6. Collaborations and Intellectual Lineage

Varadarajan was a deeply collaborative researcher.

The ISI Cohort: His lifelong friendship and collaboration with K. R. Parthasarathy, R. Ranga Rao, and S. R. S. Varadhan (a Abel Prize winner) defined a golden era of Indian mathematics.
Harish-Chandra: Although never his student in a formal sense, Varadarajan was the primary steward of Harish-Chandra's legacy. He spent years editing the Collected Papers of Harish-Chandra, ensuring that the work of the greatest Indian mathematician since Ramanujan was preserved and understood.
Donald Babbitt: A long-time collaborator at UCLA with whom he explored the intersection of differential equations and representation theory.
Students: He supervised numerous Ph.D. students who went on to hold chairs at major universities worldwide, including the likes of Jing-Song Huang and others in the field of harmonic analysis.

7. Lesser-Known Facts

Interest in Philosophy: Varadarajan was deeply interested in the history of Indian science and philosophy. He wrote several essays on the history of mathematics in India, seeking to contextualize modern discoveries within a longer historical continuum.
The "Euler" Connection: He was a great admirer of Leonhard Euler. In 2006, he published Euler Through Time: A New Look at Old Themes, a book that re-examined Euler’s work through the lens of modern mathematics.
Polymathic Tendencies: Despite being a "pure" mathematician, he was an avid reader of classic literature and was known to quote Sanskrit poetry and Western classics with equal ease during his lectures.
A "Mathematically Driven" Marriage: His wife, Veda Varadarajan, was often seen at his side during international conferences; they were known in the UCLA community for their warmth and for hosting "mathematical salons" at their home.