In Stanford’s math corner, a building housing hundreds of years of mathematical progress, Professor Kannan Soundararajan is in his element. On any given day, his blackboard quickly fills with expressions and equations, detailing material from the classes he teaches and his research interests. His bookshelf displays several groundbreaking publications by famous mathematicians, all members of the great tradition of unearthing new mathematical insights.
Soundararajan’s track record indicates his own role in continuing this tradition. He recently received the 2011 Infosys Prize in Mathematical Sciences for his work in proving a significant result called the Quantum Unique Ergodicity Conjecture.
Soundararajan’s mathematical inclinations were evident even in his childhood. In middle school, his teachers identified his talent for the subject and put him in touch with a research institute in the city of Madras (now Chennai), India.
He employed his mathematical talent to represent India at the 1991 International Math Olympiad, winning a silver medal.
Soundararajan’s teachers encouraged him throughout high school to pursue mathematics, and it was in this environment that his specific mathematic focus emerged: analytic number theory.
Analytic number theory is the study of integers using the techniques of calculus and analysis — a field of mathematics pursued by many of Soundararajan’s mentors. One of Soundararajan’s most influential instructors, Ramachandran Balasubramanian proved in 1992 that every sufficiently large integer can be written as the sum of 19 fourth powers.
Soundararajan’s mentorship by leading mathematicians, combined with his own affinity for analysis, steered him toward analytic number theory. According to Soundararajan, the guiding problem in modern number theory is the notoriously unresolved Riemann hypothesis, and much of his mathematical effort is devoted toward developing the tools needed to prove or disprove the hypothesis.
Filling much of Soundararajan’s blackboard, for instance, are L-functions, which are closely related to the key players in the Riemann hypothesis. For distilling the characteristics of these functions, and providing insight into how prime numbers are distributed, Soundararajan received the 2003 Salem Prize and the 2005 SASTRA Ramanujan Prize.
For the most part, Soundararajan describes his mathematical approach as relatively streamlined.
“My own strengths lie in attacking the problem itself and developing tools unique to solving the particular problem,” he said.
Nevertheless, he characterized “eureka” moments as the almost serendipitous instances when two disparate conjectures connect to form a proof.
This idea of looking beyond the immediate scope of a problem to make incisive connections played a key role in Soundararajan’s most recent award-winning advancement.
To the seasoned academic, the Quantum Unique Ergodicity Conjecture might sound like more of a physics problem. In fact, quantum ergodicity is a branch of physics that aims to apply quantum mechanical laws to the macroscopic scale of classical mechanics, another famous but unresolved problem.
This conjecture in particular examines how the shape of their enclosures influences waves. In place of waves, Soundararajan offered the example of pool balls enclosed on a frictionless pool table. If a pool ball is hit in certain directions, it will bounce in one consistent, limited path around the pool table. Hitting it in other directions, however, will cause the ball to bounce in a more disorderly manner, covering paths that span more of the pool table.
While Soundararajan is not a physicist, he was exposed to this problem as a graduate student at Princeton University. The experience eventually caused him to realize that a major portion of this idea could be equated to properties of shapes derived from L-functions, a subject squarely in his area of mathematical expertise. With Roman Holowinsky of the University of Toronto, he was able to prove that for a large class of enclosure shapes, the “pool ball” never gets stuck in one track, and its path evenly spans the area of the “pool table.”
In an email to The Daily, Soundararajan noted the surprising nature of this melding of physics and abstract mathematics.
“The connection with quantum chaos is a feature that has made this problem attractive to many people,” he said. “It’s a little unusual to find such a connection which has genuine interest for people working in very different areas.”
The groundbreaking nature of such results runs parallel to his continued enjoyment of the mathematical process. Soundararajan’s crucial and counterintuitive realization that the Quantum Unique Ergodicity Conjecture could be reformulated to a problem involving L-functions exemplifies his favorite aspect of mathematics.
“In working on math problems, often one starts out in the dark,” Soundararajan said. “For me, the most fun is when there is a key piece to be found, and once one finds the rest, the proof falls easily into place.”