There is no question about the fact that mathematical genius Srinivasa Ramanujan has left behind a rich legacy of problems for mathematicians to solve. In his short life of little over 32 years, he reached unimaginable heights. What is surprising is that his mathematics, done over a hundred years ago, finds applications today in areas other than pure mathematics, which were not even established during his time (22 December 1887 – 26 April 1920). Two among these are signal processing and Black Hole physics.
Signal processing
Examples of signals that are processed digitally include obvious ones like speech and music and more research-oriented ones such as DNA and protein sequences. All these have certain patterns that repeat over and over again and are called periodic patterns. For example, a DNA molecule is made up of a 4 bases (Adenine Guanine, Thymine and Cytosine).
Sometimes, a sequence, say AGT, keeps repeating several times in a region of the DNA. In real life, more complex repeating patterns may need to be identified as they bear significance to health conditions. So, in signal processing, one thing we are interested in is extracting and identifying such periodic information.
Identifying and separating the periodic portion is much like using a sieve to separate particles of different sizes. A mathematical operation akin to a sieve is used to separate out the periodic regions in the signal. Some of the best-known methods to extract periodic components in signals involve Fourier analysis. Using Ramanujan Sums for this process is much less known. “A Ramanujan Sum is a sequence like c(1), c(2), c(3) ... This sequence itself repeats periodically... It was thought, by a number of authors before me, to be useful in identifying periodic components in signals, much the same as sines and cosines are used in Fourier analysis,” says P.P. Vaidyanathan who has developed these ideas over the last decade. He is the Kiyo and Eiko Tomiyasu Professor of Electrical Engineering at the California Institute of Technology, U.S.
Communication from far
Prof. Vaidyanathan came across this work in a manner that illustrates how friendly connections play a role in the development of science. Several years ago, mathematicians H. Gopalakrishna Gadiyar and R. Padma, from VIT, Vellore, were studying the twin prime problem when they observed that some arithmetical function which captures the properties of the primes should have a Ramanujan-Fourier Series. They sent their paper to Bhaskar Ramamurthi, Director of IIT Madras, who in turn forwarded the paper to Prof. Vaidyanathan, a friend from his graduate days.
Intrigued by the Ramanujan Sum mentioned in the paper, Prof. Vaidyanathan delved deep into it and developed the concept of “Ramanujan subspaces.” These ideas were further developed by his doctoral student Srikanth Tenneti who showed that using these gave a method that worked better than Fourier analysis when the region of periodicity is short.
“A number of extensions using two- and higher-dimensional generalisations for images and video, and extensions for non-integer (whole number) periods,” are on the cards, according to Prof. Vaidyanathan.
Partitions of a number
Ramanujan’s interest in the number of ways one can partition an integer (a whole number) is famous. For instance, the integer 3 can be written as 1+1+1 or 2+1. Thus, there are two ways of partitioning the integer 3. As the integer to be partitioned gets larger and larger, it becomes difficult to compute the number of ways to partition it. The seemingly simple mathematical calculation is related to a very sophisticated method to reveal the properties of black holes, as has been established by physicist Atish Dabholkar, who is now Director, International Centre for Theoretical Physics in Trieste, Italy, and Assistant Director General of UNESCO.
Ramanujan related this counting problem to some special functions called “modular forms”. A modular form is symmetric, in the sense that it does not change, under a set of mathematical operations called “modular symmetry”. “A geometric analogy for such a function would be a circle which does not change its shape under rotations [circular symmetry],” explains Prof. Dabholkar. “Using this symmetry, Ramanujan and G.H. Hardy found a beautiful formula to compute the number of partitions of any integer.”
Nearly modular forms
In his famous letter to Hardy in 1919, Ramanujan introduced the “mock theta functions” and observed that they were “almost modular”. “A geometric analogy would be a ‘mock circle’ that is nearly circular but with a small blip,” explains Prof. Dabholkar. “It is not easy to explain precisely what a ‘blip’ is, similarly, ‘almost modular’ remained a mystery for close to a century,” he adds.
Following the work of mathematician Sanders P. Zwegers in 2002, in which he introduced “mock modular forms,” giving a precise definition of what “almost modular” means, Prof Dabholkar’s paper with Sameer Murthy and Don Zagier made the connection between mock modular forms and Black Hole physics.
Black Hole entropy
A separate concept in physics, entropy, explains why heat flows from a hot body to a cold body and not the other way around. The results of Ramanujan and Hardy on partitions and the former’s subsequent work on what are called mock theta functions have come to play an important role in understanding the very quantum structure of spacetime – in particular the quantum entropy of a type of Black Hole in string theory, according to Prof. Dabholkar.
Stephen Hawking showed that when you take into account quantum effects, a Black Hole is not quite black, it is rather like a hot piece of metal that is slowly emitting Hawking radiation. Thus, one can associate thermodynamic quantities like temperature and entropy to a Black Hole.
“Mock modular forms are beginning to appear more and more in many areas of physics... Our work has also had unexpected applications in new topics in mathematics such as ‘Umbral Moonshine’, which are quite unrelated to black holes,” explains Prof. Dabholkar.
“It is a tribute to the remarkable originality and power of Ramanujan’s genius that the ideas he created a century ago are now finding applications in such diverse contexts,” he says.
