Say you have been facing a maths problem for a very long time and neither you nor your colleagues have been able to solve it so far. Many years after the problem, a computer scientist finds an idea, teams up with a mathematician and a data scientist, and 5 years later, finds a solution. How would you react?
It’s been a glorious decade of Indians making international headlines, and this turns to be another proud feather in the cap. This is because Nikhil Srivastava solved a problem that arose in 1959, and had stayed unsolved since then.
Eminent Indian-American mathematician Nikhil Srivastava has been jointly selected for the inaugural Ciprian Foias Prize in Operator Theory by American Mathematical Society (AMS). Nikhil shared a joint statement along with his friends (i.e. the other awardees), that they would accept the award on behalf of the people whose work contributed to the resolution of the 1959 Kadison-Singer problem.
Nikhil Srivastava. Photo : Wikipedia
FIRST, WHO ARE THE WINNERS?
This isn’t Prof Nikhil Srivastava’s first big award. He has won the George Polya Prize in 2014, and the Michael and Shiela Held Prize in 2021. The Ciprian Foias Prize is the third big award to his name.
He has collaborated in this effort with two of his friends, Adam Marcus and Daniel Spielman. Adam Marcus is the Chair of Combinatorial Analysis at the École Polytechnique Fédérale de Lausanne (EPFL) in Switzerland. Nikhil himself teaches at the University of California, Berkeley; while Daniel Spielman is a Sterling Professor of Computer Science, a professor of statistics and data science, and a professor of mathematics.
WHY ARE THEY GETTING THIS PRIZE?
The award is to recognise their “highly original work’’ that broke down the understanding of matrices using certain methods (like iterative sparsification and interlacing polynomials). Their ideas have provided a tool kit that is expected to have several future applications. That is something because many best mathematicians of the world have not been able to figure it out for the last 50+ years. They also published new constructions of Ramanujan graphs that described the sparse, interconnected data networks, and their work has uncovered a new deep connection between linear algebra, polynomials and graph theory that is surprising to many.
The Ciprian Prize will be presented in January 2022 at the 2022 Joint Mathematics Meeting in Seattle, which is the largest mathematics gathering in the world. It was created in 2020 in the memory of Ciprian Foias, an influential scholar in Operator Theory and fluid mechanics. The current prize amount is US$ 5,000.
SO, EXACTLY WHAT DID SRIVASTAVA, MARCUS AND SPIELMAN DO?
Imagine a math puzzle in the form of a big ball. This ball was meant to be solved by mathematicians and is made of many compartments, some of which are easy and some are hard. Most of the problem was solved by mathematicians, except for one part. And this one part had been unsolvable for 50 years. The trinity of Adam, Nikhil and Daniel brought their own unique tools to solve this one problem that mathematicians so far were oblivious to.
THE 1959 KADISON-SINGER PROBLEM
The original problem that was posed by Kadison and Singer was this: If one had to know about the state of a quantum system, would having complete knowledge of its sub systems help understand this system? If yes, to what extent?
A practical application of this concept was the concept of matrices (yes, the math concept that had brackets and zeros and ones in certain locations). Asking the above question is equivalent to someone asking: Could matrices be broken down into more depth and simplified? If yes, how much?
This problem had a lot of applications in many fields, like engineering, maths (obvio) and quantum physics. If someone knew how to answer this, they would have cracked the biggest problem that these fields faced and hopefully develop better ways to evolve.
The answer to this Kadison problem was that, in some instances the answers were NO. But for other instances, math experts were not so sure.
SO, HOW DID SRIVASTAVA, MARCUS AND SPIELMAN ARRIVE AT THE RESULT?
When global mathematicians were busy solving their math problems and couldn’t solve questions to this Kadison-Singer problem, computer scientist Adam was trying to solve his own different problem. His problem was on similar lines, but different. Adam was trying to reduce a computer network to have less connections, but which could act as effectively as before. This would help them compress data and help in efficient computation.
On second thoughts, this seemed to be very similar to a friend's maths problem, which was also known as the Kadison-Singer problem. When Adam Marcus realised he could solve the maths and computer science problem together, he joined forces with the other two specialists, Daniel Spielman and Nikhil Srivastava to work together, and eventually solved the problem.