Maths that won Fields Medal: number theory diversified, optimal transport

India-origin Akshay Venkatesh takes number theory to new disciplines, Germany’s Peter Scholze works on ‘perfectoid spaces’

Written by Amitabh Sinha | Pune | Updated: August 3, 2018 8:58:18 am
Maths that won Fields Medal: number theory diversified, optimal transport Italy’s Alessio Figalli, right, Germany’s Peter Scholze, center, and Australian Akshay Venkatesh, pose for a photo during the 2018 International Congress of Mathematicians in Rio de Janeiro, Brazil, Wednesday, Aug. 1, 2018.(AP Photo/Silvia Izquierdo)

On Wednesday, four mathematicians, including India-born Australian Akshay Venkatesh, were awarded the Fields Medal, often described as the “Nobel prize of mathematics” the most prestigious award in the subject, at the once-in-four-years event, International Congress of Mathematicians, in Rio de Janeiro. The Fields Medal is awarded to a maximum of four mathematicians, all below the age of 40; the 60 winners so far include one more India-born mathematician, Manjul Bhargava, in 2014. A look at the work that fetched the medal to this year’s winners:

Akshay Venkatesh, 36

The Stanford University professor, whose family moved from New Delhi to Perth when he was two years old, is a number theorist but has also contributed to diverse mathematics disciplines, often using amalgamated techniques.
“Because the oeuvre of Akshay Venkatesh is so diverse, a complete overview is not possible in a short space,” says the four-page profile by the International Mathematical Union (IMU) that gives away the prize. Venkatesh himself says in a video, “I think just manipulating numbers makes me feel happy.”

A Stanford University profile says one substantial area of his work has been finding more ways in which “homogenous dynamics” can be used in number theory. “For example, he describes a ball bouncing inside a triangle when the ball doesn’t slow down. His math asks questions about what spaces the ball avoids or prefers and how this changes if the triangle’s sides are curved. He then uses those ideas to solve problems in number theory,” it says.

“Most mathematicians are either problem-solvers or theory-builders. Akshay Venkatesh is both. What is more, he is a number theorist who has developed an unusually deep understanding of several areas that are very different from number theory. This breadth of knowledge allows him to situate number theory problems in new contexts…,” the IMU profile says.

Read |  Who is Akshay Venkatesh?

Peter Scholze, 30

Scholze, who works at the University of Bonn, is one of the youngest ever winners. He was being considered a certain winner for the last few years. “Peter Scholze possesses a type of mathematical talent that emerges only rarely,” says his profile by IMU.

Scholze too is a number theorist, though he works mainly on the algebraic side. When a doctoral student, he formulated the concept of “perfectoid spaces” that is considered groundbreaking work in algebraic geometry.
One of Scholze’s works involved finding integer solutions to equations such as y^2 = x^3 – x, says Debargha  Banerjee of IISER Pune. “There are a few very simple integer solutions to this, but in general integer solutions to more complicated versions of this equation are extremely difficult to find. Scholze found a novel way of finding these solutions which are immensely important for several branches of mathematics,” he says.

Alessio Figalli, 34

The Italian mathematician has published over 150 papers by age 34, more than many accomplished mathematicians manage in their entire career. His main contribution has been in “optimal transport”, a concept that has been probed for more than 250 years now. At a basic level, it means finding the most efficient and economical ways to transport objects from one place to another. It involves complex mathematics and finds uses in physics, biology, economics and even the financial markets.

Read | Indian-origin mathematician Akshay Venkatesh wins Fields medal

Fields Medal Caucher Birkar

Caucher Birkar, 40

A Kurd from Iran, he has sought political asylum in Britain. His main contribution has been in birational geometry, a branch of algebraic geometry. Specifically, he worked on polynomial equations, which can be of infinitely many kinds, containing different variables and exponentials. Mathematicians try to find general solutions to families of such equations, and based on similarities of characteristics of such general solutions, the equations can be classified.
Birkar’s work has helped in bring out common characteristics in many of these seemingly unrelated polynomial functions and in categorising them in groups.

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