In News

• Five years ago, in 2018, Dr. Langlands was awarded the Abel Prize, one of the highest honours for mathematicians, for “his visionary program connecting representation theory to number theory”.

About the program

• This program was set in motion in 1967 when Dr. Langlands, then 30 and at Princeton University, wrote a 17-page letter to the French mathematician André Weil with a series of tentative ideas.
• The Langlands Program consists of “very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp”.
• At the heart of the Program is an attempt to find connections between two far-flung areas of mathematics: number theory and harmonic analysis.
• Number theory is the arithmetic study of numbers and the relationships between them. A famous example of such a relationship is the Pythagoras theorem: a2 + b2 = c2.
• Harmonic analysis is interested in the study of periodic phenomena. Unlike number theorists, who deal with discrete arithmetics (like integers), harmonic analysts deal with mathematical objects more continuous in nature (like waves).

Purpose of the program

• In 1824, Norwegian mathematician Niels Henrik Abel proved that it was impossible to have a general formula to find the roots of polynomial equations whose highest power is greater than 4 (e.g., x5 + 2x4 – 5x3 – 9x2 = 0).
• An example of a general formula is the quadratic formula used to solve quadratic equations.
• Around the same time, unaware of Abel’s work, French mathematician Évariste Galois arrived at the same conclusion – and went a step ahead. In 1832, he suggested that instead of trying to find the precise roots of such polynomial equations, mathematicians could focus on symmetries between roots for an alternate route.
• Consider the polynomial equation x2 – 2 = 0. The two roots of x in this equation are √2 and -√2. Now, consider a different polynomial involving one of these roots (say, √2): √22 + √2 = 2 + √2.
• This equation – of the form ɑ2 + ɑ = 2 + ɑ, where ɑ = √2 – holds true for the other root as well: (-√2)2 + (-√2) = 2 + (-√2) = 2 - √2.
• So the two roots of the polynomial x2 – 2 = 0 are symmetric. And a Galois group is a collection of symmetries of the roots of a polynomial equation.
• The Langlands Program seeks to connect every Galois group with automorphic functions, allowing mathematicians to investigate polynomial equations using tools from calculus, and build a bridge from harmonic analysis to number theory.

How program has helped?

• In 1994, Andrew Wiles and Richard Taylor applied Langlands’ conjectures to prove Fermat’s last theorem. This proof had eluded mathematicians for more than three centuries.
• The Program has also helped mathematicians create new automorphic functions from preexisting ones. Such possibilities, they understand, could be crucial to prove the Ramanujan conjectures, many of which remain unsolved.
• One conjecture of the Program is called functoriality. It posits that we can raise the coefficients of g(x) – a, a₁, a₂, etc. – to any integer to create a different automorphic function. That is, the following function, where the coefficient has been raised to an integer k, should also be automorphic:
• The Program has also evolved into its own field of mathematics. One offshoot – called Geometric Langlands – investigates connections between algebraic geometry and representation theory.
• Mathematicians have even conjectured connections between Geometric Langlands and physics.
• The Langlands Program is a mathematical exercise in translation – in building bridges across mathematical cultures with different objects and languages.

Source- The Hindu