Langlands quest to turn structures into numbers

13 February 2020

A normal day for a mathematician like Dr Kei Yuan Chan (Croucher Fellowship, 2015) is thinking of explanations for general phenomena, then trying to prove them.

Usually this begins by looking at a mathematical object from a book and then trying to compute different elements of it, such as its rotation or reflection. Then he takes what he has learned from that mathematical shape and tries to apply it to another model to generalise the phenomena. Once that step is achieved he sets to work to prove it.

This process can be used to approach many questions in mathematics. Chan, alongside the world’s top mathematicians, has been problem solving the Langlands program, a set of far-reaching conjectures linking number theory and geometry. “This program is so large, I can only hope to understand some aspects of it,” said Chan, an investigator at the Shanghai Center of Mathematical Science at Fudan University, China.

The Langlands program, first proposed by the American-Canadian mathematician Robert Langlands in 1967, seeks to prove that it is possible to translate a structure from a linear representation to numbers. The key to this is decomposing the spatial representation into individual elements. From there, mathematicians can translate that information into number theory, involving straightforward arithmetic.

Chan has approached the Langlands program from a branch of mathematics called representation theory, which studies abstract algebraic structures by linearly representing their spaces. He uses the principles of symmetry to solve unknowns and compose the space of these abstract structures.

Starting with core representations, Chan can then decompose the spaces into smaller parts to separate the elements of the composition. These individual elements are interesting in themselves, but Chan seeks to identify the homological properties that dictate the decomposition of the elements.

While the decomposed elements can be identified, the principles governing the decomposition remain undiscovered. These properties have not been predicted before. Understanding how to decompose the representative spaces is “one of the fundamental questions”, according to Chan. Achieving this would be a step towards understanding the whole Longlands program.

Chan likens it to chemistry. “A compound is built of atoms and bonds. I am trying to see what kinds of ‘atoms’ appear in maths and understand the bonds between them. In this case the ‘bonds’ are homological properties.”

The goal is to extract a distinguished element on a decomposed space. “Roughly speaking, homological property tells us the location of the targeted element, and how I should pick up the correct space for the decomposition,” Chan said.

Chan, who has been investigating deeply on those homological properties, discovered new rules in decomposing GL(n) simple spaces as GL(n-1) spaces, and concluded that a large amount of the decomposed spaces do not admit a higher extension with most of the simple GL(n-1) spaces. The new finding provides deeper insight on the structures of those spaces. Guided from the principle of the Langlands program, such structures can also be reflected from the arithmetic side, known to be a part of the celebrated Gan-Gross-Prasad conjectures.

Robert Langlands proposed the original program as a generalisation of class field theory and Chan has been working on his section for about ten years. “The program remains largely open, and at the same time it keeps expanding in various perspectives,” Chan said. “For instance, how to decompose representations in the program is known as the ‘relative Langlands program’ nowadays.”

Solving the Langlands program has promoted the development of other branches of mathematics such as algebraic geometry and harmonic analysis, Chan explained. “People try to find new techniques to handle the program. It would take time to find the long-term, real application.”

Though solving the program is not within his bounds, he enjoys approaching it from a logical perspective. “I do want to see the program established one day, maybe not the whole thing, but at least the original program,” Chan said.

Dr Kei Yuan Chan obtained his PhD from the University of Utah in 2014 under the direction of Professor Peter Trapa. He went on to spend two and half years in Professor Eric Opdam’s Algebraic Group Theory research team at the University of Amsterdam, where he was partially supported by the Croucher Foundation. This was followed by a postdoctoral fellowship in the Algebra group at the University of Georgia. He is currently a (tenure track) young investigator at the Shanghai Center for Mathematical Sciences at Fudan University. Dr Chan was awarded a Croucher Fellowship in 2015.

To view Dr Chan’s full Croucher profile please click here.