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Mathematical Science

Number Theory: Beyond Subtraction: Exploring the Non-Standard Metric and p-Adic Integers in Mathematics

Beyond Subtraction: Exploring the Non-Standard Metric and p-Adic Integers in Mathematics
Professor Kei Yuen CHANLead Researcher // Professor Kei Yuen CHAN, Assistant Professor of the Department of Mathematics
Collaborator // The University of California, Berkeley; Imperial College London; Chinese Academy of Sciences
When comparing the distance between two numbers, such as 3 and 7, simply subtracting them (7-3=4) gives us a sense of how far apart they are. This method of subtraction is a common way to measure the distance between numbers.
However, there is another method called the non-standard metric that can be used to measure distances. This metric defines distance using the prime number factorisation of the numbers. By fixing a prime number p, a number x can be written in the form Pky, where k is a non-negative integer and y is relative prime to p. For instance, if we fix p=2, we can calcuate the distances between 3 and 7 and between 4 and 7. In this case, 7-3=22 and 7-4=20 X 3. Comparing the exponents, we see that 2 is greater than 0.
In the non-standard metric, distances are compared based on the number of prime factors present in the factorisation. Since 4(22) has more prime factors than 3(20), the distance between 4 and 7 is considered longer than the distance between 3 and 7!


Initiated at Fudan University, our researcher studied this new metric known as p-adic integers. In contemporary mathematics, the Langlands programme is a significant framework that uses matrices to describe various number systems. Groups can be considered collections of matrices, and the branching law uses an inductive structure to understand the entire group based on a subset of matrices. The precise link between matrices and numbers in branching laws is referred to as the Gan-Gross-Prasad conjecture. His recent discovery involves a new method of utilising the functorial properties of parabolic inductions in branching laws, which resolves a conjecture of Gan-Gross-Prasad for Arthur-type representations.
Journal paper: Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case) (published in Crelle's Journal, 2022)

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