Congratulations to Assistant professor Max Gurevich for receiving The Kurt Mahler Prize Fund in Mathematics for the published paper: “On restriction of unitarizable representations of general linear groups and the non-generic local Gan–Gross–Prasad conjecture.”
The paper is part of a quest for understanding branching laws that govern restriction of representations (linear actions) of a group to a lower-rank subgroup.
These are curious mathematical phenomena that sometimes appear in a physical context of symmetry breaking.
The elegant classical case, where simple combinatorial branching laws fully determine the spectral behavior of the restriction operation, is that of finite permutations groups.
Given an irreducible linear action (a building block for a symmetry of a given type) of the group of permutations on n symbols, restriction to the smaller subgroup of permutations on n-1 symbols will render the irreducible situation into a reducible one. In other words, the original building blocks will now decompose into smaller ones.
All of these ‘blocks’ have combinatorial ‘names’ – they may be described by possible partitions of the integer n (or n-1). Hence, branching laws describe how a given partition decomposes into
partitions of a smaller integer.
This is a good example for one of the strengths of Lie theory, which attempts to describe continuous symmetries, that often appear in nature, through discrete terms that would be easier for us to grasp.
In his studies, the groups are infinite, coming from symmetries inherent in number theory. Their representations are infinite-dimensional, and things can become tough… In fact, general branching problems like the one described above for finite groups are considered in this setting too ‘wild’ to admit a short and inclusive solution. Yet, recent years have seen some progress on the matter.
In this paper exhibits new techniques for identifying branching laws for representations in a special class that naturally appears in number theory. As an application, they have affirmed cases of an emerging system of conjectures (Gan-Gross-Prasad), which is a leading promising direction in the wide effort of connecting geometry with number theory.
Read more in the full version of the paper:
https://lnkd.in/dE2w2TAM
Learn more about Max and his research: https://lnkd.in/dqeDU2VQ
