Mathematical Sciences Institue
Australian National University
Canberra, ACT, Australia
noah.white@anu.edu.au
Office 2.70, Hanna Neumann Building 145
![It's me!](img/noah.jpeg)
Background
I am a representation theorist interested in problems that have some connection to combinatorics, algebraic geometry and integrable systems. I arrived at ANU in 2020 after being a postdoc at University of California, Los Angeles and obtaining my PhD at the University of Edinburgh.
Research
My research has many interconnections however it can be broadly split into several areas:
Bethe ansatz in representation theory
The Bethe ansatz is a method used to construct eigenvectors for Hamiltonians in some quantum mechanical systemns such as Gaudin spin chains. It has found wide application in representation theory and helped resolve several conjectures (famously the Shapiro-Shapiro conjecture). My research concerns the specific case of the Gaudin model for (\mathfrak{gl}_n)) which I used to prove a conjecture of Etingof, showing that the monodromy of the Bethe eigenvectors recovers an action of the cactus group coming from crystal graphs. The same setting was used to resolve a conjecture of Bonnaf'e-Rouqiuer concerning a reinterpretation of Kazhdan-Lusztig cells for the symmetric group in terms of the rational Cherednik algebra and Calogero-Moser space.
- Gaudin Algebras, RSK and Calogero-Moser Cells in Type A, joint with A. Brochier and I. Gordon.
- Labelling Schubert intersections in the Grassmanian.
- The monodromy of real Bethe vectors for the Gaudin model.
Cactus group actions
Actions of the braid group play an important role in representation theory. Often these actions come with a parameter that one can send to zero. In this limit, one recovers an action of the associated cactus group. I am interested in both calculating specific examples of these actions, and understanding more precisely the relation between the braid group and the cactus group actions.
- The cactus group and Lusztig’s isomorphism, joint with R. Rouquier, in preparation.
- Affine matrix ball model and the affine cactus group, joint with N. Cheung, in preparation.
Generalised Jucys-Murphy elements
The classical Jucys-Murphy elements of the symmetric group arise from the restriction to smaller symmetric subgroups and are used to construct the irreducible representations. From many points of view, there is nothing special about the symmetric subgroups considered and one can defined generalised Jucys-Murphy elements for any chain of parabolic subgroups. I show how to calculate the eigenvalues of these operators, and how to find additional operators to diagonalise the multiplicity spaces that arise from more general restrictions. This is research in progress.
Invariants for the reflection equation algebra
Invariant functions for the general linear group in its regular representation are given by the coefficients of the characteristic polynomial (e.g. the trace and the determinant). With David Jordan I investigated the analgous problem for the reflection question algebra, a quantisation of the general linear group. Various implicit descriptions of the invariants have been given, however we were able to write down explicit formulas. This also yielded a new description of the centre of the Drinfeld-Jimbo quantum group.
- The center of the reflection equation algebra via quantum minors, joint with D. Jordan.
Teaching
My current teaching:
- [ODE][ode] offered through the Joint International Science College (ANU and Shandong University)
My previous teaching at ANU:
- Crystals: combinatorial algorithms and tensor categories, Semester 2 2021
- ODE (joint ANU-Shandong University course), Semester 1 2021, 2022, 2023
- MA1 (joint ANU-Shandong University course), Semester 2 2020
My previous teaching at UCLA:
- Math 229B Winter 2020
- Math 115A Spring 2020, Winter 2020, Fall 2019, Spring 2019, Winter 2019, Winter 2018
- Math 170A Fall 2017
- Math 32B Spring 2020, Fall 2020, Spring 2018, Winter 2017
- Math 31B Spring 2017
- Math 3B Winter 2019, Fall 2018, Fall 2017, Winter 2017, Fall 2016
Software and other
- REA, a Magma package complementing my joint paper with David Jordan. It implements the relations int he reflection equation algebra and it's adjoint action. A significant amount more unpolished code is available, feel free to get in touch if you think it might be useful.
- Reflections calculator, a simple calculator for generating root systems. Helpful for the students in my crystals class while doing their final exam!
- pyAMBC, a python implementation of the Affine Matric Ball consturction and the cactus group action on affine permutations.