# Noah White

University of Califonia,
Los Angeles

# Introduction

The motivation for my research comes from representation theory. One can often make difficult questions in representation theory simpler by reinterpreting them in terms of some geometry. Often this sheds light on the underlying combinatorics. It is this interaction which I find fascinating.

1. A particularly beautiful example are the Littlewood-Richardson coefficients. On the combinatorial side, these numbers control the multiplication of Schur polynomials. On the algebraic side they are responsible for the decomposition of tensor products of simple $\mathfrak{gl}_n$-modules. And on the geometric side they tell us how Schubert varieties intersect. This correspondence is at the heart of one of my lines of research. More specifically I am interested in the relationship between crystal bases, the Gaudin model and the algebraic Bethe ansatz for $\mathfrak{gl}_n$.

2. Reflection groups, braid groups and their Hecke algebras are fundamental objects in the representation theory of finite groups of Lie type. I am interested in the relationship between the cactus group (a relative of the braid group) and the Kazhdan-Lusztig cells of the reflection group. In type A, this overlaps with the previous line of research.

3. There are many interesting functions on the variety of matrices such as the determinant, trace, various minors, etc. The reflection equation algebra, related to the quantum group $U_q(\mathfrak{gl}_n)$, is a quantisation of the variety of matrices and I am interested in understanding noncommutative analogues of classical facts.

# Papers and preprints

Labelling Schubert intersections in the Grassmanian

The center of the reflection equation algebra via quantum minors (jt with D. Jordan)

The Monodromy of real Bethe vectors for the Gaudin model

The cactus group and Lusztig’s isomorphism (jt with R. Rouquier). In preparation.

Calogero-Moser cells in type A and the RSK correspondence (jt with A. Brochier and I. Gordon). In preparation.

# Talks

Here is a list of the talks I have given.

 Feb 2020 New Connections in Representation theory TBA Nov 2019 University of Colorado, Boulder Algebraic Lie Theory seminar Particle collisions and RSK. Nov 2019 Special Session on Geometric Methods in Representation Theory, Particle collisions and RSK. Nov 2019 Special Session on Geometry and Representation Theory of Quantum Algebras and Related Topics, The cactus group and cell representations. Apr 2018 UC Davis Algebra and discrete math seminar, Symmetries of Kazhdan-Lusztig cells. Apr 2018 UCLA Algebra seminar, A moduli of approaches to the representation theory of the symmetric groups. Nov 2017 AMS Western Sectional meeting, The center of the reflection equation algebra (and quantum group) for $GL_n$. Oct 2017 Toronto, Geometric representation theory seminar, The center of the reflection equation algebra (and quantum group) for $GL_n$. Jun 2016 Köln algebra seminar, The cactus group and the Gaudin model. Apr 2016 Max Plank Institute, Oberseminar, The cactus group and RSK. Mar 2016 IMJ Paris Diderot Groupes, Représentations et Géométrie seminar, Monodromy of the Gaudin system in type A. Aug 2015 Melbourne algebra/geometry/topology seminar, Cactus group actions on Young tableaux. Aug 2015 UQ pure maths seminar, Cactus group actions on Young tableaux. Jul 2015 Sydney algebra seminar, Cactus group actions in Schubert calculus, crystals and integrable systems. May 2015 Joint Heriot Watt, Edinburgh, Glasgow, St. Andrews PG Colloquium, Reality in algebraic geometry. Feb 2015 QMUL algebra seminar, Schubert calculus and the cactus group. Dec 2014 GEARS seminar, Glasgow, The representation theory of the symmetric groups via the action of a “maximal torus”. Mar 2014 ARTIN in Newcastle, The center of quantum $GL_n$ Sep 2014 Geometry club, Edinburgh, Galois Theory in Enumerative Geometry. Sep 2013 Geometry club, Edinburgh, Invariants of knots from representation theory.