University of Califonia,
Los Angeles
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.
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 $\mathfrak{gl}_n$.
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.
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.
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.
The center of the reflection equation algebra via quantum minors (jt with D. Jordan)
Here is a (incomplete) list of the talks I have given.
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. |