I am a member of the
Algebra research group.
Broadly speaking, I am interested in representations of Lie-theoretic
objects and related aspects of combinatorics and geometry.
The papers and talks listed below can be categorized into three
main themes.
I am very interested in the geometry of the various varieties
which generalize the closures
of nilpotent orbits, such as orbit closures in exotic nilpotent cones,
Nakajima's quiver varieties, and nilpotent
orbits for symmetric spaces. The desingularizations and intersection cohomology of these varieties usually have representation-theoretic
significance. See papers 1,3,10,13 and talks 9,10,12,16.
The subject of my PhD thesis and subsequent research was the extension
of Lusztig's results on
reductive groups over a finite field
to the case of symmetric spaces. See papers 2,4,9 and talks 1-4.
I also work in more combinatorial
areas related to hyperplane arrangements and wonderful compactifications. My interest has been in computing the characters of the
representations of symmetric groups and wreath products on the
cohomology of various varieties on which they act. See
papers 5-8,11,12,14 and talks 5-8,11,13-15.
I currently hold two research grants from the Australian Research Council, on `The geometry of exotic nilpotent cones' and `Pyramids and decomposition numbers for the symmetric and general linear groups' (joint with A. Mathas).
Spherical functions of the symmetric space
G(Fq²)/G(Fq),
MIT Lie Groups Seminar, November 2000,
pdf (15 pages).
Spherical functions and character sheaves,
University of Sydney Algebra Seminar, August 2001
(a revised version of my thesis defence),
pdf (12 pages).
Symmetric subgroup invariants in irreducible representations
of GF, when G=GLn,
Annual Meeting of the Australian Mathematical Society, September 2001,
pdf (6 pages).
Fourier transform and Kloosterman sums,
University of Sydney Algebra Seminar, October 2002,
pdf (14 pages).
Species over a finite field,
Australia/New Zealand Mathematics Convention, July 2003,
pdf (6 pages, 2 pictures missing).
Polynomial functors and representations of wreath products,
University of Sydney Algebra Seminar, August 2003,
pdf (14 pages).
Representations of wreath products on cohomology of De Concini-Procesi
compactifications,
Lie Minisymposium, November 2003,
pdf (12 pages, 4 pictures missing).
The symmetric group representation on cohomology of the
regular elements of a maximal torus of the special linear group,
Annual Meeting of the Australian Mathematical Society, September 2004,
pdf (8 pages, 19 counting overlays).
Nilpotent orbits and Kazhdan-Lusztig polynomials of type A,
Mathematical Physics and Lie Theory, December 2004,
pdf (7 pages).
Equalities of Kazhdan-Lusztig polynomials of type A,
University of Sydney Algebra Seminar, March 2005,
pdf (14 pages).
Bases for certain cohomologies with local coefficients of a hyperplane
complement, Annual Meeting of the Australian Mathematical Society, September
2005, pdf (9 pages, 24 counting overlays).
Quiver varieties and zero weight spaces, AMSI Workshop on Lie Theory,
Lattices, and Dynamics, November 2005,
pdf (11 pages).
Poset homology and the cohomology of real De Concini-Procesi models,
University of Sydney Algebra Seminar, August 2006,
pdf (17 pages, 53 counting overlays).
The cohomology of the real De Concini-Procesi model of type Bn,
Annual Meeting of the Australian Mathematical Society, September
2006, pdf (8 pages, 20 counting overlays).
The cohomology of real De Concini-Procesi models of Coxeter type,
Braids and their Ramifications, Cortona, May 2007,
pdf (14 pages, 45 counting overlays).
Orbit closures in the enhanced nilpotent cone,
Shuffles, Descents, and Representations, Nice, September 2007,
pdf (14 pages, 37 counting overlays).
Enhancing the nilpotent cone,
Australia/New Zealand Mathematics Convention, Christchurch, December 2008,
pdf (10 pages, 26 counting overlays).
Geometry of the exotic nilpotent cone, Algebraic Lie Theory Seminar,
Isaac Newton Institute, Cambridge, June 2009,
pdf (13 pages, 31 counting overlays).
Teaching
In May 2009, I was awarded a Faculty of Science Citation for Excellence in Teaching.
I have lectured the following units of study.