Anthony Henderson

I am an Associate Professor in the School of Mathematics and Statistics at the University of Sydney. Here is my curriculum vitae (pdf, 8 pages).

Postal address: A/Prof Anthony Henderson School of Mathematics and Statistics F07 University of Sydney NSW 2006 Australia Office: Room 805 Carslaw Building Email: anthony.henderson[at]sydney.edu.au

Research

• I have a long-standing interest in the geometry of nilpotent varieties and Springer fibres, and their generalizations such as orbit closures in Hilbert nullcones and Nakajima's quiver varieties. It is often the case that the (intersection) cohomology of these varieties controls decomposition numbers in representation theory. See preprints 1-3, research publications 1,3,10,13,15-17,19, expository publication 1 and talks 1-5,8.

  Current funding for research in this area: In November 2011, I was awarded a four-year Future Fellowship by the Australian Research Council for the project `Springer fibres, nilpotent cones, and representation theory'. Andrew Mathas and I hold an ARC Discovery-Projects grant on `Pyramids and decomposition numbers for the symmetric and general linear groups'.

• I have also worked 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 research publications 5-8,11,12,14,18,20 and talks 6-7.
• 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 research publications 2,4,9.

In 2011, I was the inaugural winner of the Christopher Heyde Medal of the Australian Academy of Science. This medal is for an outstanding researcher in the mathematical sciences under the age of 40, working in Australia: it is awarded annually on a three-year rotation between pure mathematics, applied mathematics, and statistics.

I am an Associate Editor of the Journal of the Australian Mathematical Society.

Preprints

1. with Pramod N. Achar: Geometric Satake, Springer correspondence, and small representations, Selecta Math. (N.S.), Online First, DOI:10.1007/s00029-013-0125-7 (33 pages).
2. with Pramod N. Achar and Simon Riche: Geometric Satake, Springer correspondence, and small representations II, available at arXiv:1205.5089v2 or here (version 2, 72 pages).
3. with Pramod N. Achar, Daniel Juteau and Simon Riche: Weyl group actions on the Springer sheaf, available at arXiv:1304.2642 or here (27 pages).

Research Publications

1. Fourier transform, parabolic induction, and nilpotent orbits, Transform. Groups 6 (2001), no. 4, 353-370.
2. Spherical functions of the symmetric space G(F_q²)/G(F_q), Represent. Theory 5 (2001), 581-614.
3. Two-row nilpotent orbits of cyclic quivers, Math. Z. 243 (2003), no. 1, 127-143.
4. Symmetric subgroup invariants in irreducible representations of G^F, when G=GL_n, J. Algebra 261 (2003), no. 1, 102-144.
5. Representations of wreath products on cohomology of De Concini-Procesi compactifications, Int. Math. Res. Not. 2004 (2004), no. 20, 983-1021.
6. Species over a finite field, J. Algebraic Combin. 21 (2005), no. 2, 147-161.
7. Bases for certain cohomology representations of the symmetric group, J. Algebraic Combin. 24 (2006), no. 4, 361-390.
8. Plethysm for wreath products and homology of sub-posets of Dowling lattices, Electron. J. Combin. 13 (2006), no. 1, Research Paper 87, 25 pp.
9. Induced characters of the projective general linear group over a finite field, J. Algebra 307 (2007), no. 1, 116-135.
10. Nilpotent orbits of linear and cyclic quivers and Kazhdan-Lusztig polynomials of type A, Represent. Theory 11 (2007), 95-121.
11. with Eric Rains: The cohomology of real De Concini-Procesi models of Coxeter type, Int. Math. Res. Not. 2008 (2008), rnn001, 29 pp.
12. The symmetric group representation on cohomology of the regular elements of a maximal torus of the special linear group, J. Aust. Math. Soc. 84 (2008), no. 1, 85-98.
13. with Pramod N. Achar: Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (2008), no. 1, 27-62. Note the corrigendum.
14. with Gus Lehrer: The equivariant Euler characteristic of real Coxeter toric varieties, Bull. London Math. Soc. 41 (2009), no. 3, 515-523.
15. Exterior powers of the reflection representation in the cohomology of Springer fibres, C. R. Math. Acad. Sci. Paris 348 (2010), no. 19-20, 1055-1058.
16. with Pramod N. Achar and Eric Sommers: Pieces of nilpotent cones for classical groups, Represent. Theory 15 (2011), 584-616.
17. with Pramod N. Achar and Benjamin F. Jones: Normality of orbit closures in the enhanced nilpotent cone, Nagoya Math. J. 203 (2011), 1-45.
18. with Michelle L. Wachs: Unimodality of Eulerian quasisymmetric functions, J. Combin. Theory Ser. A 119 (2012), no. 1, 135-145.
19. with Peter E. Trapa: The exotic Robinson-Schensted correspondence, J. Algebra 370 (2012), 32-45.
20. Rational cohomology of the real Coxeter toric variety of type A, in Configuration Spaces: Geometry, Combinatorics, and Topology, Publications of the Scuola Normale Superiore, no. 14, A. Björner, F. Cohen, C. De Concini, C. Procesi and M. Salvetti (eds.), Pisa, 2012, 313-326, available from SNS and Birkhäuser.

Expository Publications

1. Enhancing the Jordan canonical form, Austral. Math. Soc. Gaz. 38 (2011), no. 4, 206-211.
2. Representations of Lie Algebras: An Introduction Through gl_n, Australian Mathematical Society Lecture Series, no. 22, Cambridge University Press, Cambridge, 2012, available from CUP Australia, CUP America, Amazon, Amazon UK and Book Depository.

Selected Recent Talks

1. Enhancing the nilpotent cone, Louisiana State University Algebra Seminar, October 2009, pdf (15 pages, 34 counting overlays).
2. Enhancing the Jordan canonical form, University of Utah Mathematics Colloquium, November 2009, pdf (16 pages, 29 counting overlays).
3. Pieces of nilpotent cones for classical groups, London Algebra Colloquium, March 2010, pdf (16 pages, 32 counting overlays).
4. Orbit closures in nilpotent cones, Université de Caen Algebra/Geometry Seminar, June 2010, pdf (15 pages, 30 counting overlays).
5. Exotic nilpotent cones and Springer representations of Weyl groups, Séminaire Chevalley, Paris, June 2010, pdf (16 pages, 31 counting overlays).
6. Representations of reflection groups on the cohomology of varieties, Algebra and Geometry of Configuration Spaces and Related Structures, Pisa, June 2010, pdf (14 pages, 36 counting overlays).
7. The toric variety of the symmetric group, Combinatorial Representation Theory and Integrable Models, Melbourne, January 2011, pdf (13 pages, 23 counting overlays).
8. Geometric Satake, Springer correspondence, and small representations, Algebraic Groups and Representation Theory, Hong Kong, January 2013, pdf (12 pages, 32 counting overlays).

Supervision

• Andrew Crisp, PhD (current)
• Natalie Aisbett, PhD (under examination)
• Michael Sun, MSc (awarded Jan 2010)
• Michael Hendriksen, MSc (associate supervisor - current)
• Clinton Boys, PhD (associate supervisor - current)
• Graham White, MSc (associate supervisor - awarded May 2013)
• Ge Li, PhD (associate supervisor - awarded Dec 2012)
• Justin Koonin, PhD (associate supervisor - awarded Mar 2012)
• Neil Saunders, PhD (associate supervisor - awarded Feb 2011)

Undergraduate Teaching

• Semester 1, 2007-08: MATH1901 Differential Calculus (Advanced).
• Semester 1, 2005-06,08-10: Special Studies Program unit MATH1906.
• Semester 1, 2007-09: MATH2069 Discrete Mathematics and Graph Theory (joint with the Advanced version MATH2969).
• Semester 1, 2011: Special Studies Program unit MATH2916.
• Semester 2, 2011: MATH2968 Algebra (Advanced).
• Semester 2, 2006-10: MATH3966 Modules and Group Representations (Advanced).
• Semester 1, 2003-04: MATH3002 Rings and Fields, which has since been subsumed into MATH3062 Algebra and Number Theory.
• Semester 2, 2005 and Semester 1, 2011: Pure Mathematics Honours unit Lie Algebras.
• Semester 1, 2002: Pure Mathematics Honours unit Introduction to Lie Theory.
• Summer 2004 and 2007: Lie Algebras in the AMSI Summer School.

Some Links

• School of Mathematics and Statistics external front page.
• Mathematics ArXiv.
• My author profile in MathSciNet.
• My page in the Mathematics Genealogy Project.
• George Lusztig, my thesis advisor at MIT.
• Kevin McGerty, a fellow Lusztig student, now at Oxford University.
• Lenny Ng, a fellow MIT graduate student, now at Duke University.
• Tara Holm, a fellow MIT graduate student, now at Cornell University.
• Pramod Achar, a fellow MIT graduate student, now a collaborator at Louisiana State University.
• Peter Trapa, a fellow MIT graduate student, now a collaborator at the University of Utah.
• Eric Sommers, a fellow Lusztig student, now a collaborator at the University of Massachusetts Amherst.
• Daniel Juteau, a collaborator at Université de Caen.
• Simon Riche, a collaborator at Université Blaise Pascal - Clermont-Ferrand II.
• Geordie Williamson, a collaborator at Max-Planck-Institut für Mathematik in Bonn.