Before coming here I was a postdoc at the University of Oxford working with Raphaël
Rouquier. Before that I completed my PhD in Freiburg under the supervision of Wolfgang Soergel.
In June, 2011 I gave a three hour course on
localisation in equivariant cohomology at the University of
Melbourne at the invitation
of Arun Ram. Lecture
notes and exercise sheets are here.
In May 2011, I gave a four hour course called "Kazhdan-Lusztig
polynomials and intersection cohomology complexes on the flag
variety" as part of a spring school Flag Varieties, organised by
Cédric Bonnafé. Scans of the lectures and
slides are avialable here.
In 2011 I gave a number of talks about the
p-smooth locus in a variety of locations (Jussieu, Newcastle,
Aberdeen). Notes of my talk (which contain a bit more detail
than the talk itself) are here.
Some pictures to illustrate what is going on.
In 2011 I have given a number of talks about
generators and relations for Soergel bimodules (Chamonix,
Clermont-Ferrand, BMC in Leicester,
Cambridge, Sydney, Ottawa, Amherst, Jussieu). Slides are here.
Earlier last year I finished translating Cédric Bonnafé's book
"Representations du groupe SL2(Fq)". It is now available.
Teaching:
In Michaelmas term (2010) I taught Lie algebras (C2.1a). Click here for exercise sheets.
Research Interests:
I work in geometric representation theory, but love all things vaguely
geometric. Here is a list of things that I am particularly interested
in at the moment:
When is the decomposition theorem true in positive characteristic? What happens when it fails? By work of Soergel, Fiebig, Mirkovic-Vilonen and Juteau understanding the decomposition theorem has important implications for modular representation theory. Hoping to understand the failure of the decomposition theorem led to the discovery of parity sheaves. Many fundamental questions in representation theory (the Lusztig conjecture, James conjecture, the problem of calculating the characters of tilting modules for algebraic groups etc.) can all be translated into questions about parity sheaves. I spend a lot of time thinking about these questions, often together with
Daniel
Juteau and Carl
Maunter.
With Ben Elias we have recently succeeded in giving a generators and relations description of the monoidal category of Soergel bimodules (generalising this remarkable
paper of
Elias-Khovanov). This has allowed me to perform lots of calculations which were impossible before, and should have some theoretical consequences.
With Ben Webster we have given a geometric constructiong of triply graded (HOMFLYPT) homology in terms of constructible sheaves and the weight filtration. This also led to some investigations of Markov traces, and their
generalisations outside of type A. There are interesting
connections to the work
of Ben-Zvi and Nadler and that of Bezrukavnikov, Finkelberg and Ostrik.
Ben has written lots of slides about
this work. On the other hand, there are some conjectures of Vivek Shende, Alexei Oblomkov and Jake Rasmussen which relate triply graded link homology to some different geometry (punctual Hilbert schemes and affine Springer fibres). There are lots of interesting questions to think about here.
I am interested in combinatorial models of modular perverse
sheaves (this has a lot in common with the ``generators and relations for Soergel bimodules'' discussed above). In a project with Peter Fiebig we showed that one may calculate the stalks of parity sheaves on the flag variety using moment graphs (giving a positive characteristic counterpart of this idea of
Braden-MacPherson). (You can see a video of Peter
talking about this work.) Such combinatorial models also exist for toric varieties, and are much simpler than the case of Schubert varieties. Hopefully this will get written up at some stage!
With Nicolas
Libedinsky I have been thinking about Rouquier's categorification of the braid group and his conjecture about faithfulness. Ideally, one would understand what the complexes "mean" in terms of the Garside structure on Braid groups.
As should be evident from the above, I am fascinated by the big machine called
"categorification" (this has
something to say about every topic above).
Papers, preprints:
Kumar's criterion modulo p This is joint with
Daniel
Juteau and has been submitted. Here are some expository notes and
here are some pictures of
singularities intended to illustrate what's going on!
We show that the numerator in Kumar's criterion for rational
smoothness of Schubert varieties has a natural interpretation in
terms of p-smoothness. We conjecture in certain cases
the numerator calculates the order of the torsion subgroup of the
link. One consequence is that certain parts of the equivariant
multiplicity are in fact topological invariants (and not just
invariants of the singularity with T-action).
Singular Soergel bimodules
Appeared in IMRN.
Singular Soergel bimodules is a interesting 2-category
which acts (or should act) in many representation theoretic
situations. It also has an elementary definition in terms of certain
rings of invariants, for the action of a Coxeter sytem on a
polynomial ring. In this paper we classify the indecomposable
singular Soergel bimodules, and prove that it categorifies the Schur
algebroid, a natural algebroid
generalising the Hecke algebra of a Coxeter system.
Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties
This is joint with
Peter Fiebig and has been submitted.
A moment graph is a labelled graph which encodes the
``one-skeleton'' of certain algebraic torus actions on varieties. It
is an amazing fact (usually referred to as the localisation
theorem) that one can recover a lot of cohomological information
about a variety from its moment graph. In this paper we show that
the moment graph can be used to calculate the stalks of ``parity
sheaves'' (see below). We use this result, together with an
algebraic result of Fiebig, to deduce the p-smooth locus of Schubert
varieties. We also apply this to representation theory and show that
moment graphs can be used to calculate the weight spaces of tilting
modules.
The geometry of Markov traces
This is joint with
Ben Webster and
appeared in Duke Mathematical Journal.
We show that the Jones-Ocneanu trace on the Hecke algebra of type
A evaluated on a Kazhdan-Lusztig basis element is a mixed Poincare
polynomial of the B-conjugation equivariant cohomology of the
corresponding intersection cohomology complex. This then gives a
natural trace on Hecke algebras of finite type. We then show that
this trace is equal to a trace defined by Gomi in 2006. This yields
a simple geometric proof of Gomi's result, and provides a natural
framework in which to interpret his definition. Another useful (and
unexpected) biproduct of our investigations is a proof that the
Hochschild homology of Soergel bimodules in
finite type is free. (This had been observed previous by Rasmussen
in Type A, but his proof doesn't generalise.)
Parity sheaves
This is joint with Daniel
Juteau and Carl
Maunter and has been submitted.
You can see a video of Daniel
talking about this work in Cambridge and here are some slides of a talk I gave in Durham.
We introduce a new class of sheaves on certain
varieties (the "parity sheaves") which we believe will be
fundamental in attempts to use modular perverse sheaves in
representation theory. We show that one may prove a decomposition
theorem type result for certain maps (which we call "even"), and
show the role played by certain intersection forms introduced in
work of de Cataldo and Migliorini in determining the stalks of
parity sheaves. We also give lots of examples. Probably the most
important being that parity sheaves exist on the affine
Grassmannian, and (under some moderate assumptions)
correspond to tilting modules.
A geometric construction of colored HOMFLYPT homology
This is joint with
Ben Webster and
has been submitted.
You can see a video
of a talk I gave about this work in Cambridge.
This paper continues Ben and my efforts to understand various link
homology theories geometrically, in terms of constructible
sheaves. We are primarily interested in Khovanov and Rozansky's
triply graded HOMFLYPT homology, and a natural first question is
what on earth do all the gradings mean?! The crucial point is that,
on a non-proper algebraic variety one has a weight filtration before
and after pushing to a point, which may be used to construct a triple
grading. In this way we obtain a completely geometric construction
of HOMFLYPT homology, as well as various "colored"
generalisations.
Perverse sheaves and
modular representation
theory
This is joint with Daniel
Juteau and Carl
Maunter, proceedings of the summer school
"Geometric methods in representation theory", Grenoble, June 2008.
We give an overview of three applications of perverse sheaves in
modular representation theory. The basic idea is to consider sheaves of
k-vector spaces on complex
algebraic varieties, where k is a field of positive characteristic. The
corresponding categories of perverse sheaves behave like (and sometimes
are actually equivalent to) categories arising in modular
representation theory. Just as is the case for modular representations,
these categories are difficult to understand. In order to try to
convince the reader of this we give some calculations on nilpotent
cones: things are already very interesting in sl_n for n = 2, 3 and 4!
Modular intersection
cohomology complexes on flag varieties
With an appendix by
Tom Braden. Appeared
in Mathematische Zeitschrift.
The software and W-graphs referred to in this paper are available
here.
For a fixed field k of positive characteristic
almost nothing is known about intersection cohomology complexes on flag
varieties with coefficients in k. In this article we present a
combinatorial algorithm which, if successful, proves that they ''look
the same'' as in characteristic 0. Our algorithm relies on the W-graph
for which no general description is known. Thus we can only apply our
techniques in small rank. Thanks
to results of Soergel, we are able to conclude parts of the Lusztig
conjecture on modular representations of reductive groups.
In the appendix, Tom Braden gives some examples of torsion in the
stalks or costalks of intersection cohomology complexes on Schubert
varieties in type A7 and D4.
A
geometric model for Hochschild homology of Soergel bimodules
This is joint with with Ben
Webster and appeared in
Geometry and Topology.
Khovanov
has constructed a knot invariant in the homotopy category of bigraded
modules over a polynomial ring. This involves first constructing a
complex of Soergel bimodules and then taking Hochschild homology. In
this paper we show that all of this may be interpreted geometrically:
each term in the complex may be viewed is the equivariant cohomology of
a ``Bott-Samelson'' type space, and the maps in the complex are induced
from maps between Bott-Samelson varieties. Using geometric techniques
we are also able to give explicit descriptions of the Hochschild
homology of certain ``smooth'' Soergel bimodules in type A.
PhD Thesis, Essays, Software etc:
Singular Soergel bimodules
This is my PhD thesis. It defines singular Soergel bimodules in a
general framework and classifies the indecomposable bimodules (generalising
results of Soergel). Soergel bimodules and their singular variants have
many applications (including the study of category O, equivariant
perverse sheaves and knots) and the list will probably grow in the
future. One exciting possibility is a tensor category of bimodules
which is equivalent to the representation ring of an (adjoint)
semi-simple group.
Why
the Kazhdan-Lusztig basis of the Hecke
Algebra is a Cellular Basis
My honours essay at the University of Sydney supervised by Gus Lehrer.
In this essay I prove that the standard Kazhdan-Lusztig basis of the
Hecke algebra of the symmetric group is a cellular basis in the sense
of Graham and Lehrer. This involves lots of combinatorics centred
around the Robinson-Schensted correspondence. One of the corollaries of
cellularity is the fact that the cell
representations are irreducible.
The
Fundamental Example of
Bernstein and Lunts
I
have tried to write a motivated introduction to the equivariant
derived category, as well as provide the details of a proof of the
so-called "fundamental example". This relates the equivariant
intersection
cohomology of a torus stable subvariety of affine space to the
intersection cohomology of a projective variety (a quotient) and the
equivariant
stalk at 0. The proof of the fundamental example is complete, except
for the
assumption of the hard Lefschetz theorem for intersection cohomology.
Formal
Groups Work
This is the product of a project with David Kohel
at the University of
Sydney.
We have written software for Magma which calculates the formal group of
the Jacobian of a genus 2 curve.
An
Introduction to the
Birman-Wenzl-Murakami Algebra
The product of a vacation scholarship at the University of New South
Wales under the supervision of Jie Du.
I
introduce the BMW-algebra and calculate some canonical bases in
small dimensions.
Six lectures on Deligne-Lusztig theory
Notes from lectures given by Raphael Rouquier in Oxford, Hilary Term,
2010. (Notes from Lecture 5 by David Craven.)
