gexordie.williaxmson@maxths.ox.ac.uk
(remove three letters!)
My CV is available here and you
can send me a message here.
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?
By work of Soergel, Mirkovic-Vilonen and Juteau understanding the
decomposition theorem has important implications for modular
representation theory. Also, when the decomposition theorem fails it is
often very difficult to calculate the stalks of simples. I have been
working on these types of questions with Daniel
Juteau and Carl
Maunter.
With Ben Webster
I am trying to understand Khovanov-Rozansky link homology and its
coloured generalisations using constructible sheaves and the weight
filtration. There are many interesting and unresolved questions here.
Ben has written lots of slides about
this work.
I am interested in understanding Kazhdan-Lusztig theory without
weights or the decomposition
theorem. Partly motivated by positive characteristic considerations,
and partly motivated by a desire to understand why Kazhdan-Lusztig
polynomials have positive coefficients. In this direction I obtained
closed
formulas for Kazhdan-Lusztig polynomials for affine A2, but
unfortunately haven't written it up.
I am interested in combinatorial models of modular perverse
sheaves. In a project with Peter Fiebig
we are trying to understand the relationship between certain
equivariant sheaves on (Kac-Moody) flag varieties and sheaves on the
moment graph. The aim is a positive characteristic counterpart of this idea of
Braden-MacPherson. (You can see a video of Peter
talking about this work.)
Soergel bimodules and their singular variants provide many
interesting questions
(this was essentially my PhD thesis). In particular, I believe
the work of Nicolas
Libedinsky and the recent work of Matthew Dyer is not as well
known as it should be.
More generally I am fascinated by the big machine called
"categorification" (this has
something to say about every topic above).
Papers, preprints:
Parity sheaves
This is joint with Daniel
Juteau and Carl
Maunter.
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.
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 tiple
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 and has been submitted.
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!
Intersection
cohomology complexes on low rank flag varieties
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
tecniques in small rank. Thanks
to results of Soergel, we are able to conclude parts of the Lusztig
conjecture on modular representations of reducitive groups.
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 indecomposables (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.
