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 2013 I gave a talk at the Arbeitstagung in
memory of Hirzebruch. Slides and a video is available here.
In April and May 2013 I gave an IMPRS course "Topology of algebraic
varieties and perverse sheaves".
Here is my Oberwolfach report from
the meeting "algebraic groups" in April, 2013.
In March 2013, Ben Elias and I gave a master
class on Soergel bimodules and Kazhdan-Polynomials.
Videos of all lectures are here. Lecture
notes are and exercises are available here.
In Feb '13 I gave two seminartalks and a colloquium at MIT about my
recent work with Ben Elias.
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.
Teaching:
In Michaelmas term (2010) I taught Lie algebras (C2.1a). Click here for exercise sheets.
In the Wintersemester 2012-13 I held the Algebra II lecture.
Research Interests:
I work in geometric representation theory, and love all things vaguely
geometric. I spend most of my time thinking about:
perverse sheaves and the decomposition theorem,
structures (e.g. weights and Hodge theory) that help understand perverse sheaves,
combinatorial models for perverse sheaves (e.g. Soergel bimodules and
moment graphs), and objects that look geometric but aren't
(e.g. the coinvariant algebra of H3),
categorification, diagrammatic
algebra, "generators and relations",
Kazhdan-Lusztig theory and its modular versions
(p-canonical basis etc.)
geometric questions motivated by modular representation theory,
especially non-trivial geometric behaviour predicted by modular representation
theory,
braid group actions and link homology.
Papers, preprints:
Diagrammatics for Coxeter
groups and their braid groups Preliminary version and is joint with Ben
Elias.
We give a generators and relations description of the 2-groups
associated to Coxeter groups and their braid groups. This gives nice
criteria for a braid group to act on a category in terms of
generalized Zamolodchikov relations.
A reducible characteristic variety in type A Preliminary version, will be submitted to the proceedings of
Vogan's birthday conference.
We give an example of a characteristic cycle of an intersection
cohomology D-module of a Schubert variety in SL(12)/B
which has two components in the same two-sided cell. This implies
an example as in the title of the paper. We came across this example
trying to answer a related question, namely whether the image of the
p-canonical basis in an irreducible representation of the
Hecke algebra of the symmetric group always coincides with the
Kazhdan-Lusztig basis. Alas it doesn't!
Parity sheaves and tilting modules This is joint with Daniel
Juteau and Carl
Maunter and has been submitted.
We show that the geometric Satake equivalence relates parity sheaves
and tilting modules under explicit and mild restrictions on the
characteristic of the field of coefficients. This allows geometric
proofs of the stability of tilting modules under tensor product and
restriction to a Levi subgroup (under the same bounds). Recently,
Achar and Rider have used this result to give a proof
of the Mirkovic-Vilonen conjecture in almost all cases.
Kazhdan-Lusztig conjectures and shadows of Hodge theory This is joint with Ben
Elias and has been submitted.
We give a gentle and motivated introduction to Soergel modules and their "Hodge theory". It is an expanded
version of a talk I gave at the Arbeitstagung in memory of
Hirzebruch. It can be seen as providing geometric background for our
paper The Hodge theory of Soergel bimodules below.
Appendix to
Modular perverse sheaves on flag varieties I: tilting and parity
sheaves by Pramod Achar and Simon Riche.
This is joint with Pramod
Achar and Simon Riche and
has been submitted.
This paper and it sequel provides a (beautiful!) Koszul duality between
tilting modules and parity sheaves (building on work
of Bezrukavnikov and Yun in the characteristic zero setting). In the
appendix we establish some basic properties of modular tilting sheaves on
the flag variety.
Schubert calculus and torsion
This is a preliminary version.
We observe that certain numbers occuring in Schubert calculus for
SL_{n} also occur as entries in intersection forms controlling
decompositions of Soergel bimodules and parity sheaves in higher
rank. These numbers grow exponentially in the rank. This observation
gives many counter-examples to Lusztig's conjecture on the characters
of simple rational modules for SL_{n} over a field of positive
characteristic. We also explain how to use our results to get counter-examples to the James conjecture.
Soergel calculus
This is joint with Ben
Elias and is a preliminary version.
The category of Soergel bimodules provides the most concrete
incarnation of the Hecke category, the basic object of
Kazhdan-Lusztig theory. We present the monoidal category of Soergel
bimodules by generators and relations. We give a diagrammatic
treatment of Libedinsky's "light leaves" morphisms, and show that
they give a basis for morphisms. This allows us to give a new proof
of Soergel's classification of the indecomposable Soergel bimodules.
On cubes of Frobenius extensions This is joint work with Ben Elias and is a preliminary version.
We prove some relations between induction and
restriction functors for hypercubes of Frobenius extensions.
We discovered these relations whilst trying to understand
singular Soergel bimodules. There is one relation which is still missing a proof. We posted this paper because we need the (proven) relations elsewhere.
The Hodge theory of Soergel bimodules
This is joint with Ben Elias
and will appear in the Annals of Mathematics.
In geometric situations Soergel bimodules can be realised
as the equivariant intersection cohomology of Schubert varieties,
and hence have interesting real Hodge theory (hard Lefschetz,
Hodge-Riemann bilinear relations etc). Inspired by work of de Cataldo
and Migliorini giving Hodge theoretic proofs of the decomposition
theorem
we prove that these Hodge theoretic properties always
hold for Soergel bimodules, whether they come from Schubert varieties or not!
This gives structures strong enough to deduce Soergel's
conjecture, and hence the Kazhdan-Lusztig positivity conjecture. The result can also be used to give the first algebraic proof
of the Kazhdan-Lusztig conjectures on characters of simple highest weight
modules over complex semi-simple Lie algebras.
On an analogue of the James conjecture.
Appeared in Representation Theory.
We give a counterexample to the most optimistic analogue of the
James conjecture for simply laced Khovanov-Lauda-Rouquier
algebras. The basic idea is to recycle counterexamples known for
Schubert varieties (due mostly to Braden and Polo). The bridge is
provided by recent results of
Maksimau. There are interesting connections to the reducibility of
the characteristic variety, using work of Kashiwara and Saito and a
result with Vilonen below.
Modular Koszul duality
This is joint with
Simon Riche and
Wolfgang Soergel
and will appear in Compositio.
Classical Koszul duality (due to Beilinson, Ginzburg and Soergel)
relates category O and the derived category of Bruhat constructible
sheaves of complex vector spaces on the flag variety. Modular Koszul
duality relates "modular category O" (a subquotient of rational
representations of a reductive group) and the derived category of
constructible sheaves on the flag variety, this time with
coefficients of positive characteristic. The key difficulty (which
turns a simple idea into a sixty page paper) is establishing the
formality of the dg-algebra of extensions of parity sheaves on the
flag variety.
Characteristic cycles and decomposition numbers
This is joint with
Kari Vilonen
and will appear in Math. Res. Let.
There are a number of false conjectures around characteristic cycles
and decomposition numbers: e.g. Kazhdan and Lusztig's conjecture
that characteristic varieties for Schubert varieties are
irreducible; Kleshchev-Ram's conjecture that decomposition numbers
for for KLR algebras are trivial in finite type; various people's
hopes that stalks and costalks of IC sheaves on flag varieties have
torsion only in bad characteristic. In this article we prove that the
topological side (a decomposition number is non-trivial)
implies the analytic side (the characteristic variety is
reducible). This is a consequence of the trivial observation that the
characteristic cycle of a sheaf commutes with base change of
coefficients.
Standard objects in
2-braid groups
This is joint with
Nicolas
Libedinsky and will appear in Proceedings of the LMS.
The 2-braid group is a categorification of the braid group. It has
been around for a few decades in highest weight representation
theory (long before it aquired a name). A few years ago Khovanov showed
that it may be used to construct HOMFLYPT homology, and since then
there has been growing interest in its type A incarnation. Rouquier
has emphasised that a study of the morphisms in the 2-braid group
should have many applications in representation theory (new proofs
of Kazhdan-Lusztig type conjectures, understanding of t-structures
in modular representation theory, construction of Spetses, etc.) In
this paper we make a first step in the study of morphisms spaces in
2-braid groups, namely we consider "standard" and
"costandard" objects and show that they satisfy a vanishing
condition conjectured by Rouquier.
Kumar's criterion modulo p This is joint with
Daniel
Juteau and will appear in Duke Mathematical Journal. 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
will appear in Annales de l'Institut Fourier.
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 will appear in Journal
of the AMS.
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, Séminaires et Congrès 24-II (2012), 313-350.
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.)