Susumu Ariki (Research Institute for Mathematical Sciences, Kyoto University)

Abstract:
In 1995, Dipper, James and Murphy made a precise conjecture for
when the bilinear form of a cell module of the Hecke algebra of
type *B* vanishes. Their aim to classify simple modules was
achieved by a different approach, but the conjecture has remained
open. Based on jointwork with Kreiman and Tsuchioka, Jacon and I have
settled the conjecture affirmatively. The proof is achieved by
first converting the original problem to a crystal theoretic problem,
then by inventing a non-crystal theoretic method to control its path model
in the sense of Littelmann.

Cedric Bonnafé (Université de Franche-Comté)

Abstract:
Joint work with N. Jacon: Let *W _{n}* denote the Weyl group of type

Jim Carrell (University of British Columbia)

Abstract:
Let *G* be a semi-simple algebraic group and *B* a Borel
subgroup of *G*. The closure of a *B*-orbit in the flag variety
*G*/*B* is called a Schubert variety. Every Schubert variety *X*
is a projective variety. By the celebrated work of Kazhdan and
Lusztig, the singular locus of *X* in the sense of rational
smoothness determines where certain Kazhdan-Lusztig
polynomials are nontrivial and hence is of great importance
in representation theory. On the other hand, the
singular locus of *X* in the sense of algebraic geometry is
also of interest. In this talk, we will survey how the
singular loci are determined and how they are related. We
will also describe a partial desingularization via blowing up
in the rationally smooth setting.

Charlie Curtis (University of Oregon)

Abstract:
The main result is the calculation of the intersection *BwB* ∩ *yBx*^{−1}*B*,
for *w, x, y* in the Weyl group, in terms of the root structure. The set was originally
used by Iwahori to obtain the structure constants of the Iwahori Hecke algebra. As applications
a formula is obtained for the structure constants of the Hecke algebras of Gelfand-Graev
representations, and a new formula for the polynomials *R _{x,y}* of Kazhdan and Lusztig.

François Digne (Université de Picardie)

Abstract:
Broué's conjectures on blocks with abelian defect imply,
in the case of a finite group of Lie type, that the Hecke algebra
of some reflection group *W* should act on the cohomology of a
Deligne-Lusztig variety. One tries to define such an action
by making the braid group of *W* act on the variety
itself. This leads to some interesting conjectures in braid group
theory, involving Lehrer-Springer generalization of regular elements
in reflection groups.

Some of these conjectures have been recently solved.

Tony Dooley (University of New South Wales)

Abstract:

Matt Douglass (University of North Texas)

Abstract:
The Steinberg variety, *Z*, of a reductive complex algebraic group, *G*, is a
variety that has been used to understand representations of reductive
groups of the same type as *G*. The first part of this talk will give a
quick survey some applications and generalizations of the Steinberg
variety. The second part of the talk will be devoted to computing the
total Borel-Moore homology of *Z*.

Jie Du (University of New South Wales)

Abstract:
Almost at the same time as C.M. Ringel discovered the Hall
algebra realization of the positive part of the quantum enveloping
algebras associated with semisimple complex Lie algebras, A.A. Beilinson,
R. MacPherson and G. Lustig discovered a realization
for the entire quantum gl_{n} via a geometric setting of
quantum Schur algebras (or *q*-Schur algebras). This remarkable
work has many applications. For example, it provides a crude model
for the introduction of modified quantum groups, it leads to the
settlement of the integral Schur-Weyl reciprocity and, hence, the
reciprocity at any root of unity, and it has also provided a
geometric approach to study quantum affine
sl_{n}. The BLM
work has also been used to investigate the presentations of
*q*-Schur algebras, infinitesimal quantum
gl_{n} and their
associated little *q*-Schur algebras. In this talk, I will focus
on the latest developments of the Beilinson-Lusztig-MacPherson
approach in the study of quantum
gl_{∞}, infinite
*q*-Schur algebras and their representations.

This is joint work with Qiang Fu.

Matthew Dyer (University of Notre Dame)

Abstract: We consider a certain category, the objects of which are Coxeter systems with a given "reflection representation" over a possibly non-commutative coefficient ring. The full subcategory of objects associated to a fixed Coxeter system has (trivially) a universal object. A basic observation used to establish favorable properties of the corresponding "universal coefficent rings" and "universal root systems" is that the universal objects behave functorially with respect to morphisms of Coxeter systems which are injective on the simple reflections.

Michael Eastwood (University of Adelaide)

Abstract: The classical Radon transform takes a function on the plane and integrates it over the straight lines in the plane. Its invertibility provides the mathematical basis of modern medical imaging techniques. The X-ray transform takes a function in three-space and integrates it over the straight lines, the terminology being motivated by medical imaging. As one might expect, both of these transforms are best viewed on real projective space. In this talk, I shall discuss what happens on complex projective space where the straight lines are the Fubini-Study geodesics. This is joint work with Hubert Goldschmidt.

Matt Fayers (Queen Mary)

Abstract:
Let H_{n}
denote the Iwahori-Hecke algebra of the symmetric group over a field * F*;
we are interested in computing the decomposition numbers for
H

Omar Foda (University of Melbourne)

Abstract: Elements from classical integrable systems show up with increasing regularity in recent developments in mathematical physics.

The Kyoto school approach to classical integrable systems can be used to classify and extend these results.

I wish to introduce (and give some examples of) a recently discovered connection between plane partitions and free fermion vertex operators.

The talk should be (hopefully) introductory and the example(s) to be discussed are meant to be quite explicit.

Joint work with M Wheeler (Melbourne).

Dennis Gaitsgory (Harvard University)

Abstract:
The geometric Satake equivalence states that
one can realize the Langlands dual group * ^{L}G*, or
rather the category of its representations Rep(

The category Rep(* ^{L}G*) has a natural 1-parameter
deformation to Rep(

In the talk, I'll discuss a solution to this problem, suggested
recently by Jacob Lurie. Its essence is that instead of the
*G*[[*t*]]-equivariance condition on a perverse sheaf, we should
impose a Whittaker condition. This allows for the desired
q-deformation.

If time permits, we will also discuss the implications of the above construction to both local and global quantum Langlands duality.

Victor Ginzburg (University of Chicago)

Abstract:
We introduce a class of holonomic *D*-modules on GL_{n}×
**C**^{n}. The
corresponding perverse sheaves are reminiscent of (and include as special
cases) Lusztig's character sheaves.

Simon Goodwin (University of Birmingham)

Abstract:
Let *G* be a linear algebraic group defined over
**F**_{p} and let *X* be a *G*-variety defined over
**F**_{p}. For *q* a power of *p*, we write *G*(*q*) for the
group of **F**_{q}-rational points of *G*, and *X*(*q*) for the
set of **F**_{q}-rational points of *X*. We consider questions
about uniformity in *q* of the number *k*(*G*(*q*),*X*(*q*)) of
*G*(*q*)-orbits in *X*(*q*). We are mainly concerned with the case
where *X* is a normal subgroup or overgroup of *G*, and *G* is
acting by conjugation.

Fred Goodman (University of Iowa)

Abstract: Affine and cyclotomic BMW algebras are BMW analogues of affine and cyclotomic Hecke algebras. I will survey what I know about these algebras and describe some difficulties that arise in the cyclotomic case. This is mostly joint work with Holly Hauschild Mosley.

John Graham

(Joint work with Richard Green.)

Abstract: The Hecke algebra of the symmetric group has a quotient called the Temperley Lieb algebra which is often defined using certain planar diagrams. We extend this diagram calculus to an analogous quotient of the Hecke algebra of any Weyl group in a uniform manner. We also discuss the affine case.

Ian Grojnowski (University of Cambridge)

Abstract:

Dick Hain (Duke University)

Abstract:

Jun Hu (Beijing Institute of Technology)

Abstract:
We prove an integral version of Schur-Weyl duality
between the specialized Birman-Murakami-Wenzl
algebra *B _{n}*(−

Jens Jantzen (Aarhus Universitet)

Abstract:

Alexander Kleshchev (University of Oregon)

Abstract:
In this talk we will explain an attempt to define Verma modules for finite *W*-algebras of any
type (by Brundan-Goodwin-Kleshchev). This is a step toward understanding finite dimensional modules
over *W*-algebra, at least in the standard Levy type. We then explain how everything works
nicely in type *A*, where a complete and satisfactory theory is available. We will also talk
about applications to (degenerate) cyclotomic Hecke algebras via a Schur-Weyl type duality
between these algebras and finite *W*-algebras in type *A*.

Hanspeter Kraft (University of Basel, Switzerland)

Abstract:
Given a selfdual faithful representation *V* of a reductive group *G* (in
characteristic zero)
we obtain a beautiful Galois correspondence between closed reductive subgroups of *G* and
certain subalgebras of the tensor algebra *T*(*V*), namely those which are graded,
contraction-closed and non-degenerate. (These notions will be explained in the talk.)
As a consequence, one gets – in a unified way – the well-known First Fundamental Theorems
for the classical groups and – in addition – some new FFTs for other groups.
(This is based on some unpublished work of Lex Schrijver in the case of compact groups.)

Mathai Varghese (University of Adelaide)

Abstract:
I will give new decription of bivariant *K*-theory
in terms of noncommutative correspondences, a
diagram calculus for bivariant *K*-theory to manipulate
the intersection products, and the applications to
a noncommutative Grothendieck-Riemann-Roch theorem
and duality. This represents ongoing joint work with
Brodzki, Rosenberg and Szabo.

Jean Michel (Paris VII University)

Abstract:
If the *l*-Sylow *S* of *G* is abelian, there is a derived equivalence
between the principal *l*-block *B* of *G* and the principal *l*-block
*b* of *N _{G}*(

Alex Molev (University of Sydney)

Abstract: A new version of Cherednik's fusion procedure will be discussed. We show that the matrix unit formulas for the symmetric group provided by this procedure can be derived from a construction of Murphy.

Paul Norbury (University of Melbourne)

Abstract:
The moduli space of genus *g* curves with *n* labeled points is a
Kahler manifold of finite volume. The Kahler form can be deformed to give
symplectic moduli spaces consisting of hyperbolic genus *g* surfaces with *n*
geodesic boundary components of given lengths. Mirzakhani proved that
the volumes of these moduli spaces are polynomials in the lengths of the *n*
geodesic boundary components by computing the volumes recursively in *g* and
*n*. The polynomials have deep properties - in particular their
coefficients are rational intersection numbers on the moduli space of
curves. By allowing cone angles on hyperbolic surfaces we give new
recursion relations between the volume polynomials. This has interesting
consequences for the geometry of the moduli space.

Cheryl Praeger (University of Western Australia)

Abstract:
The fundamental problem that prompted the research I will report on was that of estimating the
proportion of involutions in a finite classical group that have a "large but not too large" fixed
point space. Such involutions were required by Leedham-Green and O'Brien for a black-box classical
group recognition algorithm. Applying a theorem of Niemeyer and mine enabled Leedham-Green and O'Brien
to construct such an involution, with high probability, by examining *O*(*n*) random elements, where n is
the dimension of the classical group. In joint work with Niemeyer and Lubeck, we have reduced the
number of random elements needed to *O*(log *n*). Our methods exploit the link between *F*-stable tori and
Weyl group elements in classical groups, and also apply results about the probability distribution of
elements in finite symmetric groups.

Claudio Procesi (Universita di Roma, La Sapienza)

Abstract:

Arun Ram (University of Wisconsin, Madison)

Abstract: This talk is about the combinatorics of indexing points in affine flag varieties. It is possible to make choices so that the points are indexed by a refinement of Littelmann's path model in such a way that the Schubert cell and the Mirkovic-Vilonen slice are easily read off the "path" indexing of the point. From this, the relations for the affine Hecke algebra can be derived, both in the Iwahori-Matsumoto and in the Bernstien generators. If time permits I will discuss the action of the "root operators" on points, and/or the relation to the Kamnitzer and Baumann-Gaussent indexings of Mirkovic-Vilonen cycles.

Jacqui Ramagge (University of Wollongong)

Abstract: In 1994 Willis developed the start of a structure theory for totally disconnected locally compact groups. This has initiated a program to classify the topologically simple groups in the class.

I will explain why such groups are interesting, give an introduction to the structure theory and discuss some results, with a particular focus on Kac-Moody groups.

Hyam Rubinstein (University of Melbourne)

Abstract: This is joint work with B. Rubinstein and P. Bartlett. Machine learning combines ideas from computer science, statistics and functional analysis. However there are also geometric and topological aspects. PAC (probably approximately correct) learning was introduced by Valiant in the early 1980s. As larger sample sizes are given, then a class of concepts can be learnt more and more accurately, (called learnability of the class) if and only if the VC dimension is finite. It is well known that a mistake bound for prediction is given by the one inclusion graph density, which is bounded by the VC dimension. We prove that a better mistake bound, found experimentally by Kuzmin and Warmuth, is valid. We also extend the classical bounds to the case of multiclasses, where a finite number of possibilities occur for each sample choice, rather than two. Finally we establish the peeling conjecture of Kuzmin and Warmuth.This is that maximum classes of given VC dimension can be ordered so that one concept vertex can be removed at a time, where the vertex has degree at most the VC dimension. As corollaries, this explains why maximum classes have many compression schemes and why it is difficult to compress maximal classes by embedding them into maximum ones of the same VC dimension. This is related to the important problem, whether compressibility is equivalent to finite VC dimension (and hence learnability).

Toshiaki Shoji (Graduate School of Mathematics, Nagoya University)

Abstract:
Let H* _{n,r}*
be the Ariki-Koike algebra associated to the
complex reflection group

T. A. Springer

Abstract:
Let *G* = Sp_{2n}(**C**), acting
in *V* = **C**^{2n}.
Put V = *V* ⊕ Λ^{2} *V*.
The exotic nilcone is the nilcone (set of unstable points) in V,
relative to the action of *G*. It is used by Syu Kato in his geometric approach to
the representation theory of the multi-parameter Hecke affine algebra
associated to *G*.

My talk will be quite elementary. I shall discuss the basic properties of V.

Ross Street (Macquarie University)

(Joint work with Craig Pastro.)

Abstract:
Given a monoidal comonad *G* on a monoidal category
C,
Moerdijk has observed that the category
C^{G} of Eilenberg-Moore
*G*-coalgebras (better called *G*-comodules) is monoidal via a lifting
of the tensor product of
C.
Recall that the left internal
hom * ^{B}D* of two objects

Kai Meng Tan (National University of Singapore)

Abstract:

Michela Varagnolo

(Joint work with E. Vasserot.)

Abstract: We study the finite dimensional representations of Cherednik algebras. The representation theory of these algebras splits into two case: the one of DAHA (double affine Hecke algebras) and the one of rational DAHA (a degeneration of the first one). The main result is the classification of all finite dimensional spherical simple modules for the rational DAHA. In order to do so we embeds the category of finite dimensional modules for the rational DAHA into the category of finite dimensional modules for the DAHA. Then we use a geometric construction of simple modules for the DAHA (proved by Vasserot) via the homology of the affine Springer fibers.

Changchang Xi (Beijing Normal University)

Abstract:
In this talk I shall
first report some results on cellular algebras introduced by Graham
and Lehrer, and then introduce the so-called affine cellular
algebras, a generalization of cellular algebras, and survey the
general representation theory and structural theory of affine
cellular algebras. Typical examples of affine cellular algebras are
the affine Temperley-Lieb algebras studied by Graham and Lehrer, and
others, and the affine Hecke algebras of type *A* studied by
Lusztig, Nanhua Xi, and others.

The contents of my talk are taken from joint works with Steffen Koenig.

Nanhua Xi (Chinese Academy of Sciences)

Abstract: Some maximal elements in a baby Verma module are constructed. The elements should be helpful to understand the structure of baby Verma modules.

Amnon Yekutieli (Ben Gurion University)

Abstract:
Let *X* be a smooth algebraic variety over a field of
characteristic 0, endowed with a Poisson bracket. A quantization of
this Poisson bracket is a formal associative deformation of the
structure sheaf *O _{X}*, which realizes the Poisson bracket as its first
order commutator. More generally one can consider Poisson
deformations of

I will explain what these deformations are. Then I'll state a
theorem which says that under certain cohomological conditions on *X*,
there is a canonical quantization map (up to gauge equivalence).
This is an algebro-geometric analogue of the celebrated result of
Kontsevich (which talks about differentiable manifolds).

It appears that in general, without these cohomological conditions,
the quantization will not be a sheaf of algebras, but rather a stack
of algebroids, otherwise called a twisted associative deformation of
*O _{X}*.

In the second half of the talk I'll talk about twisted deformations and twisted quantization, finishing with a conjecture.

The work is joint with F. Leitner (BGU).

Ruibin Zhang (University of Sydney)

(Joint work with Gus Lehrer.)

Abstract:
The Temperley-Lieb algebra may be thought of
as a quotient of the Hecke algebra of type *A*, acting on tensor
space as the commutant of the natural action of U_{q}(sl_{2}) on
(**C**(*q*)^{2})^{⊗n}. We define and study a quotient of the
Birman-Wenzl-Murakami algebra, which plays an analogous role for the
3-dimensional irreducible representation of
U_{q}(sl_{2}).