Combinatorics and Geometry
Org: Ian Goulden (Waterloo)
- FRANÇOIS BERGERON, UQAM
Diagonal coinvariant space, what's up?
We will give a description of some of the current developments
surrounding the study of diagonal coinvariant space, for reflection
groups, and related spaces.
- NANTEL BERGERON, York University
Lattices, Representations and (Hopf) Algebras
We show that we can associate two graded dual structures of bialgebra
to the Grothendieck ring associated to certain tower of lattices.
This is work in progress with M. Zabrocki and H. Thomas.
- GRAHAM DENHAM, University of Western Ontario
Discrete Morse theory for infinite complexes
Ten years after its introduction, discrete Morse theory has proven to
be a wonderful tool to study the homotopy type of a finite,
combinatorially defined complex. I will describe a generalization to
a class of infinite complexes: as an example, one obtains new proofs
that certain combinatorial models for classifying spaces are, in fact,
- JOHN IRVING, Wilfrid Laurier University
Tree Pruning and Transitive Factorizations in the Symmetric
The primary focus of this talk will be the (genus 0) Hurwitz
enumeration problem, which asks for the number H0(p) of
decompositions of a given permutation p into an ordered product of
a minimal number of transpositions such that these factors act
transitively on the underlying set of symbols. (The problem is
typically phrased in terms of counting almost simple branched
coverings of the sphere by the sphere with arbitrary ramification over
one special point, but the two phrasings are equivalent.) I shall
demonstrate that transitive factorizations can be encoded as planar
edge-labelled maps with certain descent structure, and
describe a bijection that "prunes trees" from such maps. This
allows for a shift in focus from the combinatorics of factorizations
to the sometimes more manageable combinatorics of smooth maps. As a
result, we gain some combinatorial insight into the nature of
Hurwitz's famous formula for H0(p), and derive new bijections
that prove his formula in certain restricted cases.
- DAVID JACKSON, University of Waterloo, 200 University Avenue West, Waterloo,
Ontario N2L 3G1
Combinatorial aspects of the double Hurwitz numbers and
Faber's intersection numbers
Faber's intersection numbers, double Hurwitz numbers and Hurwitz
numbers contain a surprising amount of combinatorial structure.
Goulden, Vakil and I hope to gain further understanding of the Faber
intersection numbers by examining their relationship to the double
Hurwitz numbers. I shall give an account of the high points of this
work by starting with the classical case of genus zero covers of the
sphere, for it is in this case that there is a glimpse of an instance
of a basic algebraic-combinatorial structure that appears to pervade
our approach. Vakil's earlier talk has given an introduction to the
moduli space of curves and an impression of how such geometric
questions may be translated into combinatorics. I shall describe how
an approach through algebraic combinatorics may be used to determine
further properties of these numbers.
- JOEL KAMNITZER, University of California, Berkeley
We will discuss the Mirkovic-Vilonen polytopes which give a new model
for the combinatorics of representations of complex reductive groups.
These polytopes are the moment map images of the Mirkovic-Vilonen
cycles, which are varieties arising from the geometric Satake
Isomorphism. They also arise from the combinatorics of Lusztig's
canonical basis. We will discuss new results concerning a crystal
structure on the set of these polytopes.
- ALLEN KNUTSON, UC Berkeley
Multidegrees, their computation, and applications
Homogeneous polynomials can be interpreted as T-equivariant
cohomology classes on affine space. Having positive coefficients is a
sign that they are geometric, or more precisely, "effective", being
representable as the class of a T-invariant subscheme. With such a
geometric interpretation in hand, there are various ways to compute
the class in automatically positive ways.
I'll explain a couple of general recipes for doing this, one being
"geometric vertex decompositions", and apply them to matrix Schubert
varieties; one payoff will be a bunch of old and new formulæ for
double Schubert (and Grothendieck) polynomials.
It's not too much of a surprise, though, that geometry helps one
compute Schubert polynomials, as they have a geometric origin. So
I'll also talk about a very surprising application of multidegrees in
statistical mechanics, where the combinatorics predated the geometry,
and is still very mysterious.
This work is joint with Ezra Miller, Alex Yong, and Paul Zinn-Justin.
- KEVIN PURBHOO, University of British Columbia
The generalised Horn recursion
The classical statement of Horn's conjecture gives a recursively
defined set of inequalities for possible eigenvalues of triples of
Hermitian matrices (A,B,C) satisfying A+B+C=0. Reformulated,
however, this recursion tells us that there is a recursive nature to
the set of non-vanishing Littlewood-Richardson numbers. I'll discuss
a generalisation which recursively characterises the non-vanishing
Schubert intersection numbers for all minuscule flag varieties.
This is joint work with Frank Sottile.
- KONSTANZE RIETSCH, King's College, London (UK), and Waterloo
Quantum cohomology of G/P and the Peterson variety
Let G be a reductive linear algebraic group. The Peterson variety
is a projective subvariety of the Langlands dual flag variety which,
by a remarkable theorem of Dale Peterson's, has strata that are
isomorphic to Spec(qH*(G/P)) for the varying parabolic
subgroups P in G. We give a new construction of these strata
shedding some (vague) light on the why and wherefore of Peterson's
- MIKE ROTH, Queen's University, Kingston, Ontario
Varieties with positive definite intersection form
The talk will discuss the problem of finding varieties with positive
definite middle intersection form.
- MISHA SHAPIRO, Michigan State University
Single and double Hurwitz numbers
The classical problem of finding numbers of ramified covers of the two
dimensional sphere of fixed ramification types goes back to Hurwitz.
The interests to Hurwitz numbers was revived after physicists
discovered that these numbers play central role in quantum
chromodynamics. I will describe the method of computing so-called
single Hurwitz numbers. These formulas were used by Okounkov and
Pandharipande to give another proof of Witten's conjecture. We will
also discuss an approach to Hurwitz numbers of coverings with two
- GREG SMITH, Queen's Unversity
Toric varieties as fine moduli spaces
We'll discuss how to construct projective simplicial toric varieties
as the fine moduli space of representations of a quiver.
- HUGH THOMAS, University of New Brunswick
The lattice of noncrossing partitions via representation
theory of quivers
Associated to any finite reflection group, there is a combinatorial
object called the lattice of noncrossing partitions. In type A,
these are just the classical noncrossing partitions. In this talk, I
will discuss a new approach to the lattice of noncrossing partitions
for crystallographic reflection groups, using the representation
theory of quivers. This approach yields a new proof that the
noncrossing partitions do indeed form lattices for these groups (a
result proved in a type-free way for the first time earlier this year
by Brady and Watt), and also clarifies connections between noncrossing
partitions, clusters, and other related objects.
This is joint work with Colin Ingalls.
- RAVI VAKIL, Stanford University, Stanford, CA 94305, USA
Double Hurwitz numbers and the intersection theory of the
moduli spaceof curves
The moduli space of genus g curves, the "parameter space" of all
genus g Riemann surfaces, is a central object in geometry. Its
topology exhibits some remarkable structure. As an important example,
Faber's "intersection number conjecture" predicts that its
cohomology ring exhibits unsuspected combinatorial structure, related
to the symmetric group. This conjecture will soon be a theorem,
thanks to Givental's proof of the "Virasoro conjecture for projective
space" and the details soon to be provided in a book by Lee and
Pandharipande. However, this proof does not "explain" why this
combinatorial structure should be there. Ian Goulden, David Jackson,
and I hope to do this by exploiting a relationship of these
intersection numbers with double Hurwitz numbers; this programme is
complete in a large class of cases ("up to three points"). In this
lecture I will give an introduction to the moduli space of curves and
Faber's conjecture, and I will give some impression of how to
translate these geometric questions into combinatorics.
The translation uses joint work with Tom Graber.
- ALEXANDER YONG, UC Berkeley/Fields Institute
On Smoothness and Gorensteinness of Schubert varieties
The study of singularities of Schubert varieties in the flag manifold
involves interesting interplay between algebraic geometry,
representation theory and combinatorics.
Although all Schubert varieties are Cohen-Macaulay, few are smooth.
An explicit combinatorial characterization of the smooth ones was
given by Lakshmibai and Sandhya (1990). The singular locus of an
arbitrary Schubert variety was determined around 2001 by several
Gorensteinness is a measurement of the "pathology" of the
singularities of an algebraic variety; it logically sits between
smoothness and Cohen-Macaulayness. We explicitly characterize which
Schubert varieties are Gorenstein, analogous to Lakshmibai and
Sandhya's theorem. Here is the geometric interpretation: a Schubert
variety is Gorenstein if and only if it is Gorenstein at the generic
points of the singular locus. We also compute the canonical sheaf of
a Gorenstein Schubert variety as a line bundle in terms of the
I will discuss the geometric corollaries and questions that arise in
this work. This is a joint project with Alexander Woo.