| Date |
Speaker
|
SPECIAL LECTURE
January 26
3:00 |
Carolina
Benedetti (U. of Comlombia, South America)
Volumes of matroid polytopes
Given a matroid M we can associate with it its matroid polytope
$P_M$ as well as its independent set polytope I_M. In this talk
I will show an explicit way to decompose these polytopes as
a signed Minkowski sum of simplices. Using this decomposition,
which involves a lot of information of M such as its beta invariant,
I will give a nice formula to calculate the volumes of P_M and
I_M, offering a geometric point of view for beta (M). Finally,
we will see analogous results in the case of a nice class of
flag matroids, namely the cascading flag matroids. In this case
the role of beta (M) will be done by the gamma invariant. This
is joint work with Federico Ardila (San Francisco State University)
and Jeffrey Doker (UC Berkeley). |
| January 9, 2009 |
Alexander
Rossi Miller, University of Minnesota
Differential posets and Smith normal forms
In this talk I will introduce r-differential posets, then
move on to some conjectures about their structure. The main
focus will be on a conjecture asserting a strong property
for the up and down maps U and D in an r-differential poset:
DU+tI and UD+tI have Smith normal forms over Z[t]. We will
discuss the current progress of this conjecture in Young's
lattice, as well as in Young-Fibonacci posets. This is joint
work with Vic Reiner.
A preprint is available at http://arxiv.org/abs/0811.1983
|
SPECIAL LECTURE
November 24
3:00 |
Amel Kaouche, University du Québec à
Montréal (UQAM)
Imperfect gases and graph invariants
The Mayer and Ree-Hoover theories for the virial expansions
in the context of a non-ideal gas reveal certain invariants
(weights) associated to graphs. We give a special attention
to the case of the hard-core continuum gas in one dimension.
We present the method of graph homomorphisms that we apply
to compute the Mayer and Ree-Hoover weights of various classes
of graphs.
|
November 21
3:30 |
Luis Guillermo Serrano Herrera, University of Michigan
The shifted plactic monoid
We introduce a shifted analog of the plactic monoid of Lascoux
and Schützenberger, the shifted plactic monoid. It can
be defined in two different ways: via the shifted Knuth relations,
or using Haiman's mixed insertion.
Applications include: a new combinatorial derivation (and
a new version of) the shifted Littlewood-Richardson Rule;
similar results for the coefficients in the Schur expansion
of a Schur P-function; and a shifted counterpart of the theory
of noncommutative Schur functions in plactic variables.
A preprint is available at http://arXiv.org/abs/0811.2057.
|
SPECIAL LECTURE
November 17
3:00 p.m. |
Joel
Kamnitzer, University of Toronto
MV polytopes and components of quiver varieties
A number of interesting bases exist for the upper half of the
universal envelopping algebra of a semisimple Lie algebra. One
such basis is Lusztig's semicanonical basis which is indexed
by components of quiver varieties.
Another interesting basis is indexed by Mirkovic-Vilonen cycles
which lead to the combinatorics of MV polytopes. In this talk,
I will explain a natural bijection between the components of
quiver varieties and the MV polytopes. This is joint work with
Pierre Baumann. |
November 14
3:30 |
Janvier
Nzeutchap, York University and Fields Institute
Robinson-Schensted Algorithm for Shifted Tableaux, P-Schur
and Q-Schur functions (part 2) |
|
SPECIAL LECTURE
November 10
3:00 p.m.
|
Hugh
Thomas (University of New Brunswick)
Antichains in the poset of positive roots, Catalan phenomena,
and some conjectures of Panyushev
Associated to any finite crystallographic root system, there
is a certain number, the generalized Catalan number. There are
two major families of objects counted by the generalized Catalan
number: one family contains the clusters in the associated cluster
algebra and the noncrossing partitions in the associated reflection
group, and others, while the second family contains the antichains
in the poset of positive roots, regions in the Shi arrangement
inside the dominant chamber, and others. There are bijections
within each family, but no natural type-free bijection between
the families. I will report on work towards constructing such
a bijection. It turns out that a crucial ingredient is a certain
cyclic action on the antichains in the poset of positive roots
defined by Panyushev (arXiv:0711.3353). This action is non-trivial
to analyze even in type A. In the course of our construction,
we prove some of Panyushev's conjectures about his action. This
is joint work with Drew Armstrong. |
October 31
3:30 |
Janvier
Nzeutchap, York University and the Fields Institute
Robinson-Schensted Algorithm for Shifted Tableaux |
October 24
3:30 |
Janvier Nzeutchap, York University and the Fields
Institute
The Poirier-Reutenauer Hopf algebra of tableaux (part 2)
During this session, we will use many elementary examples.
1. review the definition of plactic Schur functions
2. product of tableaux and concatenation of languages (coplactic
classes)
3. products of tableaux and posets isomorphism: a suspected
bijection
4. a Yamanouchi poset
|
October 17
3:30 |
Janvier
Nzeutchap, York University and the Fields Institute
The Poirier-Reutenauer Hopf algebra of tableaux
During this session, we will use many elementary examples.
1. review some essential definitions related to this algebra
2. application 1: the Littlewood-Richardson rule
3. introduce the permurohedron, dominance and tableauhedron
orders
4. application 2: tableaux posets and Kostka numbers
5. application 3: an interpretation of homogenous symmetric
functions
6. recall a result due to Taskin: each product of two tableaux
is an interval of each tableaux poset
7. state problem 1: describe cover relations in the dominance
and tableauhedron orders
8. state problem 2: an efficient algorithm to compute product
of two Schur functions
9. research direction 1 (introduction): products of tableaux
and posets isomorphism |
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