Of
Graph Theory and
Combinatorics
2006-07
SCHEDULE
Weds, May 9 – Peter
Otto
(
"The Random Cluster Model"
7:30pm, Room 343,
Neuberger Hall
Abstract: In
1969 Kees Fortuin and Piet Kasteleyn introduced the random
cluster
model as a unification
of two of the most famous models in statistical mechanics,
namely, the bond
percolation model and the Ising/Potts model. Its connection
to bond
percolation is that the random cluster model is also a stochastic geometry
model that places the
randomness on the edges of a
graph. On the other
hand, the connection
to the Potts model is that it incorporates the q states that the
vertices of the Potts models can randomly take on.
As a result, through the random
cluster model, connectivity
questions in the edge model can be related to questions
of correlations in the corresponding vertex model. In this talk, I'll
introduce the general
random cluster model and
its coupling to the Ising/Potts model and then focus on the
models defined on the complete
graph to discuss the phase transition behavior
of these models.
Weds, Apr 18 – Brett
Stevens
(
"Primitive pentanomials, shift
register
sequences and orthogonal arrrays"
7:30pm, Room 343,
Neuberger Hall
Abstract:
Consider a maximum-length binary shift-register sequence generated by
a primitive polynomial f of degree m. Let C_n^f denote the set of all
subintervals
of this sequence
with length n, where m < n < 2m+1 , together with the zero vector of
length n. Munemasa considered the case in which the polynomial f generating the
sequence is a trinomial satisfying certain conditions. He proved that, in
this case,
C_n^f corresponds to an orthogonal array
of strength 2 that has a property very close
to being an orthogonal array of strength
3. Munemasa's result was based on his
proof that
very few trinomials of degree at most 2m are divisible by the given
trinomial f.
In this talk, we
consider the case in which the sequence is generated by a pentanomial f
satisfying
certain conditions. Our main result is that no trinomial of degree at most
2m is
divisible by the
given pentanomial
f, provided that f is not
in a finite list of exceptions we
give. As a
corollary, we get that, in this case, C_n^f corresponds to an
orthogonal array
of strength
3. This effectively minimizes the skew of the Hamming weight distribution
of
subsequences in the
shift-register sequence.
Weds, Apr 4 – Jamie Pommersheim
(
"Combinatorial
Games and Factorization
in Generalized
Power Series Rings"
7:30pm, Room 343,
Neuberger Hall
Abstract: In the
1970s, John Horton Conway introduced the surreal numbers, a
number system that
can be used to quantify the play of many combinatorial games.
The surreal numbers
form a field containing all real numbers as well as all ordinal
numbers.
Among the elements of this field, one finds omega (the first
infinite
ordinal), omega - 1 , sqrt{omega}, 1/omega (an infinitesimal)
as
well as many more
exotic numbers. Within the surreal numbers, there is
also a natural generalization of the integers, called the omnific
integers,
denoted OZ. The first
three numbers listed above are omnific integers. With
this new notion of integer,
one can revisit many of the classical questions
of number
theory. Most immediately, what are the prime numbers? Can
every omnific integer be factored uniquely into omnnific prime
numbers? A
structure theorem tells us that every surreal number can be
represented as a
certain generalized power series, so these questions are
closely related to
algebraic properties of generalized power series rings. In this
talk, after
introducing the surreal numbers, we'll explore some questions about
primes in these rings.
Tues, Feb 13
– Erin McNicholas
(
“Embedded Tree Structures of Genus
Zero One-Face Maps”
7:00pm, Room 375, Neuberger Hall
Abstract: Motivated
by questions from Random Matrix Theory, we examine a class of
three-regular graphs used to represent one-face maps. The embedded graph
of a genus zero one-face map is a planar tree, and there is a
correlation between its vertices and the primitive cycles of the
associated three-regular graph. In this talk, we examine the structure
and eigenvalue statistics of these embedded planar
trees. In particular,
we show how the Dyck path representation can be used
to recast questions
about the probabilistic structure of random planar trees into
straightforward counting problems. Using this Dyck
path approach, we
find: 1. the expected number of degree k vertices adjacent to j degree d
vertices in a random planar tree, 2. the structure of the planar tree's
adjacency matrix under a natural labeling of the vertices, and 3. an
explanation for the existence of eigenvalues with
multiplicity greater
than one in the tree's spectrum.
Tues, Nov 28 – Elise Lockwood
(
"Matchings
Polynomials are the Coolest Things EVER!"
7:00pm,
Room 377, Neuberger Hall
Abstract: The
matchings polynomial of a graph G is simply a generating
function
for the number of matchings of various sizes in G. In a famous
paper of Heilmann
and Lieb, this polynomial was used to study the properties
of
monomers and dimers, sparking a lot of research into
the Ising model and
many other
aspects of models for phase transitions in statistical mechanics. But
all
applications aside, the matchings polynomial is a fascinating mathematical
object
to study
in its own right, rich with many surprises and deep connections – an
unexpected
joy, perhaps, given its humble beginnings as a formally
defined
generating function!
Sat, Nov 11 – Combinatorial Potlatch
Richard
Brualdi
(
“The
Bruhat Order for (0,1)-Matrices”
Abstract: We
discuss the classical Bruhat order on the set of
permutations of {1,2,3,...,n} and two possible
extensions to
more
general matrices of 0's and 1's.
Gary Gordon
(
“Graph
Polynomials For You; Graph Polynomials For Me”
Abstract: The Tutte polynomial of a graph is a two-variable
polynomial
that generalizes the very well studied one-variable
chromatic
polynomial. The Tutte polynomial can be generalized
and
specialized in other ways that give combinatorial and
"practical"
information about the graph. In this talk, we’ll meet
the Tutte polynomial and some of its cousins. We’ll also see
some
applications to an expected rank polynomial that
measures
how reliable a given graph is. Some of this work
is the
product of
Experiences
for Undergraduates) program.
Matt DeVos
(
“Sumsets and Subsequence Sums”
Abstract: Two
vibrant and interesting topics in combinatorics are
sumsets and
subsequence sums in abelian groups. We will give
a survey
of these areas, featuring some recent results (joint with
L. Goddyn, B. Mohar, and R. Samal), and an underappreciated
conjecture
of Schrijver and Seymour.
The
fun begins at 10:30am,
Room
171, Cramer Hall
Combinatorial Potlatch Webpage
Weds, Nov 8 – Marni
Mishna
(
“Classifying
Lattice Walks”
7:00pm,
Room 375, Neuberger Hall
Abstract: The
objects of study here are two-dimensional lattice walks, with a
fixed set
of step directions, restricted to the first quadrant. These walks are well
studied,
both in a
general context of probabilistic models, and specifically as particular case
studies
for fixed direction sets, notably the so-called Kreweras'
walks defined by the
direction set
{NE, W, S}. The goal here is to examine two series associated to these
walks: a
simple length generating function, and a complete generating function
which
encodes endpoints of walks, and to determine combinatorial criteria which
decide
when these series satisfy algebraic series (algebraic) or certain kinds
of linear
differential equations (D-finite), or none of the above. (Indeed we have
examples
that are not-D-finite) We shall present a complete classification of all
nearest neighbour walks where the set of directions is of
cardinality three, and discuss
how this
leads to a natural, well supported, conjecture for the classification of
nearest
walks with
any direction set. If time permits we might discuss how this work might
transfer
to walks on other kinds of graphs, such as Cayley graphs.
(Work in
progress with M. Bousquet-Melou, and A. Rechnitzer)
Weds, Oct 25
– Mark Maclean
(
“Distance-Regular
Graphs for the Working Mathematician”
7:00pm,
Room 375, Neuberger Hall
Abstract: Distance-regular
graphs are a class of undirected graphs possessing a high
degree of combinatorial regularity. As a familiar example, the graphs formed
by the vertices and edges of the five platonic solids are distance-regular.
In this talk I’ll be giving an introduction to distance-regular graphs and how
we study
them. I’ll then define a condition on these graphs that I call
the taut condition. We’ll explore the algebraic and combinatorial
implications
of the
taut condition -- and discuss whether it’s worth our attention.
Back to my home page.