Of
Graph Theory and
Combinatorics
2005-06 SCHEDULE
Tues, Oct 18 – David Perkinson (
“Convex
hulls of finite group representations”
Abstract: The image of a real
representation of a finite group is a collection
of n x n matrices sitting in the Euclidean space
R^{n^2}. What is the relation between
properties of the representation and the structure of
the polytope that is the
convex hull of this collection of matrices? This talk will
report on work done
with Reed College students in several undergraduate theses.
Tues, Nov 1 – Kira
Durand and Dustin Toci (
“Competitive
Coloring on Trees and Forests”
Abstract: While it has been proven that the game chromatic number
for
any forest is at most four, the problem of classifying which forests have game
chromatic number two, three, and four has not been looked in to. After a brief
intro to competitive graph coloring, we will recap the results of our research on this
problem. A walkthrough of our proofs for the classifications of forests according to
game chromatic number will be given as well as a discussion of our related findings,
namely our search for the smallest tree with game chromatic number four.
Tues, Dec 6 – Klay
Kruczek (
"An
Introduction to Positional Games"
7:00pm,
Room 385, Neuberger Hall
Abstract: In this talk, we will learn about Positional Games, which can be
thought of as generalized Tic-Tac-Toe games played on a hypergraph, where the
winning lines in these games are the hyperedges. We will discuss the Erdos-Selfridge
Theorem (the breakthrough theorem in this area) and Tumbleweeds, which are
hypergraphs that make the bound in this theorem tight. Finally we will discuss
Player 2 pairing strategies in N^d Tic-Tac-Toe.
Tues, Feb 21 – Jenny Quinn (
"An
Alternate Approach to Alternating Identities"
Abstract: Identities involving sums of alternating terms are
frequently tackled using
the Principle of Inclusion-Exclusion. While a powerful tool, it has a tendency
to obscure any relationship among the sets being considered. My goal is
to directly and concretely understand alternating identities by finding
correspondences between odd and even sets. In fact this strategy can be
used to prove the Principle of Inclusion-Exclusion...among other things.
Tues, Mar 21 – Nathan Segerlind
" Applications of Probabilistic Combinatorics in Propositional Logic"
Abstract: Propositional logic is reasoning about true/false assertions built up from
atomic true/false assertions (variables). A propositional proof system is an efficiently
checkable method for certifying that propositional statements evaluate to true for all
assignments to the variables (that the statement is a tautology). A fundamental question
in the study of propositional proof complexity is whether or not there is a propositional
proof system P and a constant c > 0 so that for all tautologies \tau written in n symbols
there is a P proof of \tau of size at most n^c. This problem is equivalent to the NP versus
coNP problem of computational complexity, and its negative resolution implies that P \neq NP.
Motivated by this connection to computational complexity and an interest in unconditional
results for satisfiability algorithms, there has been much interest in this problem for specific
propositional proof systems. In this talk, we will discuss results for extensions of the
resolution system. The methods of probabilistic combinatorics enter in two ways. The first is in
the construction of the tautologies which are candidates for requiring large proofs. The second is
a method of converting k-DNFs into constant height decision trees, by the application of a
random partial assignment to the variables.
Tues, Apr 11 – Naiomi
Cameron (Lewis and Clark)
“Pseudo-Involutions
in the Riordan Group”
Abstract: A Riordan matrix
is an infinite, lower triangular matrix whose columns
are generating functions. The
study of the Riordan group structure (induced by
matrix multiplication) is of interest in its own right, but it can also lead
to the discovery and proof of nice combinatorial identities. In this talk,
I will investigate some combinatorial aspects of involutions in the Riordan
group. In particular, I will describe a class of Riordan matrices having
pseudo-order
two (which includes Pascal's triangle), and illustrate
how to obtain proofs of combinatorial
identities, such as
Coming soon: Charles
Trevellyan, Sean Powers!
Back to my home page.