In the classical finite projective plane, PG(2,q), there is only one non-degenerate conic – a set of points satisfying a quadratic form.
These simple structures have a rich theory. For instance, when q is odd, they can be defined synthetically, without any regard to their natural algebraic definition. In this talk, I will survey some of the broad uses of conics in the study of coding theory and cryptography. We look at optical orthogonal codes, low-density parity-check codes, and a certain cryptographic protocol. In each case, we use configurations of conics to provide new examples, and to study properties of the objects in question. Along the way, I plan to discuss the involvement of undergraduates in this line of research.

The primary combinatorial tool used in reliability testing is the the covering array, which generalizes orthogonal arrays, Sperner set systems and is related to Erdős Ko Rado-type results. In the past few years the model has been adapted to incorporate application relevant issues: known non-interactions in the system under test, mixed alphabet sizes and mixed strength. We will discuss two more recent adaptations which are surprisingly related: avoiding pairs of forbidden interactions and locating the exact location of an error when its existence is discovered. We will discuss the complexity of these questions, combinatorial characterizations of feasibility and arrays and algorithms that solve these problems in some instances. We will use graph theory, complexity theory and combinatorial group testing techniques to address these problems.

The strongly connected reliability scRel(D,p) of a digraph D is the probability that the spanning subgraph of D consisting of the operational arcs is strongly connected, given that the vertices always operate, but each arc is independently operational with probability p [0,1]. We show that the closure of the set of roots of strongly connected reliability polynomials is the entire complex plane. This is joint work done with Jason Brown [1].

The boxplot, introduced by John Tukey in 1977, is a graphic used in exploratory data analysis. It allows the viewer to quickly determine the shape of a sample, including a quick determination of which data may be considered as outliers.
For large samples, it is an easy exercise in probability theory to determine the probability that a datum genuinely arising from a known distribution will be flagged as an outlier. For small samples, things are in general much more complicated and we must fall back on simulation. This note presents the simple (and not entirely artificial) case of a sample of size 5 from a uniform distribution, which can be successfully approached as a problem in combinatorial geometry.

A T(G) triple is formed by taking a graph G and replacing every edge with a 3-cycle, where all of the new vertices are distinct from all others in G. An edge-disjoint decomposition of 3K_n into T(G) triples is called a T(G) triple system of order n. If we can decompose K_n into copies of a graph G, such that we can form a T(G) triple from each graph in the decomposition and produce a partition of the edges of 3K_n, then the resulting T(G) triple system is called perfect. We show that the spectrum of perfect T(P₂ ∪ P₂ ∪ P₂) triple systems for n at least 9 is n ≡ 1,3 mod 6, and give cyclic constructions for all admissible orders. (Joint work with Josh Manzer, University of Victoria.)  

A balanced incomplete block design (BIBD) with parameters (v,k,λ) consists of a set of blocks, each of which is a k-subset of a set V of cardinality v, such that each 2-subset of V occurs in precisely λ of the blocks of the design. Two BIBD(v,k,λ) are said to be isomorphic if there exists a bijection f from the point set of one to the point set of the other such that f maps the blocks of the first design to the blocks of the second.
The existence of designs with given parameters has been an active area of research since their initial conception. Questions about the existence of non-isomorphic designs with the same parameters also have a long history.
We report on some techniques for constructing sets of non-isomorphic designs for particular parameters.

This is joint work with David Pike.  

We will give a brief update on some recent work that the speaker has been involved with in the quest for either a counterexample or a solution to Vizing's conjecture on the Cartesian product of 2 graphs. The good (and bad) news is that despite the current economic situation, there appears to be still lots of work available in this area!

A museum is attempting to guard each of its rooms. Rather than having guards placed at every room in a dominating set, Hartnell, Rall, and Whitehead (1998) considered having one watchman walk around the museum in such a way that the visited rooms form a dominating set; i.e., the watchman’s route is a dominating walk. As the goal is to minimize the amount of time for which any room is unobserved, we are looking for a ‘minimum closed dominating walk’ for a given graph. This talk will introduce the original problem and initial results and will then focus on a variation (from Davies, Finbow, Hartnell, Li, & Schmeisser, 2003) which considers the number of watchmen required when certain time restraints are placed on the monitoring of each room.  

Codiscovered by Coxeter and Grünbaum in the 1970’s, the 11-cell is a very beautiful abstract regular 4-polytope, with automorphism group PSL₂(11). Here we give what is perhaps the most natural construction (via orthogonal groups over the ring ℤ[τ]; work with Egon Schulte), then try to sensibly visualize the thing. (Work with Egon Schulte (Northeastern University) and UNB student Danny Hay).

Spin models are matrices with nonzero complex entries satisfying certain conditions related to the Reidemeister moves in knot theory. Matrices satisfying the ‘Type II’ condition form a larger class, and have been found in connection with symmetric designs, sets of equiangular lines, strongly regular graphs, and some distance regular graphs. In this talk we investigate 2-graphs, which are essentially weighted complete graphs, and weighted strongly regular graphs, and show that type-II matrices arise in connection with these objects as well. 

Abstract: Consider the game of locating a robber on a connected graph. At each turn, the locator chooses a vertex of the graph as a probe, and is given the distance from the probe to the robber. If she can uniquely locate the robber, she wins.
Otherwise the robber may move to a vertex adjacent to his location, but not to the probe vertex. The goal for the locator is to minimize the number of probes required to locate the robber.
This is a synthesis of the metric dimension location problem, the Finding Minou internet game, and the cop and robber graph game.

Abstract:We consider two option-closed combinatorial games, introduced by Nowakowski and Ottaway [1], called Roll the Lawn and Cricket Pitch. Both games use a row of nonnegative integers (or bumps) and a roller that is placed between any two bumps or at either end. Left (Right) moves the roller to the left (right), flattening each bump it passes over by 1 unless already at 0. At least one bump must decrease in size at each move. The roller is allowed to pass over bumps of height zero in Roll the Lawn, while it may not do so in Cricket Pitch. Nowakowski and Ottaway gave values for all positions of Roll the Lawn and determined outcome classes for Cricket Pitch positions, leaving game values as an open problem. We will utilize ordinal sums and introduce some effective reductions to determine the actual values of the game of Cricket Pitch.

TETRIS is a well-known video game that originated in the 1980s. Despite its popularity and the extensive research done on polyominoes tiling rectangles, there are still many unanswered questions about the mathematical properties of TETRIS. If the game is played at a constant speed, does a winning strategy exist such that a perfect player can play indefinitely? I will provide a brief overview of the history of TETRIS and outline the major results surrounding these winning strategies to date. As well, I will present a variant (non-rectangular) well in which new winning strategies are developed. A set of variant wells will be presented, permitting the application of these new strategies.