**Andrea Burgess,** Ryerson University

*Generalized designs, packings and coverings*
In 2009, Peter Cameron introduced a common generalization of
various classes of combinatorial designs such as balanced incomplete
block designs, resolvable designs and orthogonal arrays. Generalized
covering and packing designs can be defined in analogous way.
These objects bring into this framework further classes of designs,
including covering and packing arrays, Howell designs, monogamous
cycle decompositions and Kirkman signal sets. In this talk, I
will review known results on generalized designs, packings and
coverings, and discuss interesting open questions which arise
in the search for further examples of optimal generalized packing
designs. This talk will include discussions of joint work with
Robert Bailey, Michael Cavers, Peter Danziger and Karen Meagher.

**Asia Ivic Weis**, York University

*Combinatorial Structure of Chiral Polyhedra*

Classification of regular polyhedra in euclidean 3-space was
initiated by Grünbaum in 1977 and completed by Dress by addition
of a single polyhedron in 1985. In 2005 Schulte classified the
discrete chiral polyhedra in euclidean 3-space and showed that
they belong to six families. The polyhedra in three of the families
have finite faces and the other three families consist of polyhedra
with (infinite) helical faces. We show that all the chiral polyhedra
with finite faces are combinatorially chiral. However, the chiral
polyhedra with helical faces are combinatorially regular. Moreover,
any two such polyhedra with helical faces in the same family are
isomorphic.

This is a joint work with Daniel Pellicer.

**Karen Meagher**, University of Regina

*Erdös-Ko-Rado theorem for permutations*

Two permutations in the symmetric group on $n$ vertices can be
considered to be ``intersecting'' if they both map some $i \in
\{1,...,n\}$ to the same element (we say the permutations agree
on $i$). With this definition, what is the largest set of permutations
such that any two are intersecting? In the last 10 years, several
different

proofs have been published that show that the largest such set
is either the stabilizer of a point or a coset of the stabilizer
of a point. This result is known as the EKR theorem for permutations.

There are several questions that are natural to ask once this
result is established. For example what is the largest set of
permutations that agree on a set of $t$ elements? There are other
ways to define intersection for permutations, does an EKR type
result hold for these as well? What is the largest set of intersecting
permutations in a subgroup of the symmetric group? I will present
some new results by various researchers on these questions.

**Bruce Reed**, McGill University

*How long does it take to catch a drunk miscreant?*

We discuss the answer to a question of Churchley who asked how
long it will take a cop to catch a drunk robber who moves randomly.
We begin by discussing other variants of the cop-robber paradigm.
This is joint work with Alex Scott, Colin McDiarmid, and Ross
Kang. We rely heavily on work of Komarov and Winkler.

**Wenan Zang**, University of Hong Kong

*When is the Matching Polytope Box-totally Dual Integral?*

Let $G=(V,E)$ be a graph. The *matching polytope* of $G$,
denoted by $P(G)$, is the convex hull of the incidence vectors
of all matchings in $G$. As shown by Edmonds, $P(G)$ is determined
by the following linear system $\pi(G)$:

• $x_e\ge 0$ |
for each $e\in E$; |

• $x(\delta(v))\le 1$ |
for each $v \in V$; |

• $x(E[U])\le \lfloor \frac{1}{2}|U| \rfloor$ |
for each $U \subseteq V$ with $|U|$ odd. |

Cunningham and Marsh strengthened this theorem by showing that
$\pi(G)$ is always totally dual

integral. In 1984, Edmonds and Giles initiated the study of all
graphs $G$ for which $\pi(G)$ is

box-totally dual integral. The purpose of this talk is to present
a structural characterization of

all such graphs. (Joint work with Guoli Ding and Lei Tan.)

**Bei Zeng**, Guelph University

*Symmetries of Codeword Stabilized Quantum Codes*

Symmetry is at the heart of coding theory. Codes with symmetry,
especially cyclic codes, play an essential role in both theory
and practical applications of classical error-correcting codes.
Here we examine symmetry properties for codeword stabilized (CWS)
quantum codes, which is the most general framework for constructing
quantum error-correcting codes known to date. A CWS code Q can
be represented by a self-dual additive code S and a classical
code C, i.e., Q=(S,C), however this representation is in general
not unique. We show that for any CWS code Q with certain permutation
symmetry, one can always find a S with the same permutation symmetry
as Q such that Q=(S,C). As many good CWS codes have been found
by starting from a chosen S, this ensures that when trying to
find CWS codes with certain permutation symmetry, the choice of
S with the same symmetry will suffice. A key step for this result
is a new canonical representation for CWS codes, which is in terms
of a unique decomposition as union stabilizer codes. For CWS codes,
so far mainly the standard form (G,C) has been considered, where
G is a graph state. We analyze the symmetry of the corresponding
graph of G, which in general cannot possess the same permutation
symmetry as Q. We show that it is indeed the case for toric code
on a square lattice with translational symmetry, even if its encoding
graph can be chosen to be translational invariant.

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