CanaDAM 2019
SFU Harbour Centre, 29 - 31 mai 2019

Combinatorial geometry
Président: Andrew Vince (University of Florida)

UMAKANTA PATTANAYAK, Singapore University of Technology and Design
Geometry of integer hulls of strictly convex sets  [PDF]

We study various interesting geometric properties of integer hulls of a class of strictly convex sets in $\mathbb{R}^{n}$. We discuss sufficient conditions under which minimal faces of integer hulls of costrictly convex sets (a strictly convex set is called a costrictly convex set if either it is a relatively open set or its closure is a strictly convex set) are integral affine sets. We show that under mild assumptions integer hulls of costrictly convex sets are locally polyhedral sets (a convex set is called a locally polyhedral set if its intersection with each polytope is a polytope). Moreover, we show that the convex hull of the integer points of an A-polyhedral set (a pointed polyhedral set satisfying certain additional assumptions) is a locally polyhedral set (Moussafir established this result in 2003, and we give an alternative proof of this result).

DAVID RICHTER, Western Michigan University
A Report on Orthotopes  [PDF]

For the purposes of this talk, a $d$-dimensional ``orthotope'' is a union of finitely many closed $d$-dimensional boxes aligned with the usual coordinate axes. This represents a generalization of the notion of an orthogonal polytope or an $xyz$ polyhedron, as studied by David Eppstein, Elena Mumford, et al. In this talk we focus on the problem of realizing the face lattice of a given convex polytope by an orthotope, demonstrating connections to shellability questions.

SERGEI TSATURIAN, University of Manitoba
Results in Euclidean Ramsey theory  [PDF]

A typical question in Euclidean Ramsey theory has the following form: is it true that for any colouring of Euclidean space $\mathbb{E}^n$ in two (or more) colours there exists a monochromatic copy of some fixed geometric configuration $F$?

I will focus on the asymmetric version of this question - is it true that for any colouring of $\mathbb{E}^n$ in red and blue, there exists either a red copy of $F_1$ or a blue copy of $F_2$? Most of the questions in this field are very easy to state, but even some simplest cases are still open.

I will give a brief overview of known results. I will also present my result, that deals with the case of $\mathbb{E}^2$, $F_1$ being the configuration of two points at distance 1 to each other, and $F_2$ being 5 points on a line with distance 1 between consecutive points.

ETHAN WHITE, University of British Columbia
Rigid unit-bar frameworks  [PDF]

A framework in Euclidean space consists of a set of points called joints, and line segments connecting pairs of joints called bars. A framework is flexible if there exists a continuous motion of its joints such that all adjacent joints remain at a constant distance, but the distance between at least one pair of nonadjacent joints changes. For example, a square in the plane is not rigid since it can be deformed into a family of rhombi. Loosely speaking, a framework is infinitesimally rigid if it does not wobble. Infinitesimally rigid frameworks are rigid. We will present infinitesimally rigid bipartite unit-bar frameworks in $\mathbb{R}^n$, and infinitesimally rigid bipartite frameworks in the plane with girth up to 12. Our constructions answer questions of Hiroshi Maehara. Joint work with Jozsef Solymosi.

STEPHEN J. YOUNG, Pacific Northwest National Laboratory
A Linear Time Measure for Network Analysis  [PDF]

Graph similarity metrics serve far-ranging purposes across many domains in data science. As graph datasets grow in size, scientists need comparative tools that capture meaningful differences, yet are lightweight and scalable. In this work we develop a linear time algorithm for the recently introduced Graph Relative Hausdorff metric as well as explore some extremal aspects of the measure. Additionally, we apply this measure to anomaly detection in a cybersecurity context using a synthetic dynamic graph model from Hagberg, Lemons, and Mishra as well as anonymized cybersecurity data set released by Los Alamos National Laboratory. Joint work with Sinan Aksoy, Katy Nowak, and Emilie Purvine.