CanaDAM 2017
Université Ryerson, 12 - 16 juin 2017

Geometry and Combinatorial Optimization
Org: Guyslain Naves (Aix-Marseille University)

The linear span of lattice points in the half-open unit cube  [PDF]

Let $\Lambda\subset\mathbf{R}^n$ be a lattice which contains the integer lattice $\mathbf{Z}^n$. We characterize the space of linear functions $\mathbf{R}^n\rightarrow\mathbf{R}$ which vanish on the lattice points of $\Lambda$ lying in the half-open unit cube $[0,1)^n$. We also find an explicit formula for the dimension of the linear span of $\Lambda\cap[0,1)^n$.

ROBERT DAVIS, Michigan State University
Detecting the Integer Decomposition Property in Reflexive Simplices  [PDF]

The Ehrhart series of a lattice polytope $P$ is a rational function encoding the number of lattice points in nonnegative integer dilates of $P$. The numerator of this series is the (Ehrhart) $h^*$-polynomial of $P$, and its list of coefficients, called the $h^*$-vector, are nonnegative integers. A wide-open conjecture claims that all Gorenstein polytopes with the integer decomposition property (IDP) have unimodal $h^*$-vectors. Without requiring IDP, the conjecture is false even in the case of reflexive simplices. In this talk, we recast the conjecture for reflexive simplices in the language of number theory according to a classification via arithmetic sequences. We then prove the conjecture for certain families of reflexive simplices, and see that there exist simplices within these families that meet only a necessary (but not sufficient) condition for possessing IDP.

GUYSLAIN NAVES, Marseille University
Packing and covering with balls on Busemann surfaces  [PDF]

We prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $\delta$, the minimum number of closed balls of radius $\delta$ with centers at $\mathcal S$ and covering the set $S$ is at most 19 times the maximum number of disjoint closed balls of radius $\delta$ centered at points of $S$: $\nu(S)\le \rho(S)\le 19\nu(S)$, where $\rho(S)$ and $\nu(S)$ are the covering and the packing numbers of $S$ by ${\delta}$-balls.

Tours, Colouring or Somewhere In Between  [PDF]



Atlantic Association for Research in the Mathematical Sciences Centre de recherches mathmatiques The Fields Institute Pacific Institute for the Mathematical Sciences Socit mathmatique du Canada Université Ryerson Office of Naval Research Science and Technology