CanaDAM 2019
SFU Harbour Centre, 29 - 31 mai 2019

Discrete geometry
Org: Agelos Georgakopoulos (University of Warwick, UK)

Bridges between embedded graphs and the geometry of surfaces  [PDF]

In this talk, we will illustrate on precise problems how the continuous (Riemannian) geometry of surfaces can shed light on the combinatorics of embedded graphs, and vice versa. We will showcase connections between on the first side geodesics, optimal homotopies and sweep-outs, and on the other side non-trivial cycles (edge-width), the complexity of a planar searching problem (homotopy height) and branch decompositions of planar graphs. This talk does not assume any knowledge in differential geometry or topology.

Based on joint works with Erin Chambers, Gregory Chambers, Eric Colin de Verdière, Alfredo Hubard, Francis Lazarus, Tim Ophelders and Regina Rotman.

NÓRA FRANKL, London School of Economics
Nearly k-distance sets  [PDF]

Determining the largest cardinality $m_k(d)$ of a $k$-distance set in $\mathbb{R}^d$ is a well known, hard problem. We study an approximate version: A set $S$ is an $\varepsilon$-nearly $k$-distance set if there are $1\leq t_1\leq t_2\leq \dots \leq t_k$ such that every distance determined by $S$ fall in the $\varepsilon$-neighbourhood of some $t_i$. Let $M_k(d)$ be the largest $N$, such that for any $\varepsilon>0$, there is an $\varepsilon$-nearly $k$-distance set in $\mathbb{R}^d$. We prove that $m_k(d)=M_k(d)$ if $k\leq 3$, or $d\geq d(k)$, and that $M_k(d)=\Theta_d(k^d)$. We also study other related problems.

JOHN HASLEGRAVE, University of Warwick
Spanning surfaces in 3-uniform hypergraphs  [PDF]

A classical result of Dirac gives the best possible minimum degree condition to guarantee a Hamilton cycle in a graph. Several extensions to hypergraphs exist, but these all look for one-dimensional cyclic structures. A natural alternative viewpoint is to generalise a spanning topological circle to a spanning topological sphere. I will give an asymptotically tight analogue of Dirac’s theorem for such structures in 3-uniform hypergraphs, answering a question of Gowers. The result is not specific to the sphere, but applies to any given surface. This is joint work with Agelos Georgakopoulos (Warwick), Richard Montgomery (Birmingham) and Bhargav Narayanan (Rutgers).

ANDREW VINCE, University of Florida
A Combinatorial Construction of Self Similar Tilings  [PDF]

A method, based on rooted trees, for the construction of tilings of Euclidean space with the following properties will be discussed. There are finitely many tiles up to congruence. The tilings are self-similar and quasiperiodic, but not periodic.