Research

Discrete Mathematics

Monday, April 19, 2021

Title: Triangle Mysteries are Abelian
Speaker: Lon Mitchell
Time: 2:00pm‐3:00pm
Place: Zoom Meeting

Abstract

A simple card game designed to showcase randomness instead shows a pattern: a “triangle mystery”. Abelian Group Theory, with some nice connections to Combinatorics, provides an explanation of the mystery. One might think that non-Abelian groups could provide additional mysteries, but, by widening the search to include semigroups and quasigroups, we find that all triangle mysteries must be Abelian.

Monday, April 5, 2021

No seminar this week.

Monday, February 22, 2021

No seminar this week.

Monday, February 8, 2021

Title: Asymptotic Homogeneity (a Quasicrystal Property)
Speaker: Greg McColm
Time: 2:00pm‐3:00pm
Place: Zoom Meeting

Abstract

A pattern is a set of bounded and connected subsets of a Euclidean space $$\mathbb{R}_d$$. Call a pattern periodic if it admits a symmetry group of translations spanning $$\mathbb{R}_d$$. The cut-and-project method can be applied to periodic patterns to get aperiodic patterns. Given a pattern $$\mathbb{P}$$ in $$\mathbb{R}_d$$ and given $$\mathbf{x}, \mathbf{y}\in\mathbb{R}_d$$ and real $$r>0$$, let $$x \sim_r y$$ mean that there is an isometry from the $$r$$-ball $$\mathrm{B}_r(\mathbf{x})$$ onto the $$r$$-ball $$\mathrm{B}_r(\mathbf{y})$$ preserving $$\mathbb{P}$$. It is a theorem of Hof and Schlottmann that given a cut-and-project pattern, for every $$\mathbf{x}\in\mathbb{R}_d$$, the set of $$\mathbf{y}\in\mathbb{R}_d$$ satisfying $$\mathbf{x}\sim_r\mathbf{y}$$ is uniformly dense in $$\mathbb{R}_d$$ — one of the underlying properties of many models of quasicrystals. We outline a (relatively) elementary proof of this fact.

Monday, February 1, 2021

Title: Quasiperiodicity
Speaker: Greg McColm
Time: 2:00pm‐2:50pm
Place: Zoom Meeting

Abstract

Several popular models of nonperiodic crystals are “quasiperiodic” in some sense. There are many uses of the term “quasiperiodic,” but we trespass on the Analysis Seminar by starting with sums of periodic functions, as in Bohr’s notion of aperiodic functions, and projections of periodic functions. Properties of these functions correspond to properties of families of cut-and-project structures.